arXiv:0704.0778v2  [math.AG]  10 Sep 2008
FROBENIUS SPLITTING AND GEOMETRY OF
G-SCHUBERT VARIETIES
XUHUA HE AND JESPER FUNCH THOMSEN
Abstract. Let X be an equivariant embedding of a connected
reductive group G over an algebraically closed field k of positive
characteristic. Let B denote a Borel subgroup of G. A G-Schubert
variety in X is a subvariety of the form diag(G) · V , where V is a
B × B-orbit closure in X . In the case where X is the wonderful
compactification of a group of adjoint type, the G-Schubert vari-
eties are the closures of Lusztig’s G-stable pieces. We prove that
X admits a Frobenius splitting which is compatible with all G-
Schubert varieties. Moreover, when X is smooth, projective and
toroidal, then any G-Schubert variety in X admits a stable Frobe-
nius splitting along an ample divisors. Although this indicates that
G-Schubert varieties have nice singularities we present an example
of a non-normal G-Schubert variety in the wonderful compactifi-
cation of a group of type G2. Finally we also extend the Frobenius
splitting results to the more general class of R-Schubert varieties.
1. Introduction
Let G denote a connected and reductive group over an algebraically
closed field k, and let B denote a Borel subgroup of G. An equi-
variant embedding X of G is a G × G-variety which contains G =
(G × G)/diag(G) as an open G × G-invariant subset, where diag(G)
is the diagonal image of G in G × G. Any equivariant embedding X
of G contains finitely many B × B-orbits. In recent years the geom-
etry of closures of B × B-orbits has been studied by several authors.
The most general result was obtained in [H-T2] where it was proved
that B × B-orbit closures are normal, Cohen-Macaulay and have (F -
)rational singularities (actually, even stronger results were obtained).
In the present paper we will study (closed) subvarieties inX of the form
diag(G) ·V, where V denotes the closure of a B×B-orbit. Subvarieties
of equivariant embeddings of G of this form will be called G-Schubert
varieties.
When G is a semisimple group of adjoint type there exists a canonical
equivariant embedding X of G which is called the wonderful compact-
ification. The wonderful compactifications are of primary interest in
this paper. Actually, this work arose from the question of describing
the closures of the so-called G-stable pieces of X. The G-stable pieces
makes up a decomposition of X into locally closed subsets. They were
introduced by Lusztig in [L] where they were used to construct and
http://arxiv.org/abs/0704.0778v2
2 XUHUA HE AND JESPER FUNCH THOMSEN
study a class of perverse sheaves which generalizes his theory of charac-
ter sheaves on reductive groups. More precisely, these perverse sheaves
are the intermediate extensions of the so-called “character sheaves”
on a G-stable piece. This motivates the study of closures of G-stable
pieces which turns out to coincide with the set of G-Schubert varieties.
Before discussing the closures of G-stable pieces in details, let us
make a short digression and discuss some other motivations for study-
ing G-stable pieces and G-Schubert varieties (in wonderful compactifi-
cations):
(1) When G is a simple group, the boundary of the closure of the
unipotent subvariety of G in the wonderful compactification X,
is a union of certain G-Schubert varieties (see [He] and [H-T]).
Thus knowing the geometry of these G-Schubert varieties will
help us to understand the geometry of the closure of the unipo-
tent variety within X.
(2) Let Lie(G) denote the Lie algebra of a simple group G over a
field of characteristic zero. Let ≪,≫ denote a fixed symmet-
ric non-degenerate ad-invariant bilinear form. Let <,> be the
bilinear form on Lie(G)⊕ Lie(G) defined by
< (x, y), (x′, y′) >=≪ x, x′ ≫ − ≪ y, y′ ≫ .
In [E-L], Evens and Lu showed that each splitting Lie(G) ⊕
Lie(G) = l ⊕ l′, where l and l′ are Lagrangian subalgebras of
Lie(G)⊕ Lie(G), gives rise to a Poisson structure Πl,l′ on X. If
moreover, one starts with the Belavin-Drinfeld splitting, then
all the G-stable pieces/G-Schubert varieties and B×B−-orbits
of X are Poisson subvarieties, where B− is a Borel subgroup
opposite to B. Thus to understand the Poisson structure on
X corresponding to the Belavin-Drinfeld splitting, one needs
to understand the geometry of the G-stable pieces/G-Schubert
varieties. If we start with another splitting, then we obtain
a different Poisson structure on X and in order to understand
these Poisson structures, one needs to study the R-stable pieces
[L-Y] instead (see Section 12), which generalize both the G-
stable pieces and the B × B−-orbits.
The main technical ingredient in this paper is the positive character-
istic notion of Frobenius splitting. Frobenius splitting is a powerful tool
which has been proved to be very useful in obtaining strong geometric
conclusions for e.g. Schubert varieties and closures of B × B-orbits in
equivariant embeddings. In the present paper we obtain two types of
results related to G-Schubert varieties over fields of positive character-
istic. First of all, if we fix an equivariant embedding X of a reductive
group G then we prove that all G-Schubert varieties in X are simul-
taneously compatibly Frobenius split by a Frobenius splitting of X .
Secondly, concentrating on a single G-Schubert variety X, in a smooth
projective and toroidal embedding X , we prove that this admits a
stable Frobenius splitting along an ample divisor. Statements of this
form put strong conditions on the intertwined behavior of cohomology
groups of line bundles on X and its G-Schubert varieties. As this is re-
lated to geometric properties it therefore seems natural to expect that
G-Schubert varieties should have nice singularities. It therefore comes
as a complete surprise that G-Schubert varieties, in general, are not
even normal. We only provide a single example of this phenomenon
(in the wonderful compactification of a group of type G2), but expect
that this absence of normality is the general picture.
In obtaining the Frobenius splitting result mentioned above, we have
developed some general theory of how to construct Frobenius splitting
of varieties of the form G×PX (see Section 4.2 for the definition). This
part of the paper is influenced by the theory of B-canonical Frobenius
splitting as discussed in [B-K, Chap.4]; in particular the proof of [B-K,
Prop.4.1.17]. The presentation we provide is more general and makes
it possible to extract even better result from the ideas of B-canonical
Frobenius splittings. This theory is presented in Chapter 5 in a general-
ity which is more than necessary for obtaining the described Frobenius
splitting results for G-Schubert varieties. However, we hope that this
theory could be useful elsewhere and we certainly consider it to be of
independent interest.
This paper is organized in the following way. In Section 2 we intro-
duce notation, and in Section 3 we briefly define Frobenius splitting
and explain its fundamental ideas. Section 4 is devoted to some results
on linearized sheaves which should all be well known. In Section 5 we
study the Frobenius splitting of varieties of the form G ×P X for a
variety X with an action by a parabolic subgroup P . The main idea
is to decompose the Frobenius morphism on G×P X into maps associ-
ated to the Frobenius morphism on the base G/P and the fiber X of the
natural morphism G×P X → G/P . In Section 6 we relate B-canonical
Frobenius splittings to the material in Section 5. Section 7 contains
applications of Section 5 to general G × G-varieties. In section 8 we
define the G-stable pieces and G-Schubert varieties. In Section 9 we
apply the material of the previous sections to the class of equivariant
embeddings and obtain Frobenius splitting results for G-Schubert vari-
eties. Section 10 contains results related to cohomology of line bundles
on G-Schubert varieties. Section 11 contains an example of a non-
normal G-Schubert variety. Finally Section 12 contains generalizations
and variations of the previous sections.
We would like to thank the referee for a careful reading of this paper
and for numerous suggestions concerning the presentation.
4 XUHUA HE AND JESPER FUNCH THOMSEN
2. Notation
We will work over a fixed algebraically closed field k. The charac-
teristic of k will depend on the application. By a variety we mean a
reduced and separated scheme of finite type over k. In particular, we
allow a variety to have several irreducible components.
2.1. Group setup. We letG denote a connected linear algebraic group
over k. We fix a Borel subgroup B and a maximal torus T ⊂ B. The
notation P is used for a parabolic subgroup of G containing B. The
set of T -characters is denoted by X∗(T ) and we identify this set with
the set X∗(B) of B-characters.
2.2. Reductive case. In many cases we will specialize to the case
where G is reductive. In this case we will also use the following no-
tation : the set of roots determined by T is denoted by R ⊆ X∗(T )
while the set of positive roots determined by (B, T ) is denoted by R+.
The simple roots are denoted by α1, . . . , αl, and we let ∆ = {1, . . . , l}
denote the associated index set. The simple reflection associated to the
simple root αi is then denoted by si. The Weyl group W = NG(T )/T is
generated by the simple reflections si, for i ∈ ∆. The length of w ∈ W
will be denoted by l(w). For J ⊂ ∆, let WJ denote the subgroup of
W generated by the simple reflection associated with the elements in
J , and let W J (resp. JW ) denote the set of minimal length coset rep-
resentatives for W/WJ (resp. WJ\W ). The element in W of maximal
length will be denoted by w0, while w
0 is used for the same kind of
element in WJ . For any w ∈ W , we let ẇ denote a representative of w
in NG(T ). For J ⊂ ∆, let PJ ⊃ B denote the corresponding standard
parabolic subgroup and P−J ⊃ B
− denote its opposite parabolic. Let
LJ = PJ ∩P
J be the common Levi subgroup of PJ and P
J containing
T . Let UJ (resp. U
J ) denote the unipotent radical of PJ (resp. P
When J = ∅ we also use the notation U and U− for UJ and U
J respec-
tively. When G is semisimple and simply connected we may associate
a fundamental character ωi to each simple root αi. The sum of the
fundamental characters is then denoted by ρ. Then ρ also equals half
the sum of the positive roots.
3. The relative Frobenius morphism
In this section we collect some results related to the Frobenius mor-
phism and to the concept of Frobenius splitting. Compared to other
presentations on the same subject, this presentation differs only in its
emphasis on the set HomOX′
(FX)∗OX ,OX′
(to be defined below) and
not just the set of Frobenius splittings. Thus, the obtained results are
only small variations of already known results as can be found in e.g.
[B-K].
3.1. The Frobenius morphism. By definition a variety X comes
with an associated morphism
pX : X → Spec(k),
of schemes. Assume that the field k has positive characteristic p > 0.
Then the Frobenius morphism on Spec(k) is the morphism of schemes
Fk : Spec(k) → Spec(k),
which on the level of coordinate rings is defined by a 7→ ap. As k
is assumed to be algebraically closed the morphism Fk is actually an
isomorphism and we let F−1k denote the inverse morphism. Composing
pX with F
k we obtain a new variety
p′X : X → Spec(k),
with underlying scheme X . In the following we suppress the morphism
pX from the notation and simply use X as the notation for the variety
defined by pX . The variety defined by p
X is then denoted by X
The relative Frobenius morphism on X is then the morphism of
varieties :
FX : X → X
which as a morphism of schemes is the identity map on the level of
points and where the associated map of sheaves
X : OX′ → (FX)∗OX ,
is the p-th power map. A key property of the Frobenius morphism is
the relation
(1) (FX)
′ ≃ Lp
which is satisfied for every line bundle L on X (here L′ denotes the
corresponding line bundle on X ′).
3.2. Frobenius splitting. A variety X is said to be Frobenius split if
the OX′-linear map of sheaves :
X : OX′ → (FX)∗OX ,
has a section; i.e. if there exists an element
s ∈ HomOX′
(FX)∗OX ,OX′
such that the composition s ◦F
X is the identity endomorphism of OX′ .
The section s will be called a Frobenius splitting of X .
6 XUHUA HE AND JESPER FUNCH THOMSEN
3.3. Compatibility with line bundles and closed subvarieties.
Fix a line bundle L on X and a closed subvariety Y in X with sheaf
of ideals IY . Let Y
′ denote the closed subvariety of X ′ associated to Y
with sheaf of ideals denoted by IY ′ . The kernel of the natural morphism
HomOX′
(FX)∗L,OX′
→ HomOX′
(FX)∗(L⊗ IY ),OY ′
induced by the inclusion L ⊗ IY ⊂ L and the projection OX′ → OY ′ ,
will be denoted by EndLF (X, Y ). The associated space of global sections
will be denoted by EndLF (X, Y ). When Y = X we simply denote
EndLF (X, Y ) (resp. End
F (X, Y )) by End
F (X) (resp. End
F (X)). The
sheaf EndLF (X, Y ) is a subsheaf of End
F (X) consisting of the elements
compatible with Y . Moreover, there is a natural morphism
EndLF (X, Y )|Y → End
F (Y ),
where the notation |Y means restriction to Y .
If Y1, Y2, . . . , Ym is a collection of closed subvarieties of X then the
notation EndLF (X, Y1, . . . , Ym) (or sometimes End
F (X, {Yi}
i=1)) will de-
note the intersection of the subsheaves EndLF (X, Yi) for i = 1, . . . , m.
The set of global sections of the sheaf EndLF (X, Y1, . . . , Ym) will be de-
noted by EndLF (X, Y1, . . . , Ym).
When L = OX we remove L from all of the above notation. In
particular, the vectorspace EndF (X) denotes the set of morphisms from
(FX)∗OX to OX′ and thus contains the set of Frobenius splittings of X .
A Frobenius splitting s of X contained in EndF (X, {Yi}i) is said to be
compatible with the subvarieties Y1, . . . , Ym. When s is compatible in
this sense it induces a Frobenius splitting of each Yi for i = 1 . . . , m.
In this case we also say that s compatibly Frobenius splits Y1, . . . , Ym.
In concrete terms, this is equivalent to
(FX)∗IYi
⊂ IY ′i .
for all i.
Lemma 3.1. Let Y and Z denote closed subvarieties in X and let s
denote a global section of EndLF (X,Z, Y ).
(1) s ∈ EndLF (X, Y1) for every irreducible component Y1 of Y .
(2) If the scheme theoretic intersection Z ∩ Y is reduced then s is
contained in EndLF (X, Y ∩ Z).
Proof. Let Y1 denote an irreducible component of Y and let
J = s
(FX)∗(IY1 ⊗ L)
⊂ OX′ .
Let U denote the open complement (in X ′) of the irreducible compo-
nents of Y ′ which are different from Y ′1 . Then IY ′1 coincides with IY ′
on U and consequently J|U ⊂ (IY ′)|U as s is compatible with Y . In
particular, J|U ⊂ (IY ′1 )|U . We claim that this implies that J ⊂ IY ′1 :
let V denote an open subset of X ′ and let f be a section of J over
V . As J is a subsheaf of OX′ , we may consider f as a function on V ,
and it suffices to prove that f vanishes on Y ′1 ∩ V . If Y
1 ∩ V is empty
then this is clear. Otherwise, U ∩ V ∩ Y ′1 is a dense subset of Y
1 and it
suffices to prove that f vanishes on this set. But this follows from the
inclusion J|U ⊂ (IY ′1 )|U . As a consequence s is compatible with Y1. The
second claim follows as the sheaf of ideals of the intersection Z ∩ Y is
IY + IZ . �
The condition that Z ∩ Y is reduced, in Lemma 3.1, only ensures
that Z ∩ Y is a variety. When L = OX and s is a Frobenius splitting
this is always satisfied [B-K, Prop.1.2.1].
3.4. The evaluation map. Let k[X ′] denote the space of global reg-
ular functions on X ′. Evaluating an element s : (FX)∗OX → OX′ of
EndF (X) at the constant global function 1 on X defines an element in
k[X ′] which we denote by evX(s). This defines a morphism
evX : EndF (X) → k[X
with the property that evX(s) = 1 if and only if s is a Frobenius
splitting of X .
3.5. Frobenius D-splittings. Consider an effective Cartier divisor D
on X , and let σD denote the associated global section of the associated
line bundle OX(D). A Frobenius splitting s of X is said to be a Frobe-
nius D-splitting if s factorizes as
s : (FX)∗OX
(FX)∗σD
−−−−−→ (FX)∗OX(D)
−→ OX′ ,
for some element sD in End
OX(D)
. We furthermore say that the
Frobenius D-splitting s is compatible with a subvariety Y if sD is com-
patible with Y . The following result assures that, in this case, the
compatibility with closed subvarieties agrees with the usual definition
[R, Defn.1.2].
Lemma 3.2. Assume that s defines a Frobenius D-splitting of X.
Then sD is compatible with Y if and only if (i) s compatibly Frobe-
nius splits Y and (ii) the support of D does not contain any irreducible
components of Y .
Proof. The if part of the statement follows from [R, Prop.1.4]. So
assume that sD is compatible with Y . Then sD induces a morphism
sD : (FY )∗OX(D)|Y → OY ′ ,
satisfying sD((σD)|Y ) is the constant function 1 on Y
′. As a conse-
quence (σD)|Y does not vanish on any of the irreducible components
of Y . This proves part (ii) of the statement. Part (i) is clearly satis-
fied. �
8 XUHUA HE AND JESPER FUNCH THOMSEN
It follows that if s is compatible with Y and, moreover, defines a
Frobenius D-splitting of X then D ∩ Y makes sense as an effective
Cartier divisor on Y and, in this case, s induces a Frobenius D ∩ Y -
splitting of Y .
3.6. Stable Frobenius splittings along divisors. Let X(0) = X
and define recursively X(n) = (X(n−1))′ for n ≥ 1. Composing the
Frobenius morphisms on X(i) for i = 0, . . . , n, we obtain a morphism
X : X → X
with an associated map of sheaves
♯ : OX(n) → (F
X )∗OX .
Let, as in Section 3.5, D denote an effective Cartier divisor on X with
associated canonical section σD of OX(D). We say that X admits a
stable Frobenius splitting along D if there exists a positive integer n
and an element
s ∈ HomO
X )∗OX(D),OX(n)
such that the composed map
OX(n)
−−−−→ (F
X )∗OX
−−−−−−→ (F
X )∗OX(D)
−→ OX(n) ,
is the identity map on OX(n) . The element s is called a stable Frobenius
splitting of X along D. When Y is a closed subvariety of X we say
that the stable Frobenius splitting s is compatible with Y if
X )∗(IY ⊗ OX(D))
⊂ IY (n).
Notice that this condition necessarily implies that the support of D
does not contain any of the irreducible components of Y (cf. proof
of Lemma 3.2). Notice also that if X admits a Frobenius D-splitting
which is compatible with Y then X admits a stable Frobenius splitting
along D which is compatible with Y . The following is well known (see
e.g. [T, Lem.4.4])
Lemma 3.3. Let D1 and D2 denote effective Cartier divisors on X and
let Y denote a closed subvariety of X. Then X admits stable Frobenius
splittings along D1 and D2 which are compatible with Y if and only if
X admits a stable Frobenius splitting along D1+D2 which is compatible
with Y .
The following result explains one of the main applications of (stable)
Frobenius splitting. Remember that a line bundle L is nef if L⊗M is
ample whenever M is ample.
Proposition 3.4. Assume that X admits a stable Frobenius splitting
along an effective Cartier divisor D. Then there exists a positive integer
n such that for each line bundle L on X we have an inclusion of abelian
groups
Hi(X,L) ⊂ Hi(X,Lp
⊗ OX(D)).
In particular, if D is ample and L is nef, then Hi(X,L) = 0 for i > 0.
Moreover, if the stable Frobenius splitting of X is compatible with a
subvariety Y , D is ample and L is nef then the restriction morphism
H0(X,L) → H0(Y,L),
is surjective.
Proof. Argue as in the proof [R, Prop.1.13(i)]. �
3.7. Duality for FX. By duality (see [Har2, Ex.III.6.10]) for the finite
morphism FX we may to each quasi-coherent OX′-module F associate
an OX -module denoted by (FX)
!F and satisfying
(FX)∗(FX)
F = HomOX′
(FX)∗OX ,F
Actually, as FX is the identity on the level of points we may define
!F as the sheaf of abelian groups
HomOX′
(FX)∗OX ,F
with OX -module structure defined by
(g · φ)(f) = φ(gf),
for g, f ∈ OX and φ ∈ HomOX′
(FX)∗OX ,F
. When F = OX we
will also use the notation End!F (X) for (FX)
!OX . This sheaf is par-
ticularly nice when X is smooth as (FX)
!OX then coincides with the
line bundle ω
X , where ωX denotes the dualizing sheaf of X (see e.g.
[B-K, Sect.1.3]). If Y1, Y2, . . . , Ym is a collection of closed subvarieties
of X then End!F (X, Y1, . . . , Ym) (or End
F (X, {Yi}
i=1)) will denote the
subsheaf of End!F (X) consisting of the elements mapping the sheaf of
ideals IYi to IY ′i for all i = 1, . . . , m. We say that End
F (X, {Yi}
i=1) is
the subsheaf of elements compatible with Y1, . . . , Ym.
More generally, duality for FX implies that we have a natural iden-
tification
(FX)∗HomOX
G, (FX)
≃ HomOX′
(FX)∗G,F
whenever G (resp. F) is a quasicoherent sheaf on X (resp. X ′). This
leads to the identification
HomOX
G, (FX)
≃ HomOX′
(FX)∗G,F
where a morphism η : G → (FX)
!F is identified with the composed
morphism
η′ : (FX)∗G
(FX)∗η
−−−−→ (FX)∗(FX)
F ≃ HomOX′
(FX)∗OX ,F
Here the latter map is the natural evaluation map at the element 1
in OX . From now on we will specialize to the case where F = OX′
10 XUHUA HE AND JESPER FUNCH THOMSEN
and G equals a line bundle L on X . In this case, an element in
HomOX
L,End!F (X)
may also be considered as a global section of
the sheaf End!F (X)⊗ L
−1. For later use we emphasize
Lemma 3.5. Let η be an element in HomOX
L,End!F (X)
and let
η′ denote the corresponding element in HomOX′
(FX)∗L,OX′
by the
above identification. Then η′ factors through the morphism
(FX)∗L
(FX)∗η
−−−−→ (FX)∗End
F (X).
Moreover, the element η′ is compatible with a collection of closed sub-
varieties Y1, . . . , Ym of X if and only if the image of η is contained in
End!F (X, Y1, . . . , Ym).
Proof. The first part of the statement follows directly from the discus-
sion above. To prove the second statement we may assume that m = 1.
We use the notation Y = Y1. Let σ denote a section of L over an open
subset U of X , and consider s = η(σ) as a map
s : OX(U) → OX′(U
That s is compatible with Y means that s(f) vanishes on Y ′ whenever
f vanishes on Y for a function f on U . Alternatively, the evaluation
of f · s at 1, which coincides with η′(f · σ), should vanish on Y ′. In
particular, the image of η is contained in End!F (X, Y ) if and only if
the restriction of η′ to (FX)∗
IY ⊗ L
maps into IY ′ . This ends the
proof. �
We will also need the following remark
Lemma 3.6. Let D denote a reduced effective Cartier divisor on X
and L denote a line bundle on X. Let M = OX((p − 1)D) ⊗ L and
assume that we have an OX-linear morphism η : M → End
F (X). Let
σD denote the canonical section of OX(D) and consider the map
ηD : L → End
F (X),
induced by σ
D . Then the element
η′D ∈ HomOX′
(FX)∗L,OX′
induced by ηD, is compatible with the support of D. In particular, the
image of ηD is contained in End
F (X,D).
Proof. Notice that η′D is the composition
η′D : (FX)∗L
(FX)∗σ
−−−−−−→ (FX)∗M
−→ OX′ ,
where η′ is the element corresponding to η. Hence, the restriction of
η′D to L⊗ OX(−D) coincides with the map
(FX)∗
L⊗ OX(−D)
) (FX)∗σ
−−−−−→ (FX)∗M
−→ OX′ .
But the restriction of η′ to (cf. (1))
(FX)∗
OX(−pD)⊗M
≃ OX′(−D
′)⊗ (FX)∗M,
maps by linearity into OX′(−D
′). The in particular part follows by
Lemma 3.5. �
3.8. Push-forward operation. Assume that f : X → Z is a mor-
phism of varieties satisfying that the associated map f ♯ : OZ → f∗OX
is an isomorphism. Let f ′ : X ′ → Z ′ denote the associated morphism.
Then f ′∗ induces a morphism
f ′∗EndF (X) → EndF (Z).
If Y ⊂ X is a closed subset then the subsheaf f ′∗EndF (X, Y ) is mapped
to EndF (Z, f(Y )), where f(Y ) denotes the variety associated to the
closure of the image of Y . On the level of global sections this means
that every Frobenius splitting s of X induces a Frobenius splitting f ′∗s
of Z such that when s is compatible with Y then f ′∗s is compatible
with f(Y ). Likewise
Lemma 3.7. With notation as above, let L denote a line bundle on
Z and let s be an element of End
f∗(L)
. Then f ′∗s is an element of
EndLF
. Moreover, if s is compatible with a closed subvariety Y of
X then f ′∗s is compatible with f(Y ).
Proof. This follows easily from the fact that the sheaf of ideals of f(Y )
coincides with f∗IY [B-K, Lem.1.1.8]. �
4. Linearized sheaves
In this section we collect a number of well known facts about lin-
earized sheaves. The chosen presentation follows rather closely the
presentation in [Bri, Sect.2].
Let H denote a linear algebraic group over the field k and let X
denote a H-variety with H-action defined by σ : H × X → X . We
let p2 : H × X → X denote projection on the second coordinate. A
H-linearization of a quasi-coherent sheaf F on X is an OH×X -linear
isomorphism
φ : σ∗F → p∗2F,
satisfying the relation
(2) (µ× 1X)
∗φ = p∗23φ ◦ (1H × σ)
as morphisms of sheaves on H × H × X . Here µ : H × H → H
(resp. p23 : H × H × X → H × X) denotes the multiplication on
H (resp. the projection on the second and third coordinate). Based
on the fact that σ∗OX = p
2OX we see that the sheaf OX admits a
canonical linearization. In the following we will always assume that
OX is equipped with this canonical linearization.
12 XUHUA HE AND JESPER FUNCH THOMSEN
A morphism ψ : F → F′ of H-linearized sheaves is a morphism of
OX -modules commuting with the linearizations φ and φ
′ of F and F′,
i.e. φ′ ◦ σ∗(ψ) = p∗2(ψ) ◦ φ.
Linearized sheaves on X form an abelian category which we denote
by ShH(X).
4.1. Quotients and linearizations. Assume that the quotient q :
X → X/H exists and that q is a locally trivial principal H-bundle.
Then for G ∈ Sh(X/H), q∗G is naturally a H-linearized sheaf on X .
This defines a functor q∗ : Sh(X/H) → ShH(X). On the other hand,
for F ∈ ShH(X), q∗F has a natural action of H . Define a functor
qH∗ : ShH(X) → Sh(
X/H) by qH∗ (F) = (q∗F)
H the subsheaf of H-
invariants of q∗F. It is known that the functor q
∗ : Sh(X/H) → ShH(X)
is an equivalence of categories with inverse functor qH∗ .
In general, if H is a closed normal subgroup of G and X is a G-
variety such that the quotient X/H exists (as above), then X/H is a G/H-
variety and the functor q∗ : ShG/H(X/H) → ShG(X) is an equivalence
of categories with inverse functor qH∗ : ShG(X) → ShG/H(
X/H).
4.2. Induction equivalence. Consider now a connected linear alge-
braic group G and a parabolic subgroup P in G. Let X denote a
P -variety. Then G×X is a G× P -variety by the action
(g, p)(h, x) = (ghp−1, px),
for g, h ∈ G, p ∈ P and x ∈ X . Then the quotient, denoted by
G ×P X , of G × X by P exists and the associated quotient map q :
G×X → G×P X is a locally trivial principal P -bundle. The quotient
of G × X by G also exists and may be identified with the projection
p2 : G×X → X . In particular, we may apply the above consideration
to obtain equivalences of the categories ShP (X), ShG×P (G × X) and
ShG(G×P X). Notice that under this equivalence a P -linearized sheaf
F on X corresponds to the G-linearized sheaf IndGP (F) = (q∗p
P . In
particular, the space of global sections of IndGP (F) equals
IndGP (F)(G×P X) =
p∗2F(G×X)
k[G]⊗k F(X)
= IndGP (F(X)),
where the second equality follows by the Künneth formula. This also
explains the notation IndGP (F). Similarly, starting with a G-linearized
sheaf G on G×P X then the associated P -linearized line bundle on X
equals G′ = ((p2)∗q
∗G)G. However, by [Bri, Lemma 2(1)] the latter also
equals the simpler pull back i∗G by the P -equivariant map
i : X → G×P X,
sending x to q(1, x). In particular, we conclude that the functor i∗ :
ShG(G ×P X) → ShP (X) is an equivalence of categories with inverse
functor IndGP . Notice also that the space of global sections of G is
G-equivariantly isomorphic to
G(G×P X) = Ind
(i∗G)(X)
which follows by (3) above.
4.3. Duality. Assume that the field k has positive characteristic p > 0.
Regard X ′ as a H-variety in the canonical way and let F denote a H-
linearized sheaf on X ′. The sheaf (FX)
!F, defined in Section 3.7, is
then naturally a H-linearized sheaf on X . Moreover, the induced H-
linearization of (FX)∗(FX)
!F coincides with the natural H-linearization
HomOX′
(FX)∗OX ,F
When X is smooth the sheaf (FX)
!OX′ is canonically isomorphic to
X (cf. Section 3.7). We may use this isomorphism to define a H-
linearization of ω
X . Alternatively we may consider the natural H-
linearization of the dualizing sheaf ΩX of X and use this to define a
H-linearization of ω
X . It may be checked that the two stated ways
of defining a H-linearization of ω
X coincide.
5. Frobenius splitting of G×P X
Let G denote a connected linear algebraic group over an algebraically
closed field k of characteristic p > 0. Let P denote a parabolic subgroup
of G and let X denote a P -variety. In this section we want to consider
Frobenius splittings of the quotient Z = G×PX of G×X by P . We let
π : Z → G/P denote the morphism induced by the projection of G×X
on the first coordinate. When g ∈ G and x ∈ X we use the notation
[g, x] to denote the element in Z represented by (g, x).
5.1. Decomposing the Frobenius morphism. The Frobenius mor-
phism FZ admits a decomposition FZ = Fb ◦ Ff where Fb (resp. Ff )
is related to the Frobenius morphism on the base (resp. fiber) of π.
More precisely, define Ẑ and the morphisms π̂ and Fb as part of the
fiber product diagram
(4) Ẑ
Fb //
// (G/P)′
A local calculation shows that we may identify Ẑ with the quotient
G ×P X
′, where the P -action on the Frobenius twist X ′ of X is the
natural one. With this identification π̂ : G ×P X
′ → G/P is just the
14 XUHUA HE AND JESPER FUNCH THOMSEN
map [g, x′] 7→ gP . It also follows that the natural morphism (induced
by the Frobenius morphism on X)
Ff : G×P X → G×P X
makes the following diagram commutative
Fb //
// (G/P)′
5.2. Let M denote a P -linearized line bundle on X and let MZ =
IndGP (M) denote the associated G-linearized line bundle on Z. The
main aim of this section is to construct global sections of the sheaf
F (Z) = HomOZ′
(FZ)∗MZ ,OZ′
To this end we fix a P -character λ and let L denote the associated line
bundle on G/P (cf. Section 4). The pull back π̂∗L of L to Ẑ is then
denoted by LẐ . We then define the following sheaves
F (Z)f := HomOẐ
(Ff)∗MZ ,OẐ
F (Z)b := HomOZ′
(Fb)∗LẐ ,OZ′
with spaces of global sections denoted by End
F (Z)f and End
F (Z)b.
Notice that whenM is substituted with the P -linearized twistM(−λ) :=
M⊗ k−λ then
M(−λ)Z = MZ ⊗ π
∗(L−1) = MZ ⊗ (Ff)
and thus by the projection formula
(5) End
M(−λ)Z
F (Z)f = HomOẐ
(Ff)∗MZ ,LẐ
Sections of End
F (Z) are then constructed as compositions of global
sections of the sheaves End
M(−λ)Z
F (Z)f and End
F (Z)b. More precisely,
v ∈ HomO
(Ff)∗MZ ,LẐ
u ∈ HomOZ′
(Fb)∗LẐ ,OZ′
are global sections of the latter sheaves, then the composition u◦(Fb)∗v
defines a global section of End
F (Z).
5.3. An equivariant setup. We now give equivariant descriptions of
the sheaves End
F (Z)f and End
F (Z)b.
5.3.1. A description of End
F (Z)f . Now End
F (Z)f is a G-linearized
sheaf on Ẑ = G×P X
′. Let Y denote a P -stable subvariety of X and
let ZY = G ×P Y denote the associated subvariety of Z with sheaf of
ideals IZY ⊂ OZ . Let ẐY denote the subvariety G ×P Y
′ of Ẑ. Then
there is a natural morphism of G-linearized sheaves
F (Z)f → HomOẐ
(Ff )∗(MZ ⊗ IZY ),OẐY
induced by the inclusion IZY ⊂ OZ and the projection OẐ → OẐY . We
let End
F (Z,ZY )f denote the kernel of the above map and arrive at a
left exact sequence of G-linearized sheaves
0 → EndMZF (Z,ZY )f → End
F (Z)f → HomOẐ
(Ff)∗(MZ⊗IZY ),OẐY
In particular, the space of global sections of End
F (Z,ZY )f is identified
with the set of elements in End
F (Z)f which map (Ff )∗(MZ ⊗ IZY ) to
⊂ OẐ . Using the observations in Section 4.2 we can give another
description of the space of global sections of End
F (Z,ZY )f . Let i
X ′ → G×P X
′ denote the morphism i′(x′) = [1, x′]. Then the functor
i′ is exact on the category of G-linearized sheaves. We want to apply
this fact on the left exact sequence (6) above : notice first that
(i′)∗End
F (Z)f = HomOX′
(i′)∗(Ff )∗MZ ,OX′
where, moreover, (i′)∗(Ff)∗MZ = (FX)∗M. Thus (i
′)∗End
F (Z)f =
EndMF (X). Similarly,
(i′)∗HomO
(Ff)∗(MZ ⊗ IZY ),OẐY
= HomOX′ ((FX)∗(M⊗ IY ),OY ′).
In particular, we see that the P -linearized sheaf on X ′ corresponding to
the G-linearized sheaf End
F (Z,ZY )f equals the kernel of the natural
EndMF (X) → HomOX′ ((FX)∗(M⊗ IY ),OY ′),
i.e. it equals EndMF (X, Y ). By Section 4.2 the space of global sections
F (Z,ZY )f of End
F (Z,ZY )f is then G-equivariantly isomorphic
IndGP
EndMF (X, Y )).
Applying the above conclusions to the sheaf M(−λ) we find:
Proposition 5.1. There exists a G-equivariant isomorphism
M(−λ)Z
F (Z)f ≃ Ind
EndMF (X)⊗ kλ)
such that when Y is a closed P -stable subvariety of X then the subset of
elements of End
M(−λ)Z
F (Z)f which map (Ff)∗(MZ⊗IZY ) to (IẐY ⊗LẐ) ⊂
LẐ (cf. equation (5)) is identified with
M(−λ)Z
F (Z,ZY )f ≃ Ind
EndMF (X, Y )⊗ kλ).
16 XUHUA HE AND JESPER FUNCH THOMSEN
5.3.2. A description of End
F (Z)b. As π
′ in the fibre-diagram (4) is
flat the natural morphism (π′)∗(FG/P )∗L → (Fb)∗π̂
∗L is an isomor-
phism ([Har2, Prop.III.9.3]). Thus there is a natural isomorphism of
G-linearized sheaves
F (Z)b ≃ (π
′)∗HomO(G/P )′
(FG/P )∗L,O(G/P)′
= (π′)∗EndLF (
G/P).
Let V denote a closed subvariety of G/P . Then EndLF (
G/P , V ) is the
kernel of the natural map
EndLF (
G/P) → HomO(G/P )′
(FG/P )∗(IV ⊗ L),OOV ′
In particular, (π′)∗
EndLF (
G/P , V )
maps into the kernel of the induced
morphism
(7) End
F (Z)b → (π
′)∗HomO
(G/P )′
(FG/P )∗(IV ⊗ L),OOV ′
Let q : G→ G/P denote the quotient map. Then π̂−1(V ) identifies with
the quotient q−1(V )×P X
′. Moreover, as π′ is locally trivial it follows
that π̂∗(IV ) = Iq−1(V )×PX′. In particular, the sheaf
(π′)∗HomO(G/P )′
(FG/P )∗(IV ⊗ L),OOV ′
is isomorphic to
HomOZ′
(Fb)∗(Iq−1(V )×PX′ ⊗ LẐ),O(q−1(V )×PX)′
Thus we see that the kernel of (7) is the subsheaf End
F (Z, π
−1(V ))b
of elements which map (Fb)∗(Iq−1(V )×PX′ ⊗ LẐ) to I(q−1(V )×PX)′ . The
global sections of this subsheaf is denote by End
F (Z, π
−1(V ))b. In
conclusion
Proposition 5.2. The map π′ induces a G-equivariant morphism
(π′)∗ : EndLF (
G/P) → End
F (Z)b.
Moreover, when V is a closed subvariety of G/P then (π′)∗ maps the
subset EndLF (
G/P , V ) into End
F (Z, q
−1(V )×P X)b.
The following is also useful.
Lemma 5.3. Let Y denote a closed P -stable subvariety of X and fix
notation as above. Then each element of End
F (Z)b maps (Fb)∗(IẐY ⊗
LẐ) to I(ZY )′.
Proof. It suffices to show that the natural morphism
HomOZ′
(Fb)∗LẐ ,OZ′
→ HomOZ′
(Fb)∗(IẐY ⊗ LẐ),O(ZY )′
is zero. By linearity, this will follow if the natural morphism
I(ZY )′ ⊗ (Fb)∗LẐ → (Fb)∗(IẐY ⊗ LẐ),
is an isomorphism, which can be checked by a local calculation. �
5.4. Conclusions. By Proposition 5.1 an element v in the vectorspace
IndGP
EndMF (X) ⊗ kλ
defines an element in End
M(−λ)Z
F (Z)f . More-
over, by Proposition 5.2, an element u ∈ EndLF (
G/P) defines an element
(π′)∗(u) in End
F (Z)b. Thus by the discussion in Section 5.2 we obtain
a G-equivariant map
M,λ : End
G/P)⊗ IndGP
EndMF (X)⊗ kλ
→ End
F (G×P X),
defined by
M,λ(u⊗ v) = (π
′)∗u ◦ (Fb)∗v.
We can now prove
Theorem 5.4. Let X denote a P -variety and M denote a P -linearized
line bundle on X. Let L denote the equivariant line bundle on G/P
associated to the P -character λ. Then the G-equivariant map Φ1
defined above, satisfies
(1) When Y is a P -stable closed subvariety of X then the restriction
of Φ1
M,λ to the subspace :
EndLF (
G/P)⊗ IndGP
EndMF (X, Y )⊗ kλ
maps to End
F (G×P X,G×P Y ).
(2) When V denotes a closed subvariety of G/P then the restriction
of Φ1
M,λ to the subspace
EndLF (
G/P , V )⊗ IndGP
EndMF (X)⊗ kλ
maps to End
F (G ×P X, q
−1(V ) ×P X), where q : G → G/P
denotes the quotient map.
Proof. The first statement follows from Proposition 5.1 and Lemma
5.3. The second statement follows from Proposition 5.2 and Lemma
5.5 below. �
Lemma 5.5. Let V denote a closed subset of G/P . Then every element
of End
M(−λ)Z
F (Z)f will map (Ff)∗(MZ ⊗ Iπ−1(V )) to I(π̂)−1(V ) ⊗ LẐ .
Proof. It suffices to prove that the natural morphism
I(π̂)−1(V ) ⊗ (Ff)∗MZ → (Ff)∗
Iπ−1(V ) ⊗MZ
is an isomorphism, which can be checked by a local calculation. �
5.5. Identify IndGP
with the space of global sections of MZ (cf.
Equation (3)). Then we can define a G-equivariant morphism
(8) End
F (G×P X)⊗ Ind
→ EndF (G×P X),
by mapping s⊗σ, for σ a global section of MZ and s : (FZ)∗MZ → OZ′ ,
to the element
(FZ)∗OZ
(FZ )∗σ
−−−−→ (FZ)∗MZ
−→ OZ′ ,
18 XUHUA HE AND JESPER FUNCH THOMSEN
in EndF (G×PX). Combining Φ
M,λ with the morphism in (8) we obtain
a G-equivariant map
ΦM,λ : End
G/P)⊗ IndGP
EndMF (X)⊗kλ
⊗ IndGP
→ EndF (Z),
where an element u⊗ v⊗ σ in the domain is mapped to the composed
(9) (FZ)∗OZ
(FZ)∗σ
−−−−→ (FZ)∗MZ
(Fb)∗v
−−−→ (Fb)∗LẐ
(π′)∗u
−−−→ OZ′.
Notice that by Lemma 3.5 the map u ∈ EndLF (
G/P) factors as
(10) (FG/P )∗L
(FG/P )∗u
−−−−−→ (FG/P )∗ω
→ O(G/P)′ ,
where u! is some global section of the line bundle Ľ := ω
⊗L−1 as-
sociated to u (cf. Section 3.7), and the rightmost map is the evaluation
map with domain (FG/P )∗ω
= EndF (G/P). It follows that we may
extend (9) into a commutative diagram
(FZ)∗OZ
(FZ)∗σ //
(Fb)∗π̂
∗(u!)
(FZ)∗MZ
(Fb)∗v //
(Fb)∗π̂
∗(u!)
(Fb)∗LẐ
(π′)∗u
(Fb)∗π̂
∗(u!)
(FZ)∗π
∗Ľ // (FZ)∗(MZ ⊗ π
∗Ľ) //// (Fb)∗(π̂
rrrrrrrrrrr
where all the vertical maps are induced by multiplication by π̂∗(u!).
Likewise the lower horizontal maps are induced from the upper hori-
zontal maps by multiplication with π̂∗(u!). The triangle on the right is
induced from (10) by pull-back to Z ′.
Theorem 5.6. Let X denote a P -variety and M denote a P -linearized
line bundle on X. Let L denote the equivariant line bundle on G/P
associated to the P -character λ. Then the G-equivariant map ΦM,λ,
defined above, satisfies
(1) When Y is a P -stable closed subvariety of X then the restriction
of ΦM,λ to the subspace :
EndLF (
G/P)⊗ IndGP
EndMF (X, Y )⊗ kλ
⊗ IndGP
maps to EndF (G×P X,G×P Y ).
(2) When V denotes a closed subvariety of G/P then the restriction
of ΦM,λ to the subspace :
EndLF (
G/P , V )⊗ IndGP
EndMF (X)⊗ kλ
⊗ IndGP
maps to EndF (G ×P X, q
−1(V ) ×P X), where q : G → G/P
denotes the quotient map.
Moreover, let u ∈ EndLF (
G/P), v ∈ IndGP
EndMF (X) ⊗ kλ
and σ ∈
IndGP
. Then the element ΦM,λ(u⊗ v ⊗ σ) factorizes both as
(FZ)∗OZ
(FZ)∗σ
−−−−→ (FZ)∗MZ
−→ OZ′,
and as
(FZ)∗OZ
(FZ)∗(σ⊗π
−−−−−−−−→ (FZ)∗(MZ ⊗ π
−→ OZ′,
where s1 and s2 satisfies
i) If v is contained in IndGP
EndMF (X, Y ) ⊗ kλ
then s1 and s2 are
compatible with G×P Y .
ii) If u is contained in EndLF (
G/P , V ) then s1 is compatible with q
−1(V )×P
Proof. Part (1) and (2) follows directly from Theorem 5.4 and the defi-
nition of ΦM,λ. The existence of s1 and s2 follows by the diagram (11).
Finally the claims about the compatibility of s1 and s2 follows from
Theorem 5.4 and Lemma 5.3. �
5.6. We will now describe when an element in the image of ΦM,λ defines
a Frobenius splitting of Z. For this we consider the composed map
evZ◦ΦM,λ. Recall that an element s ∈ EndF (Z) is a Frobenius splitting
of Z if and only if evZ(s) is the constant function 1 on Z
Let u ∈ EndLF (
G/P), v ∈ IndGP
EndMF (X)⊗kλ
and σ ∈ IndGP
By Equation (9) the image of u⊗v⊗σ under evZ ◦ΦM,λ coincides with
the global section of OZ′ determined by the composed map
(12) OZ′
−→ (FZ)∗OZ
(FZ )∗σ
−−−−→ (FZ)∗MZ
(Fb)∗v
−−−→ (Fb)∗LẐ
(π′)∗u
−−−→ OZ′ .
We may divide this composition into two parts. The first part
−→ (FZ)∗OZ
(FZ)∗σ
−−−−→ (FZ)∗MZ
(Fb)∗v
−−−→ (Fb)∗LẐ
is defined by σ and v and defines a global section of LẐ . The corre-
sponding map
M,λ : Ind
EndMF (X)⊗ kλ
⊗ IndGP
→ IndGP
k[X ′]⊗ kλ
is the map induced by the morphism
(13) EndMF (X)⊗M(X) → k[X
mapping s : (FX)∗M → OX′ and τ a global section of M, to s(τ).
Notice that we here identify IndGP
k[X ′]⊗ kλ
with the space of global
sections of LẐ (cf. Equation (3)). The second part takes a global
section τ̃ of LẐ and an element u in End
G/P) to the global section of
OZ′ defined by
−→ (Fb)∗OẐ
(Fb)∗τ̃
−−−→ (Fb)∗LẐ
(π′)∗u
−−−→ OZ′.
20 XUHUA HE AND JESPER FUNCH THOMSEN
The corresponding map is
Φλ : End
G/P)⊗ IndGP
k[X ′]⊗ kλ
→ k[Z ′],
which maps u⊗ τ̃ , to ((π′)∗u)(τ̃) (cf. Proposition 5.2). The restriction
of Φλ :
(14) φλ : End
G/P)⊗ IndGP
is the map corresponding to Φλ in case X is the one point space Spec(k)
(in which case k[X ′] is just k). In combination this defines us a com-
mutative diagram
EndLF (
G/P)⊗ IndGP
EndMF (X)⊗ kλ
⊗ IndGP
Id⊗Φ2
ΦM,λ // EndF (Z)
EndLF (
G/P)⊗ IndGP
k[X ′]⊗ kλ
) Φλ // k[Z ′]
EndLF (
G/P)⊗ IndGP
φλ // k
evG/P
33ggggggggggggggggggggggggggggggg
wheremλ is the natural map which makes the lower part of the diagram
commutative. Notice that when k[X ′] = k, e.g. if X ′ is a complete and
irreducible variety, then φλ and Φλ coincides. Let χ denote the P -
character associated to the canonical G-linearization of ω−1G/P (cf. Sec-
tion 4.3). Then as noted earlier (Section 5.5) the G-module EndLF (
coincides with the space of global sections of Ľ = ω
⊗L−1 and thus
coincides with
(16) EndLF (
G/P) = IndGP
(p− 1)χ− λ
where we abuse notation and write (p− 1)χ− λ for the 1-dimensional
P -representation associated with the character (p− 1)χ−λ. It follows
that mλ is the natural multiplication map
(17) mλ : Ind
(p− 1)χ− λ
⊗ IndGP
→ IndGP
(p− 1)χ
which is surjective if the domain is nonzero, i.e. if L and ω
⊗ L−1
are effective line bundles on G/P [R-R, Thm.3].
The commutativity of the diagram (15) then implies:
Proposition 5.7. Let Ξ denote an element in the domain of ΦM,λ,
and assume that the image (Id⊗Φ2
M,λ)(Ξ) is contained in the subspace
EndLF (
G/P)⊗IndGP
(cf. diagram (15)). Then ΦM,λ(Ξ) is a Frobenius
splitting of Z if and only if φλ((Id ⊗ Φ
M,λ)(Ξ)) equals the constant 1.
In particular, if EndLF (
G/P) ⊗ IndGP
is nonzero and IndGP
contained in the image of Φ2
M,λ, then Z admits a Frobenius splitting.
Proof. The first part of the proof is just a restatement of the fact that
the diagram (15) is commutative. The second part follows by the sur-
jectivity of mλ and the fact that G/P admits a Frobenius splitting.
Corollary 5.8. Assume that X is irreducible and complete. If both
IndGP
and IndGP
(p − 1)χ − λ
are nonzero and Φ2
M,λ is surjective,
then Z admits a Frobenius splitting.
5.7. In many concrete situation the existence of a P -invariant ele-
ment in EndMF (X) ⊗ kλ is given. Notice that this is equivalent to a
G-invariant element v in IndGP
EndMF (X) ⊗ kλ
and thus ΦM,λ defines
a G-equivariant map
(18) EndLF (
G/P)⊗ IndGP
→ EndF (Z),
u⊗ σ 7→ ΦM,λ(u⊗ v ⊗ σ).
Similarly Φ2
M,λ defines a G-equivariant morphism
(19) IndGP
→ IndGP
k[X ′]⊗ kλ
which makes the diagram
(20) EndLF (
G/P)⊗ IndGP
// EndF (Z)
EndLF (
G/P)⊗ IndGP
k[X ′]⊗ kλ
) Φλ // k[Z ′]
commutative. We also note
Corollary 5.9. Assume that X is irreducible and complete and let v
denote a P -invariant element of EndMF (X)⊗ kλ. If the induced map
M,λ)|v⊗IndGP (M(X))
: IndGP
→ IndGP
is surjective then Z admits a Frobenius splitting. In particular, if
IndGP
is an irreducible G-representation then for Z to be Frobenius
split it suffices that the latter map is nonzero.
Proof. Apply Corollary 5.8. �
6. B-Canonical Frobenius splittings
In this section we continue the study of the Frobenius splitting prop-
erties of Z = G ×P X . The notation is kept as in Section 5 but we
restrict ourselves to the case where G is a connected, semisimple and
simply connected linear algebraic group. Moreover, we fix P = B,
M = OX and λ = −(p − 1)ρ. Recall that, in this setup, the dualizing
sheaf ωG/B is the G-linearized sheaf associated to the B-character 2ρ.
22 XUHUA HE AND JESPER FUNCH THOMSEN
Thus, with the notation in Section 5.6, we have χ = −2ρ. Recall also
the G-equivariant identity (see (16))
(21) EndLF (
G/B) ≃ IndGB((p− 1)χ− λ) = Ind
B(λ) = Ind
B((1− p)ρ).
The latter G-module is called the Steinberg module of G and will be
denoted by St. The Steinberg module is a simple and selfdual G-
module. A B-canonical Frobenius splitting ofX is then a B-equivariant
(22) θ : St⊗ k(p−1)ρ → EndF (X),
containing a Frobenius splitting in its image. Notice that a B-canonical
Frobenius splitting of X is not a Frobenius splitting as defined in Sec-
tion 3.2. However, there exists a unique nonzero lowest weight vector
v− of St such that θ(v−) is a Frobenius splitting in the sense of Sec-
tion 3.2. Moreover, as St is a simple G-module the map θ is uniquely
determined by θ(v−), and we may thus identify θ with θ(v−). In this
way θ(v−) will also be called a B-canonical Frobenius splitting of X .
The importance of B-canonical Frobenius splittings was first ob-
served by O. Mathieu in connection with good filtrations of G-modules.
We refer to [B-K, Chapter 4] for a general reference on B-canonical
Frobenius splittings.
6.1. Consider a B-canonical Frobenius splitting as in (22). By Frobe-
nius reciprocity this defines a map
St → IndGB
EndF (X)⊗ kλ
and as IndGB
k[X ]
contains k we may consider the inducedG-equivariant
morphism
θ̃ : St → IndGB
EndF (X)⊗ kλ
⊗ IndGB
k[X ]
Composing θ̃ with the map Φ2
M,λ of Section 5.6 we end up with a map
M,λ ◦ θ̃ : St → Ind
k[X ′]⊗ kλ
We claim
Lemma 6.1. The composed map Φ2
M,λ ◦ θ̃ is an isomorphism on its
image IndGB
Proof. We first prove that the image of Φ2
M,λ◦θ̃ is contained in Ind
For this let EndF (X)c denote the inverse image of k ⊂ k[X
′] under
the evaluation map evX . It suffices to prove that the image of θ is
contained in EndF (X)c. Notice that EndF (X)c is a B-submodule of
EndF (X) containing the set of Frobenius splittings of X . In particular,
the image of the lowest weight space of St under θ is contained in
EndF (X)c. Moreover, as St is an irreducible G-module it is generated
by the lowest weight space as a B-module. Thus, the image of θ will
be contained in the B-module EndF (X)c.
Now Φ2
M,λ ◦ θ̃ is a map from St to Ind
B(λ) = St. Thus, by Frobenius
reciprocity, it suffices to prove that Φ2
M,λ ◦ θ̃ is nonzero which is the
case as θ contains a Frobenius splitting in its image. �
Using Lemma 6.1 we can now combine the diagram (15) with the
map Φ2
M,λ ◦ θ̃ and obtain a commutative and G-equivariant diagram
(23) St⊗ St
≃ Id⊗(Φ2
Θ // EndF (G×B X)
evZ // k[Z ′]
EndLF (
G/B)⊗ IndGB
φλ // k
evG/B
88qqqqqqqqqqqqqq
where Θ is the map induced by θ̃ and ΦM,λ. By Proposition 5.7 it
follows that Θ(Ξ), for Ξ in St⊗ St, is a Frobenius splitting of Z if and
only if the image of Ξ under φλ and Id ⊗ (Φ
M,λ ◦ θ̃) equals 1. The
latter map from St⊗St to k will be denoted by φ. By construction φ is
G-equivariant. Moreover, mλ is surjective and evG/B is nonzero (as G/B
admits a Frobenius splitting) and thus φ is nonzero. As St is a simple
G-module it follows that
(24) φ : St⊗ St → k,
defines a nondegenerate G-invariant bilinear form on St. By Frobenius
reciprocity such a form is uniquely determined up to a nonzero con-
stant. In particular, this provides a very useful way to construct lots
of Frobenius splittings of Z.
Corollary 6.2. Let θ : St⊗ k(p−1)ρ → EndF (X) denote a B-canonical
Frobenius splitting of X. Then the induced morphism (defined above)
Θ : St⊗ St → EndF (G×B X),
satisfies the following
(1) The image Θ(ν) of an element ν in St⊗ St defines a Frobenius
splitting of G×BX up to a nonzero constant if and only if φ(ν)
is nonzero.
(2) If the image of θ is contained in EndF (X, Y ) for a B-stable
closed subvariety Y of X, then the image of Θ is contained in
EndF (G×B X,G×B Y ).
(3) Let v denote an element of St = EndLF (
G/B) which is compatible
with a closed subvariety V of G/B. For any element v′ ∈ St we
Θ(v ⊗ v′) ∈ EndF (G×B X, q
−1(V )×B X),
with q : G→ G/B denoting the quotient map.
24 XUHUA HE AND JESPER FUNCH THOMSEN
(4) Any element of the form Θ(v ⊗ v′) factorizes as
(FZ)∗OZ
(FZ )∗π
−−−−−→ (FZ)∗π
−→ OZ′ ,
where Z = G×B X and L is the line bundle on G/B associated
to the B-character (1 − p)ρ. Moreover, if the image of θ is
contained in EndF (X, Y ) then s is compatible with G×B Y .
Proof. All statements follows directly from Theorem 5.6 and the con-
siderations above. �
The first part (1) and (2) of the above result is well known (see e.g.
[B-K, Ex. 4.1.E(4)]). However, the second part (3) and (4) seems to
be new.
6.2. B-canonical Frobenius splitting when G is not semisim-
ple. Although Corollary 6.2 is only stated for connected, semisimple
and simply connected groups it also applies in other cases : assume
that G is a connected linear algebraic group containing a connected
semisimple subgroup H such that the induced map H/H∩B → G/B is
an isomorphism. E.g. this is satisfied for any parabolic subgroup of a
reductive connected linear algebraic group. Let qsc : Hsc → H denote
a simply connected cover of H . Then X admits an action of the par-
abolic subgroup Bsc := q
sc (B ∩ H) of Hsc. Furthermore, the natural
morphism
Hsc ×Bsc X → G×B X,
is then an isomorphism. We then say that X admits a B-canonical
Frobenius splitting if X , as a Bsc-variety, admits a Bsc-canonical Frobe-
nius splitting. In this case we may apply Corollary 6.2 to obtain Frobe-
nius splitting properties of G×B X .
6.3. Restriction to Levi subgroups. Return to the situation where
G is connected, semisimple and simply connected. Let J be a subset of
the set of simple roots ∆ and let GJ denote the commutator subgroup
of LJ . Then GJ is a connected, semisimple and simply connected linear
algebraic group with Borel subgroup BJ = GJ ∩B and maximal torus
TJ = T ∩ GJ . We let StJ denote the associated Steinberg module.
Notice that StJ = Ind
((1− p)ρJ ) where ρJ denotes the restriction of
ρ to BJ . The following should be well known but we do not know a
good reference.
Lemma 6.3. There exists a GJ -equivariant morphism
StJ → St,
such that the B−J -invariant line of StJ maps surjectively to the B
invariant line of St. In particular, if X is a G-variety admitting a B-
canonical Frobenius splitting then X admits a BJ -canonical Frobenius
splitting as a GJ -variety.
Proof. Let M denote the T -stable complement to the B-stable line in
St. Then M is B−-invariant and thus also B−J -invariant. The trans-
late ẇJ0M is then invariant under BJ and we obtain a BJ -equivariant
morphism
St → St/(ẇJ0M) ≃ k(1−p)ρJ .
By Frobenius reciprocity this defines a GJ -equivariant map St → StJ
such that the B-stable line of St maps onto the BJ -stable line of StJ .
Now apply the selfduality of StJ and St to obtain the desired map.
This proves the first part of the statement.
The second part follows easily by composing the obtained morphism
StJ → St with the B-canonical Frobenius splitting
St → EndF (X)⊗ k(1−p)ρ,
of X and noticing that the restriction of ρ to BJ is ρJ .
7. Applications to G×G-varieties
In this section we consider a linear algebraic group G satisfying the
conditions of Section 6.2, i.e. we assume that G contains a closed
connected semisimple subgroup H such that H/H∩B → G/B is an iso-
morphism. We also let Hsc denote the simply connected version of H
and let Bsc denote the associated Borel subgroup.
7.1. A well known result. Consider for a moment (i.e. in this sub-
section) the case where G is semisimple and simply connected. Re-
member that the G-linearized line bundle on G/B associated to the B-
character 2ρ coincides with the dualizing sheaf ωG/B. Let L denote the
line bundle on G/B associated to the B-character (1−p)ρ and recall from
Section 6 the notation St = IndGB((1−p)ρ) for the Steinberg module. As
the Steinberg module is a selfdual G-module we may fix a G-invariant
nonzero element v∆ in the tensorproduct St⊗ St. We may think of v∆
as a global section of the line bundle L⊠ L on (G/B)2 = G/B × G/B.
Identify G/B × G/B with G×B G/B by the isomorphism
G×B G/B → G/B × G/B,
[g, hB] 7→ (gB, ghB),
and let D denote the subvariety of G/B × G/B corresponding to G ×B
∂(G/B), where ∂(G/B) denotes the union of the codimension 1 Schubert
varieties in G/B. Then, by [B-K, proof of Thm.2.3.8], the zero scheme
of v∆ equals (p− 1)D. Consider then the natural morphism :
η : (L⊠ L)⊗ (L⊠ L) → ω
(G/B)2
= End!F ((
G/B)2)
and define
ηD : (L⊠ L) → End
G/B)2),
26 XUHUA HE AND JESPER FUNCH THOMSEN
as in Lemma 3.6, using the identification L ⊠ L = O(G/B)2
(p − 1)D
Then by Lemma 3.6 the image of ηD is contained in End
G/B)2, D)
and thus the associated element
η′D ∈ HomO((G/B)2)′
(F(G/B)2)∗(L⊠ L),O((G/B)2)′
is compatible with D. It follows
Lemma 7.1. The element in
EndL⊠LF ((
G/B)2) ≃ St⊠ St
defined by v∆ is compatible with the diagonal diag(G/B) in G/B × G/B.
Proof. We have to prove that η′D, defined above, is compatible with the
diagonal diag(G/B). As η′D is compatible with D it suffices to show that
EndL⊠LF ((
G/B)2, D) is contained in EndL⊠LF ((
G/B)2, diag(G/B)
. This fol-
lows by an application of Lemma 3.1 and an argument as at the end of
the proof of [B-K, Thm.2.3.1]. �
7.2. We return to the setup as in the beginning of this section. We
want to apply the results of the preceding sections to the case when
the group equals G×G. So let X denote a B ×B-variety and assume
that X admits a Bsc ×Bsc-canonical Frobenius splitting defined by
θ : (St⊠ St)⊗ (k(p−1)ρ ⊠ k(p−1)ρ) → EndF (X),
which is compatible with certain B×B-stable subvarieties X1, . . . , Xm,
i.e. the image of θ is contained in EndF (X,Xi) for all i. Then
Theorem 7.2. The variety (G × G) ×(B×B) X admits a diag(Bsc)-
canonical Frobenius splitting which compatibly Frobenius splits the sub-
varieties diag(G)×diag(B) X and (G×G)×(B×B) Xi for all i.
Proof. It suffices to consider the case where G = Hsc (cf. discussion
in Section 6.2). By Corollary 6.2 there exists a G × G-equivariant
morphism
Θ : (St⊠ St)⊗ (St⊠ St) → EndF ((G×G)×(B×B) X),
satisfying certain compatibility conditions. Let v∆ ∈ St ⊠ St be a
nonzero diag(G)-invariant element and let v ∈ St ⊠ St be arbitrary.
Then by Corollary 6.2 and Lemma 7.1 the element Θ
v∆ ⊗ v
is com-
patible with diag(G) ×diag(B) X and (G × G) ×(B×B) Xi for all i. In
particular, if we define the diag(G)-equivariant morphism
Θ∆ : St⊗ St → EndF ((G×G)×(B×B) X),
by Θ∆(v) = Θ
v∆ ⊗ v
, then every element in the image of Θ∆ is
compatible with diag(G) ×diag(B) X and (G × G) ×(B×B) Xi for all i.
Consider k(p−1)ρ as the highest weight line in St. Then the restriction
of Θ∆ to St⊗ k(p−1)ρ defines a diag(B)-canonical Frobenius splitting of
(G×G)×(B×B) X with the desired properties. �
Notice that by the general machinery of canonical Frobenius split-
tings (see e.g. [B-K, Prop.4.1.17]) the existence of a Frobenius splitting
of diag(G)×diag(B) X follows if X admits a diag(Bsc)-canonical Frobe-
nius splitting. In the above setup X only admits a Bsc ×Bsc-canonical
Frobenius splitting which is less restrictive. However, in contrast to the
situation when X admits a diag(Bsc)-canonical Frobenius splitting, the
present Frobenius splitting is not necessarily compatible with subvari-
eties of the form BẇB ×B X , with w denoting an element of the Weyl
group and BẇB denoting the closure of BẇB in G.
8. G-Schubert varieties in equivariant Embeddings
From now on, unless otherwise stated, we assume that G is a con-
nected reductive group.
8.1. Equivariant embeddings. Consider G as a G × G-variety by
left and right translation. An equivariant embedding X of G is then
a normal irreducible G × G-variety containing an open dense subset
which is G × G-equivariantly isomorphic to G. In particular, we may
identify G with an open subset of X , and the complement X \ G is
then called the boundary. As G is an affine variety the boundary is of
pure codimension 1 in X [Har, Prop.3.1]. Any equivariant embedding
of G is a spherical variety (with respect to the induced B × B-action)
and thus X contains finitely may B ×B-orbits.
8.2. Wonderful compactifications. When G = Gad is of adjoint
type there exists a distinguished equivariant embedding X of G which
is called the wonderful compactification (see e.g. [B-K, 6.1]).
The boundary X \ G is a union of irreducible divisors Xj , j ∈ ∆,
which intersect transversely. For a subset J ⊂ ∆, we denote the inter-
section ∩j∈JXj by XJ . As a (G×G)-variety, XJ is isomorphic to the
variety (G × G) ×P−
×P∆\J
Y, where Y denotes the wonderful com-
pactification of the group of adjoint type associated to L∆\J . Here the
×P∆\J -action on Y is defined by the quotient maps P∆\J → L∆\J
and P−
→ L∆\J . In particular, X∆ is G×G-equivariantly isomorphic
to the variety G/B− × G/B.
8.3. Toroidal embeddings. Let Gad denote the group of adjoint
type associated to G, and let X denote the wonderful compactifica-
tion of Gad. An embedding X of the reductive group G is then called
toroidal if the canonical map G→ Gad admits an extension X → X.
8.4. G-Schubert varieties. By a G-Schubert variety in an equivari-
ant embedding X we will mean a subvariety of the form diag(G) · V ,
for some B × B-orbit closure V . Notice that diag(G) · V is the image
of diag(G)×diag(B) V under the proper map
diag(G)×diag(B) X → X,
28 XUHUA HE AND JESPER FUNCH THOMSEN
[g, x] 7→ g · x,
and thus G-Schubert varieties are closed diag(G)-stable subvarieties of
If G = Gad and X = X is the wonderful compactification then a G-
Schubert variety in X∆ is diag(G)-equivariantly isomorphic to a variety
of the form G×BX(w), where X(w) denotes a Schubert variety in G/B.
In particular, this explains the name G-Schubert varieties as this is the
name used for varieties of the form G×B X(w).
In the rest of this section, we will relate G-Schubert varieties to
closures of so-called G-stable pieces. Our primary interest are G-stable
pieces in wonderful compactifications but below we will also describe
the toroidal case in general.
8.5. G-stable pieces in the wonderful compactification. LetG =
Gad denote a group of adjoint type and let X denote its wonderful com-
pactification. Let J ⊂ ∆ and identify XJ with (G× G)×P−
×P∆\J
as in Section 8.2. Using this identification it easily follows that there
exists a unique element in XJ which is invariant under U
J × UJ and
diag(LJ). We denote this element by hJ and note that as an element
of (G×G)×P−
×P∆\J
Y it equals [(e, e), eJ ], where e (resp. eJ) denotes
the identity element of G (resp. the adjoint group associated to L∆\J ).
For w ∈ W∆\J , we then let
XJ,w = diag(G)(Bw, 1) · hJ ,
and call XJ,w a G-stable piece of X. A G-stable piece is a locally closed
subset of X and by [L, section 12] and [He, section 2], we can use them
to decompose X as follows
w∈W∆\J
XJ,w.
Moreover, by the proof of [He2, Theorem 4.5], any G-Schubert variety
is a finite union of G-stable pieces. In particular, we may think of
G-Schubert varieties as closures of G-stable pieces.
8.6. G-stable pieces in arbitrary toroidal embeddings. We fix a
toroidal embedding X of G. The irreducible components of the bound-
ary X \G will be denoted by X1, . . . , Xn. For each G×G-orbit closure
Y in X we then associate the set
KY = {i ∈ {1, . . . , n} | Y ⊂ Xi},
where by definition KY = ∅ when Y = X . Then by [B-K, Prop.6.2.3],
Y = ∩i∈KYXi. Moreover, we define
I = {KY ⊂ {1, . . . , n} | Y a G×G-orbit closure in X },
and write XK := ∩i∈KXi for K ∈ I. Then (XK)K∈I are the set of
closures of G×G-orbits in X . Let now πX : X → X denote the given
extension of G → Gad. Then the closure of πX(XK) equals XP (K) for
some unique subset P (K) of ∆. This defines a map P : I → P(∆),
where P(∆) denotes the set of subsets of ∆.
As in [H-T2, 5.4], for K ∈ I we may choose a base point hK in the
open G×G-orbit of XK which maps to hP (K). By [H-T2, Proposition
5.3], XK is then naturally isomorphic to (G×G)×P−
×P∆\J
L∆\J · hK ,
where J = P (K) and L∆\J · hK is a toroidal embedding of a quotient
(L∆\J )/H by some subgroup H of the center of L∆\J .
For K ∈ I and w ∈ W∆\p(K), we then define
XK,w = diag(G)(Bw, 1) · hK ,
and call XK,w a G-stable piece of X . One can then show, in the same
way as in [He2, 4.3], that
w∈W∆\P (K)
XK,w.
Also similar to the proof of [He2, Theorem 4.5], for any B × B-orbit
closure V in X , the G-Schubert variety diag(G) · V is a finite union
of G-stable pieces. In particular, G-Schubert varieties are closures of
G-stable pieces.
9. Frobenius splitting of G-Schubert varieties
In this section, we assume that X is an equivariant embedding of G.
Let Gsc denote a simply connected cover of the semisimple commutator
subgroup (G,G) of G. We fix a Borel subgroup Bsc of Gsc which is
compatible with the Borel subgroup B in G. Similarly we fix a maximal
torus Tsc ⊂ Bsc.
Let X1, . . . , Xn denote the boundary divisors of X . The closure
within X of the B × B-orbit Bsjw0B ⊂ G will be denoted by Dj .
Then Dj is of codimension 1 in X . The translate (w0, w0)Dj of Dj will
be denoted by D̃j.
By earlier work we know
Theorem 9.1. [H-T2, Prop.7.1] The equivariant embedding X admits
a Bsc × Bsc-canonical Frobenius splitting which compatibly Frobenius
splits the closure of every B ×B-orbit.
As a direct consequence of Theorem 7.2 we then obtain
Corollary 9.2. The variety (G × G) ×(B×B) X admits a diag(Bsc)-
canonical Frobenius splitting which is compatible with all subvarieties
of the form (G×G)×(B×B) Y and diag(G)×diag(B)Y , for a B×B-orbit
closure Y in X.
Proposition 9.3. The equivariant embedding X admits a diag(Bsc)-
canonical Frobenius splitting which compatibly splits all G-Schubert va-
rieties in X.
30 XUHUA HE AND JESPER FUNCH THOMSEN
Proof. By Corollary 9.2 the variety Z = diag(G) ×diag(B) X admits
a diag(Bsc)-canonical Frobenius splitting which is compatible with all
subvarieties of the form diag(G)×diag(B) Y , with Y denoting a B ×B-
orbit closure in X . As X is a diag(G)-stable we may identify Z with
G/B ×X using the isomorphism
G×B X → G/B ×X,
[g, x] 7→ (gB, gx).
In particular, we see that the morphism
π : Z = diag(G)×diag(B) X → X,
[g, x] 7→ g · x,
is projective and that π∗(OZ) = OX . As a consequence (see Section 3.8)
the diag(Bsc)-canonical Frobenius splitting of Z induces a diag(Bsc)-
canonical Frobenius splitting of X which is compatible with all subva-
rieties of the form
π(diag(G)×diag(B) Y ) = diag(G) · Y,
i.e. with all the G-Schubert varieties in X . This ends the proof. �
As a direct consequence of Proposition 9.3, we conclude the following
vanishing result (see [B-K, Theorem 1.2.8]).
Corollary 9.4. Let X denote a projective equivariant embedding of G.
Let X denote a G-Schubert variety in X and let L denote an ample line
bundle on X. Then
Hi(X,L) = 0, i > 0.
Moreover, if X̃ ⊂ X is another G-Schubert variety, then the restriction
H0(X,L) → H0(X̃,L),
is surjective.
Later (i.e. Cor. 10.5) we will generalize the vanishing part of this
result to nef line bundle.
9.1. F-splittings along ample divisors. In this subsection we as-
sume that X is toroidal. The following structural properties of toroidal
embeddings can all be found in [B-K, Sect.6.2]. Let X0 denote the com-
plement in X of the union of the subsets BsiB− for i ∈ ∆. If we let T̄
denote the closure of T in X , then X0 admits a decomposition defined
by the following isomorphism
(25) U × U− × (T̄ ∩X0) → X0, (x, y, z) 7→ (x, y) · z.
Moreover, every G×G-orbit in X intersects (T̄ ∩X0) in a unique orbit
under the left action of T . Notice here that as T is commutative the
T × T -orbits and the (left) T -orbit in T will coincide.
Lemma 9.5. Let X denote a projective toroidal equivariant embedding
of G and let Y denote a G × G-orbit closure in X. Let K denote the
subset of {1, . . . , n} consisting of those j such that Y is contained in
the boundary component Xj. Then
Y ∩ (
j /∈K
(1, w0)Di),
has pure codimension 1 in Y and contains the support of an ample
effective Cartier divisor on Y .
Proof. Let XK = ∪j /∈KXj . We claim that Y \X
K coincides with the
open G × G-orbit Y0 of Y . Clearly Y0 is contained in Y \ X
K . On
the other hand, let U be a G×G-orbit in Y \XK . Then Xj contains
U if and only if j /∈ K. But every G × G-orbit closure in X is the
intersection of those Xj which contain it [B-K, Prop.6.2.3]. It follows
that the closure of Y0 and U coincide and thus U = Y0.
As X is normal we may choose a G × G-linearized very ample line
bundle L on X . Then H0(Y,L) is a finite dimensional (nonzero) rep-
resentation of G ×G, and it thus contains a nonzero element v which
is B × B−-invariant up to constants. The support of v is then the
union of B × B−-invariant divisors on Y . As Y0 ∩ (T̄ ∩X0) is a single
T × T -orbit it follows that
Y0 ∩X0 ≃ U × U
− × (Y0 ∩ (T̄ ∩X0)),
is an affine variety and a single B×B−-orbit. In particular, the support
of v is contained in
Y \ (Y0 ∩X0) = Y ∩ (X
(1, w0)Di).
This shows the second part of the statement. The first part follows as
Y0 ∩X0 is affine [Har, Prop.3.1]. �
Let now X denote a smooth projective toroidal embedding of G. As
the line bundles OX(Di) and OX(D̃i) are isomorphic it follows by [B-K,
Prop.6.2.6] that
(26) ω−1X ≃ OX
(Di + D̃i) +
Recall that a X is normal and G is semisimple and simply connected,
any line bundle on X will admit a unique G2sc = Gsc×Gsc-linearization.
In particular, if we let τi denote the canonical section of the line bundle
OX(Di), then we may consider τi as a B
sc = Bsc×Bsc-eigenvector of the
space of global sections of OX(Di). As in the proof of [B-K, Prop.6.1.11]
we find that the associated weight of τi equals ωi ⊠ −w0ωi, where ωi
denotes the i-th fundamental weight. Similarly, we may consider the
canonical section σj of OX(Xj) as a G
sc-invariant element.
32 XUHUA HE AND JESPER FUNCH THOMSEN
Let V denote a B ×B-orbit closure in X . As V is B ×B-stable the
subset Y = (G×G) ·V is closed in X . Thus we may consider Y as the
smallest G×G-invariant subvariety of X containing V . Now define K
as in Lemma 9.5 and let M denote the line bundle
M = OX
(p− 1)(
D̃i +
j /∈K
By Equation (26) and Lemma 3.6 it then follows that multiplication
with τ
i , for i ∈ ∆, and σ
j , for j ∈ K, defines a morphism of
B2sc-linearized line bundles
M → End!F
X, {Di, Xj}i∈∆,j∈K
⊗ kλ⊠λ,
where λ = (1 − p)ρ. By [H-T2, Prop.6.5] and Lemma 3.1 any element
in End!F (X) which is compatible with the closed subvarieties Di, i ∈ ∆,
and Xj, j ∈ K, is also compatible with V and Y . In particular, we
have defined a B2sc-equivariant map
(27) η : M → End!F (X, Y, V
⊗ kλ⊠λ,
which, by Lemma 3.5, is the same as a B2sc-invariant element η
EndMF
X, Y, V
⊗ kλ⊠λ. In particular, this defines us an element
(28) v ∈ Ind
EndMF
X, Y, V
⊗ kλ⊠λ
which is G2sc-invariant. We are then ready to use the ideas explained
in Section 5.7. First we use (18) to construct a morphism
(29) EndL⊠LF
(Gsc/Bsc)
⊗M(X) → EndF
G2sc ×B2sc X
(u, σ) 7→ ΦM,λ⊠λ(u⊗ v ⊗ σ),
where L is the Gsc-linearized line bundle on Gsc/Bsc associated to the
character λ = (1− p)ρ. Notice that we here have used that M(X) is a
G2sc-module.
Lemma 9.6. There exists a G2sc-equivariant map
(30) St⊠ St → M(X),
which maps the B−sc×B
sc-invariant line in St⊠St to a nonzero multiple
of the global section
j /∈K
j ∈ M(X),
where τ̃i denotes the canonical section of OX(D̃i).
Proof. As OX(D̃i) and OX(Di) are isomorphic as line bundles we may
consider the element
j /∈K
as a global section ofM. Then σ is a B2sc-eigenvector inM(X) of weight
(p − 1)ρ ⊠ (p − 1)ρ. In particular, σ induces a Bsc × Bsc-equivariant
k(p−1)ρ ⊠ k(p−1)ρ → M(X).
Applying Frobenius reciprocity and the selfduality of the Steinberg
module St, this defines the desired map
St⊠ St → M(X),
with the stated properties. �
Combining the map (29) with the map (30) in Lemma 9.6 we obtain
a G2sc-equivariant map
(31) Θ : EndL⊠LF
(Gsc/Bsc)
St⊠ St
→ EndF
G2sc ×B2sc X
We will now study when the map (31) describes a Frobenius splitting
of G2sc ×B2sc X . Consider the G
sc-equivariant map
(32) M(X) → St⊠ St,
σ 7→ Φ2
M,λ⊠λ(v ⊗ σ),
defined as the map (19) in Section 5.7. We claim
Lemma 9.7. The composition of the map (30) in Lemma 9.6 and the
map in (32) is an isomorphism on St⊠ St.
Proof. By Frobenius reciprocity it suffices to show that the described
composed map is nonzero. In particular, it suffices to show that
M,λ⊠λ(v ⊗ σ̃) 6= 0,
where σ̃ denotes the global section of M defined in Lemma 9.6. For
this we use the fact that the global section
(τiτ̃i)
X defines a Frobenius splitting of X (see e.g. [B-K, proof of
Thm.6.2.7]). As a consequence η(σ̃) is a Frobenius splitting ofX , where
η is the map defined in (27). Equivalently , the natural G2sc-equivariant
morphism
EndMF (X)⊗M(X) → k[X
′] = k,
defined in (13), will map η′ ⊗ σ̃ to 1. This induces a commutative
diagram
(33) Ind
EndMF
⊗ kλ⊠λ
⊗M(X)
M,λ⊠λ
++VVV
St⊗ St
EndMF (X)⊗ kλ⊠λ ⊗M(X)
// kλ⊠λ
34 XUHUA HE AND JESPER FUNCH THOMSEN
where the image of v⊗ σ̃ under the diagonal map is nonzero. This ends
the proof. �
Proposition 9.8. Let Θ denote the map defined in (31). The image
Θ(ν) of an element ν defines, up to a nonzero constant, a Frobenius
splitting of G2sc ×B2sc X if and only if the image of ν under the map
(34) φλ⊠λ : End
(Gsc/Bsc)
St⊠ St
defined in Section 5.6, is nonzero.
Proof. Apply Proposition 5.7 and Lemma 9.7.
With the identification EndL⊠LF
(Gsc/Bsc)
≃ St⊠ St the map φλ⊠λ,
defined in (34), must necessarily (up to a nonzero constant) be the
G2sc-invariant form on St⊠ St mentioned in Section 6.1. Let v∆ denote
the diag(G)-invariant element in EndL⊠LF
(Gsc/Bsc)2
defined in Section
7.1. Then the diag(G)-equivariant map
St⊗ St → k,
ν 7→ φλ⊠λ(v∆ ⊗ ν),
is nonzero and thus it must coincide (up to a nonzero constant) with
the Gsc-invariant form φ on St defined in (24).
Proposition 9.9. Fix notation as above and let D denote the effective
Cartier divisor
(p− 1)
(1, w0)Di +
j /∈K
on X. Then X admits a Frobenius D-splitting which is compatible with
the subvariety Y and the G-Schubert variety diag(G) · V .
Proof. Consider the diag(G)-equivariant morphism
Θ∆ : St⊠ St → EndF
G2sc ×B2sc X
ν 7→ Θ(v∆ ⊗ ν),
where Θ is the map in (31). By Lemma 9.8 the image Θ∆(ν) of an
element ν ∈ St⊗ St is a Frobenius splitting, up to a nonzero constant,
if and only if φ(ν) is nonzero. Here φ is the the map defined in (24).
Let v+ (resp. v−) denote a nonzero B (resp. B
−)-eigenvector of St
and let ν = v+ ⊗ v−. After possibly multiplying v+ with a constant
we may assume that s = Θ∆(ν) defines a Frobenius splitting of Z =
G2sc ×B2sc X . As v is compatible with Y and V (cf. (28)) it follows by
Theorem 5.6 and Lemma 7.1 that s factorizes as
(35) s : (FZ)∗OZ
(FZ )∗σ
−−−−→ (FZ)∗MZ
−→ OZ′,
where s1 is compatible with the subvarieties G
sc×B2sc V , G
sc×B2sc Y and
diag(Gsc)×diag(Bsc) X . Here MZ is the G
sc-linearized line bundle on Z
associated with the B2sc-linearized line bundle M on X as explained in
Section 5.2, and σ is the global section of MZ defined as the image of ν
under the map (30) in Lemma 9.6. Notice that as M is a G2sc-linearized
line bundle on the G2sc-variety X we may identify the global sections of
M and MZ . Actually , as X is a G
sc-variety the morphism
G2sc ×B2sc X →
Gsc/Bsc × Gsc/Bsc ×X,
[(g1, g2), x] 7→ (g1B, g2B, (g1, g2) · x),
is an isomorphism. Moreover, under this isomorphism, the line bundle
MZ is just the pull back of M under projection pX on the third coor-
dinate. Thus, by Lemma 9.6 it follows that σ is the pull back from X
of the effective Cartier divisor
D = (p− 1)
(1, w0)Di +
j /∈K
Applying the functor (pX)∗ to (35) we obtain the Frobenius D-splitting
(pX)∗s : (FX)∗OX
(FX)∗σD
−−−−−→ (FX)∗O(D)
(pX)∗s1
−−−−→ OX′
ofX where (pX)∗s1 is compatible with the subvarieties pX(G
sc×B2scY ) =
Y and pX(diag(Gsc)×diag(Bsc) V ) = diag(G) · V (by Lemma 3.7). This
ends the proof. �
Corollary 9.10. Let X denote a G-Schubert variety in a smooth pro-
jective toroidal embedding of a reductive group G. Then X admits a
stable Frobenius splitting along an ample divisor.
Proof. Apply Proposition 9.9, Lemma 9.5 and Lemma 3.3. �
10. Cohomology of line bundles
The main aim of this section is to obtain a generalizing the vanishing
part of Corollary 9.4 to nef line bundles. The concept of a rational
morphism is here central and for this we use [B-K, Sect.3.3] as a general
reference. First we recall :
Definition 10.1. A morphism f : Y → Z of varieties is a called a ra-
tional morphism if the induced map f ♯ : OZ → f∗OY is an isomorphism
and Rif∗OY = 0, i > 0.
The following criterion for a morphism to be rational will be very
useful ([R, Lem.2.11]).
Lemma 10.2. Let f : Y → Z denote a projective morphism of ir-
reducible varieties and let Ŷ denote a closed irreducible subvariety of
Y . Consider the image Ẑ = f(Ŷ ) as a closed subvariety of Z. Let L
denote an ample line bundle on Z and assume
(1) f ♯ : OZ → f∗OY is an isomorphism.
(2) Hi(Y, f ∗Ln) = Hi(Ŷ , f ∗Ln) = 0, for i > 0 and n≫ 0.
36 XUHUA HE AND JESPER FUNCH THOMSEN
(3) The restriction map H0(Y, f ∗Ln) → H0(Ŷ , f ∗Ln) is surjective
for n≫ 0.
Then the induced map f̂ : Ŷ → Ẑ is a rational morphism.
10.1. Toric variety. An equivariant embedding Z of the (reductive)
group T is called a toric variety (wrt. T ). Notice that, as T is commu-
tative, we may consider the T ×T -action on Z as just a T -action. The
following result should be well known but, as we do not know a good
reference, we include a proof.
Lemma 10.3. Let f : Y → Z denote a projective surjective morphism
of equivariant embeddings of T . Let T · z denote a T -orbit in Z and let
T · y denote a T -orbit in f−1(T · z) of minimal dimension. Then the
map T · y → T · z, induced by f , is an isomorphism.
Proof. Let T · z and T · y denote the closures of T · z and T · y in Z
and Y respectively. Then the induced map
f̂ : T · y → T · z,
is a projective morphism. Moreover, by the minimality assumption
on T · y, the inverse image f̂−1(T · z) equals T · y. In particular, the
induced morphism : T · y → T · z is projective. But any T -orbit in a
toric variety (wrt. to T ) is isomorphic to a torus T1 satisfying that the
cokernel of the induced map of character groups X∗(T1) → X
∗(T ) is a
free abelian group ([Ful, Sect.3.1]). In particular, the varieties T ·y and
T · z are tori and the cokernel of the induced map of character groups
X∗(T · z) → X∗(T · y) is a free abelian group. But T · y → T · z is
an affine projective morphism and thus it must be a finite morphism.
Thus the cokernel of X∗(T · z) → X∗(T · y) is a finite group and, as it
is already a free group, it must be trivial. This ends the proof as tori
are determined by their character groups. �
Lemma 10.4. Let X denote a projective embedding of a reductive
group G and let Y denote a G × G-orbit closure of X. Then there
exists a smooth toroidal embedding X̂ of G, a projective G-equivariant
morphism f : X̂ → X and a G×G-orbit closure Ŷ in X̂ such that the
induced morphism f : Ŷ → Y is a rational morphism.
Proof. Assume first that X is toroidal. By [B-K, Prop.6.2.5] there ex-
ists a smooth toroidal embedding X̂ of G with a projective morphism
f : X̂ → X . Let X0 denote the open subset of X introduced in the
beginning of Section 9.1, and let X̂0 denote the corresponding sub-
set of X̂ . Then the inverse image f−1(X0) coincides with X̂0 [B-K,
Prop.6.2.3(i)]. Let T (resp. T̂ ) denote the closure of T in X (resp. X̂).
Then T and T̂ are toric varieties [B-K, Prop.6.2.3], and the induced
map f : T̂ → T is a projective morphism of toric varieties. Thus also
the induced map
X̂0 ∩ T̂ → X0 ∩ T ,
is a projective morphism of toric varieties. As mentioned in Section 9.1
every G×G-orbit in X will intersect X0 ∩ T in a unique T -orbit. We
let T · x denote the open T -orbit in the intersection of Y with X0 ∩ T .
By Lemma 10.3 we may find a T -orbit T · x̂ in X̂0 ∩ T̂ which by f
is isomorphic to T · x, and we then define Ŷ to be the closure of the
G×G-orbit through x̂. By the isomorphism (25) we then conclude that
f induces a projective birational morphism Ŷ → Y . By [H-T2, Cor.8.4]
the orbit closure Y is normal and thus, by Zariski’s main theorem, we
conclude f∗OŶ = OY . By Lemma 10.2 (used on the morphism Ŷ → Y
and the closed non-proper subvariety Ŷ of Ŷ ) it now suffices to prove
Hi(Ŷ , f ∗L) = 0, i > 0,
for a very ample line bundle L on Y . This follows from [H-T2, Prop.7.2]
and ends the proof in the case when X is toroidal.
Consider now an arbitrary projective equivariant embedding X of
G. Let X̂ denote the normalization of the closure of the image of the
natural G×G-equivariant embedding
G→ X ×X,
where X denotes the wonderful compactification of Gad. Then X̂ is a
toroidal embedding of G with an induced projective equivariant mor-
phism f : X̂ → X . Let Ŷ denote any G×G-orbit closure in X̂. Then
f : Ŷ → f(Ŷ ) is a rational morphism [H-T2, Lem.8.3]. In particular,
we may find a G × G-orbit closure Ŷ of X̂ with an induced rational
morphism f : Ŷ → Y . Finally we may apply the first part of the proof
to Ŷ and X̂ and use that a composition of rational morphisms is again
a rational morphism. �
Corollary 10.5. Let X denote a projective embedding of a reductive
group G and let X denote a G-Schubert variety in X. Let Y = (G×G)·
X denote the minimal G × G-orbit closure of X containing X. When
L is a nef line bundle on X then
Hi(X,L) = 0, i > 0.
Moreover, when L is a nef line bundle on Y then the restriction mor-
phism
H0(Y,L) → H0(X,L),
is surjective.
Proof. Assume first that X is smooth and toroidal. Then by Propo-
sition 9.9, Lemma 9.5 and Lemma 3.3 the variety Y admits a stable
Frobenius splitting along an ample divisor which is compatibly with X.
Thus the statement follows in this case by Proposition 3.4.
38 XUHUA HE AND JESPER FUNCH THOMSEN
Let now X denote an arbitrary projective equivariant embedding of
G. Choose, using Lemma 10.4, a smooth projective toroidal embedding
X̂ with a projective equivariant morphism f : X̂ → X onto X , and a
G × G-orbit closure Ŷ in X̂ with an induced rational morphism onto
Y . Let V denote a B×B-orbit closure in Y such that X = diag(G) ·V .
As Y is the minimal G × G-orbit closure containing X it follows that
V will intersect the open G×G-orbit of Y . In particular, there exists
a B×B-orbit closure V̂ in X̂ which intersects the open G×G-orbit of
Ŷ and which maps onto V . In particular,
X̂ := diag(G) · V̂ ,
is a G-Schubert variety in X̂ which by f maps onto X. Moreover, Ŷ is
the minimal G×G-orbit closure containing X̂.
We claim that the induced morphism X̂ → X is a rational morphism.
To prove this we apply Lemma 10.2 to the rational morphism f : Ŷ →
Y . Choose an ample line bundle M on Y . Then it suffices to prove
(36) Hi(Ŷ , f ∗Mn) = Hi(X̂, f ∗Mn) = 0, i > 0, n > 0,
and that the restriction map
(37) H0(Ŷ , f ∗Mn) → H0(X̂, f ∗Mn),
is surjective for n > 0. But Mn is an ample, and thus nef, line bundle
on Y and therefore the pull back f ∗Mn is a nef line bundle on Ŷ ([Laz,
Ex. 1.4.4]). As X̂ is smooth and toroidal, the conclusion of the first
part of this proof then shows that conditions (36) and (37) are satisfied.
Now both X̂ → X and Ŷ → Y are rational morphisms. In particular,
we have identifications
Hi(Ŷ , f ∗L) ≃ Hi(Y,L), i ≥ 0,
Hi(X̂, f ∗L) ≃ Hi(X,L), i ≥ 0,
for any line bundle L on Y or, in the second equation, on X . When L is
a nef line bundle the pull back f ∗L is also nef ([Laz, Ex. 1.4.4]). Thus
as we have already completed the proof of the statement for smooth
toroidal embeddings, in particular for X̂ , this now ends the proof. �
By the proof of the above result we also find that any G-Schubert
variety X in a projective equivariant embedding of G, will admit a G-
equivariant rational morphism f : X̂ → X by a G-Schubert variety X̂
of some smooth projective toroidal embedding of G.
Remark 10.6. When X = X is the wonderful compactification of a
group G of adjoint type and L is a nef line bundle on X, then the
restriction morphism
H0(X,L) → H0(Y,L),
to any closed G×G-stable irreducible subvariety Y of X is surjective.
In particular, also the restriction morphism
H0(X,L) → H0(X,L),
to any G-Schubert variety X is surjective by the above result. We do
not know if the latter is true for arbitrary equivariant embeddings.
11. Normality questions
The obtained Frobenius splitting properties of G-Schubert varieties
in Section 9 and the cohomology vanishing results in Corollary 10.5
should be expected to have strong implications on the geometry of
these varieties. However, in this section we provide an example of a G-
Schubert variety in the wonderful compactification of a group of type
G2 which is not even normal. In fact, it seems that there are plenty of
such examples.
11.1. Some general theory. We keep the notations as in Section
8.5. For J ⊂ ∆ and w ∈ W∆\J , we let XJ,w denote the closure of XJ,w
in X. Let
K = max{K ′ ⊂ ∆ \ J ;wK ′ ⊂ K ′}.
By [He2, Prop. 1.12], we have a diag(G)-equivariant isomorphism
diag(G)×diag(PK) (PKẇ, PK)hJ ≃ XJ,w
induced by the inclusion of (PKẇ, PK)hJ in X. Let V denote the
closure of (PKẇ, PK)hJ within X. Then V is the closure of a B × B-
orbit and we find that the induced map
(38) f : diag(G)×diag(PK) V → XJ,w,
is a birational and projective morphism. Thus, by Zariski’s Main The-
orem, a necessary condition for XJ,w to be normal is that the fibers of
f are connected. Actually, in positive characteristic, connectedness of
the fibers is also sufficient forXJ,w to be normal. This follows asXJ,w is
Frobenius split (Prop. 9.3) and thus weakly normal [B-K, Prop.1.2.5].
11.2. An example of a non-normal closure. Let now, further-
more, G be a group of type G2. Let α1 denote the short simple root
and α2 denote the long simple root. The associated simple reflections
are denoted by s1 and s2. Let J = {α2} and w = s1s2 ∈ W
∆\J . In this
case K = ∅ and we obtain a birational map
f : diag(G)×diag(B) V ≃ XJ,w
where V is the closure of (Bẇ,B)hJ . By [Sp, Prop. 2.4], the part of
V which intersect the open G×G-orbit of XJ equals
(Bẇ′, B)hJ ∪
ws1≤w′
(Bẇ′, Bṡ1)hJ .
40 XUHUA HE AND JESPER FUNCH THOMSEN
In particular, x := (v̇, 1)hJ is an element of V , where v = s2s1s2. We
claim that the fiber of f over x is not connected. To see this let y
denote a point in the fiber over x. Then we may find g ∈ G and x̃ ∈ V
such that
y = [g, x̃].
By (39), x̃ = (bẇ′, b′)hJ for some b ∈ B, b
′ ∈ P∆\J and w
′ ≥ w. Then
(gbẇ′, gb′)hJ = (v̇, 1)hJ .
It follows that (v̇−1gbẇ′, gb′) lies in the stabilizer of hJ . In particular,
gb′ ∈ P∆\J and thus also g ∈ P∆\J . If g ∈ B then y = [1, x]. So assume
that g = u1(t)ṡ1 where u1 is the root homomorphism associated to α1.
Assume that t 6= 0. Then we may find b1 ∈ B and s ∈ k such that
g = u−1(s)b1 where u−1 is the root homomorphism associated to −α1.
x̃ = (g−1, g−1)(v̇, 1)hJ
= (b−11 u−1(−s)v̇, g
−1)hJ
= (b−11 v̇, g
−1)hJ
∈ (Bv̇, Bṡ1)hJ
where the third equality follows as v̇−1u−1(−s)v̇ is contained in the
unipotent radical of P−
. But (Bv̇, Bṡ1)hJ has empty intersection
with V (by (39)) which contradicts the assumption that t 6= 0. It
follows that the only possibilities for y are [1, x] and [ṡ1, (ṡ
1 v̇, ṡ
1 )hJ ].
As (ṡ−11 v̇, ṡ
1 ) is contained in V (by (39)) we conclude that the fiber
of f over x consists of 2 points; in particular the fiber is not connected
and thus XJ,w is not normal.
Remark 11.1. It seems likely that normalizations of G-Schubert vari-
eties should have nice singularities : If we let ZJ,w denote the normal-
ization of the closure of XJ,w, then the map (38) induces a birational
and projective morphism
f̃ : diag(G)×diag(PK) V → ZJ,w.
We expect that f̃ can be used to obtain global F -regularity of ZJ,w
(see [S] for an introduction to global F -regularity). In fact, by the
results in [H-T2] the B×B-orbit closure V is globally F -regular. Thus
diag(G)×diag(PK) V is locally strongly F -regular, and as
f̃∗Odiag(G)×diag(PK )V
= OZJ,w ,
it seems likely that ZJ,w is also locally strongly F -regular. Moreover,
similarly to Corollary 9.10 one may conclude that ZJ,w admits a stable
Frobenius splitting along an ample divisor. Thus ZJ,w is globally F -
regular if it is locally strongly F -regular. At the moment we do not
know if ZJ,w is locally strongly F -regular.
12. Generalizations
Fix notation as in Section 2. An admissible triple of G × G is by
definition a triple C = (J1, J2, θδ) consisting of J1, J2 ⊂ ∆, a bijection
δ : J1 → J2 and an isomorphism θδ : LJ1 → LJ2 that maps T to T
and the root subgroup Uαi to the root subgroup Uαδ(i) for i ∈ J1. To
each admissible triple C = (J1, J2, θδ), we associate the subgroup RC of
G×G defined by
RC = {(p, q) : p ∈ PJ1, q ∈ PJ2, θδ(πJ1(p)) = πJ2(q)},
where πJ : PJ → LJ , for a subset J ⊂ ∆, denotes the natural quotient
Let X denote an equivariant embedding of the reductive group G.
A RC-Schubert variety of X is then a subset of the form RC · V for
some B × B-orbit closure V in X . When G = Gad is a group of
adjoint type and X = X is the associated wonderful compactification
the set of RC-Schubert varieties coincides with closures of the set of RC-
stable pieces. By definition [L-Y, section 7], a RC-stable piece in the
wonderful compactification X of Gad is a subvariety of the form RC ·Y ,
where Y = (Bv1, Bv2) · hJ for some J ⊂ ∆, v1 ∈ W
J and v2 ∈
(notation as in Section 8.5). Notice that when J1 = J2 = ∆ and θδ
is the identity map then a RC-stable piece is the same as a G-stable
piece. On the other hand, when J1 = J2 = ∅, then a RC-stable piece
is the same as a B × B-orbit. Moreover, any RC-Schubert variety is a
finite union of RC-stable pieces [L-Y, Section 7].
The following is a generalization of Proposition 9.3 and Proposition
Proposition 12.1. Let C = (J1, J2, θδ) denote an admissible triple of
G×G and let X denote an equivariant embedding of G. Then X admits
a Frobenius splitting which compatible splits all RC-Schubert varieties
in X. If, moreover, X is a smooth, projective and toroidal embedding
and Y = XK = (G × G) · V , for some B × B-orbit closure V in X,
then X admits a Frobenius splitting along the Cartier divisor
D = (p− 1)
(wJ10 , 1)D̃i +
j /∈K
which is compatibly with Y and RC · V .
Proof. As the proof is similar to the proof of Proposition 9.3 and Propo-
sition 9.9 we only sketch the proof. In the following GJ , for a subset
J ⊂ ∆, denotes the commutator of the Levi subgroup in Gsc associated
to J . The Borel subgroup GJ ∩Bsc of GJ is denoted by BJ . Define XC
to be the G2J1-variety which as a variety is X but where the action is
twisted by the morphism
GJ1 ×GJ1
−−−→ GJ1 ×GJ2.
42 XUHUA HE AND JESPER FUNCH THOMSEN
Then the BJ1 × BJ2-canonical Frobenius splitting of X defined by
Theorem 9.1 and Lemma 6.3 induces a B2J1-canonical Frobenius split-
ting of XC. In particular, all subvarieties of XC which corresponds
to B × B-orbit closures in X will be compatibly Frobenius split by
this canonical Frobenius splitting. Now apply an argument as in the
proof of Proposition 9.3 and use the identification of RC · V ⊂ X with
diag(GJ1) · V ⊂ XC. This ends the proof of the first statement.
Assume now that X is a smooth, projective and toroidal embedding
and consider the B2sc-equivariant morphism
η : M → End!F (X, Y, V )⊗ k(1−p)ρ⊠(1−p)ρ,
defined in (27). Let YC and VC be defined similar to XC. Then η induces
a B2J1-equivariant morphism
ηC : M → End
F (XC, YC, VC)⊗ k(1−p)ρJ1⊠(1−p)ρJ1 .
Similar to the definition of v in (28) we obtain from ηC an element
vC ∈ Ind
EndF (XC, YC, VC)⊗ k(1−p)ρJ1⊠(1−p)ρJ1
and from this a G2J1-equivariant morphism
(40) End
(GJ1/BJ1)
⊗M(XC) → EndF
G2J1 ×B2J1
similar to (29). Here LJ1 is the line bundle on
GJ1/BJ1 associated to the
character (1 − p)ρJ1. Combining Lemma 6.3 and Lemma 9.6 we also
obtain a map
(41) StJ1 ⊠ StJ1 → M(XC),
with properties similar to the ones described in Lemma 9.6. As in (32)
we may also use vC to construct a morphism
M(XC) → StJ1 ⊠ StJ1 ,
such that the composition with (41) is an isomorphism on StJ1 ⊠ StJ1 .
Finally we may construct
ΘC : End
(GJ1/BJ1)
⊗ (StJ1 ⊠ StJ1) → EndF
G2J1 ×B2J1
similar to (31). In particular, a statement equivalent to Proposition
9.8 is satisfied for ΘC. Let v
+ (resp. v
− ) denote a highest (resp.
lowest) weight vector in StJ1 and let v
∆ denote the diag(GJ1)-invariant
element of End
(GJ1/BJ1)
. Imitating the proof of Proposition
9.9 we then find that ΘC(v
∆ ⊗ (v
+ ⊗ v
− )) is a Frobenius splitting of
G2J1×B2J1
XC (up to a nonzero constant). Moreover, the push forward of
this Frobenius splitting to X has the desired properties. We only have
to note that the effective Cartier associated to the image of vJ1+ ⊗ v
under the map (41) equals
D = (p− 1)
(wJ10 , 1)D̃i +
j /∈K
This ends the proof. �
We may also argue as in Corollary 10.5 to obtain
Corollary 12.2. Let X denote a projective embedding of a reductive
group G and let V denote the closure of a B × B-orbit in X. Let
Y = (G×G) · V and XC = RC · V . When L is a nef line bundle on XC
Hi(XC,L) = 0, i > 0.
Moreover, when L is a nef line bundle on Y then the restriction mor-
phism
H0(Y,L) → H0(XC,L),
is surjective.
Remark 12.3. In the case where k = C and X is the wonderful com-
pactification, the subvarieties (wJ10 , 1)D̃i, Xj and all the RC-Schubert
varieties are Poisson subvarieties with respect to the Poisson structure
on X corresponding to the splitting
Lie(G)⊕ Lie(G) = l1 ⊕ l2,
where l1 = Lie(RC) and l2 is a certain subalgebra of Ad(w
0 )Lie(B
Lie(B−). See [L-Y2, 4.5].
References
[Bri] M. Brion, Multiplicity-free subvarieties of flag varieties, Contemp. Math.
331 (2003), 13–23.
[B-K] M. Brion and S. Kumar, Frobenius Splittings Methods in Geometry and
Representation Theory, Progress in Mathematics (2004), Birkhäuser,
Boston.
[B-T] M. Brion and J. F. Thomsen, F -regularity of large Schubert varieties,
Amer. J. Math. 128 (2006), 949–962.
[E-L] S. Evens and J.-H. Lu, On the variety of Lagrangian subalgebras, I, II,
Ann. Sci. cole Norm. Sup. (4) 34 (2001), no. 5, 631–668; 39 (2006), no. 2,
347–379.
[Ful] W. Fulton, Introduction to Toric Varieties, Ann. Math. Studies, 131
(1993), Princeton University Press.
[Har] R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in
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[Har2] R. Hartshorne, Algebraic Geometry, GTM 52 (1977), Springer-Verlag.
[He] X. He, Unipotent variety in the group compactification, Adv. in Math. 203
(2006), 109-131.
[He2] X. He, The G-stable pieces of the wonderful compactification, Trans. Amer.
Math. Soc. 359 (2007), 3005-3024.
44 XUHUA HE AND JESPER FUNCH THOMSEN
[H-T] X. He and J. F. Thomsen, On the closure of Steinberg fibers in the won-
derful compactification, Transformation Groups, 11 (2006), no. 3, 427-438.
[H-T2] X. He and J.F.Thomsen, Geometry of B×B-orbit closures in equivariant
embeddings, math.RT/0510088.
[Laz] R. Lazarsfeld, Positivity in Algebraic Geometry I, classical setting: line
bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzge-
biete. 3. Folge (2004), Springer-Verlag, Berlin.
[L] G. Lusztig, Parabolic character sheaves, I, II, Mosc. Math. J. 4 (2004),
no. 1, 153–179; no. 4, 869–896.
[L-Y] J.-H Lu and M. Yakimov, Partitions of the wonderful group compactifica-
tion, math.RT/0606579.
[L-Y2] J.-H Lu and M. Yakimov, Group orbits and regular partitions of Poisson
manifolds, math.SG/0609732.
[M-R] V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology van-
ishing for Schubert varieties, Ann. of Math. 122 (1985), 27–40.
[R] A. Ramanathan, Equations defining Schubert varieties and Frobenius split-
ting of diagonals, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 61–90.
[R-R] S. Ramanan and A, Ramanathan, Projective normality of flag varieties
and Schubert varieties, Invent. Math. 79 (1985), 217–224.
[S] K. E. Smith, Globally F -regular varieties: Applications to vanishing the-
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[Sp] T. A. Springer, Intersection cohomology of B×B-orbits closures in group
compactifications, J. Alg. 258 (2002), 71–111.
[T] J. F. Thomsen, Frobenius splitting of equivariant closures of regular con-
jugacy classes Proc. London Math. Soc. 93 (2006), 570–592.
Department of Mathematics, Stony Brook University, Stony Brook,
NY 11794, USA
E-mail address : hugo@math.sunysb.edu
Institut for matematiske fag, Aarhus Universitet, 8000 Århus C,
Denmark
E-mail address : funch@imf.au.dk

  Let $X$ be an equivariant embedding of a connected reductive group $G$ over
an algebraically closed field $k$ of positive characteristic. Let $B$ denote a
Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the
form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In
the case where $X$ is the wonderful compactification of a group of adjoint
type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable
pieces. We prove that $X$ admits a Frobenius splitting which is compatible with
all $G$-Schubert varieties. Moreover, when $X$ is smooth, projective and
toroidal, then any $G$-Schubert variety in $X$ admits a stable Frobenius
splitting along an ample divisors. Although this indicates that $G$-Schubert
varieties have nice singularities we present an example of a non-normal
$G$-Schubert variety in the wonderful compactification of a group of type
$G_2$. Finally we also extend the Frobenius splitting results to the more
general class of $\mathcal R$-Schubert varieties.

Introduction
Let G denote a connected and reductive group over an algebraically
closed field k, and let B denote a Borel subgroup of G. An equi-
variant embedding X of G is a G × G-variety which contains G =
(G × G)/diag(G) as an open G × G-invariant subset, where diag(G)
is the diagonal image of G in G × G. Any equivariant embedding X
of G contains finitely many B × B-orbits. In recent years the geom-
etry of closures of B × B-orbits has been studied by several authors.
The most general result was obtained in [H-T2] where it was proved
that B × B-orbit closures are normal, Cohen-Macaulay and have (F -
)rational singularities (actually, even stronger results were obtained).
In the present paper we will study (closed) subvarieties inX of the form
diag(G) ·V, where V denotes the closure of a B×B-orbit. Subvarieties
of equivariant embeddings of G of this form will be called G-Schubert
varieties.
When G is a semisimple group of adjoint type there exists a canonical
equivariant embedding X of G which is called the wonderful compact-
ification. The wonderful compactifications are of primary interest in
this paper. Actually, this work arose from the question of describing
the closures of the so-called G-stable pieces of X. The G-stable pieces
makes up a decomposition of X into locally closed subsets. They were
introduced by Lusztig in [L] where they were used to construct and
http://arxiv.org/abs/0704.0778v2
2 XUHUA HE AND JESPER FUNCH THOMSEN
study a class of perverse sheaves which generalizes his theory of charac-
ter sheaves on reductive groups. More precisely, these perverse sheaves
are the intermediate extensions of the so-called “character sheaves”
on a G-stable piece. This motivates the study of closures of G-stable
pieces which turns out to coincide with the set of G-Schubert varieties.
Before discussing the closures of G-stable pieces in details, let us
make a short digression and discuss some other motivations for study-
ing G-stable pieces and G-Schubert varieties (in wonderful compactifi-
cations):
(1) When G is a simple group, the boundary of the closure of the
unipotent subvariety of G in the wonderful compactification X,
is a union of certain G-Schubert varieties (see [He] and [H-T]).
Thus knowing the geometry of these G-Schubert varieties will
help us to understand the geometry of the closure of the unipo-
tent variety within X.
(2) Let Lie(G) denote the Lie algebra of a simple group G over a
field of characteristic zero. Let ≪,≫ denote a fixed symmet-
ric non-degenerate ad-invariant bilinear form. Let <,> be the
bilinear form on Lie(G)⊕ Lie(G) defined by
< (x, y), (x′, y′) >=≪ x, x′ ≫ − ≪ y, y′ ≫ .
In [E-L], Evens and Lu showed that each splitting Lie(G) ⊕
Lie(G) = l ⊕ l′, where l and l′ are Lagrangian subalgebras of
Lie(G)⊕ Lie(G), gives rise to a Poisson structure Πl,l′ on X. If
moreover, one starts with the Belavin-Drinfeld splitting, then
all the G-stable pieces/G-Schubert varieties and B×B−-orbits
of X are Poisson subvarieties, where B− is a Borel subgroup
opposite to B. Thus to understand the Poisson structure on
X corresponding to the Belavin-Drinfeld splitting, one needs
to understand the geometry of the G-stable pieces/G-Schubert
varieties. If we start with another splitting, then we obtain
a different Poisson structure on X and in order to understand
these Poisson structures, one needs to study the R-stable pieces
[L-Y] instead (see Section 12), which generalize both the G-
stable pieces and the B × B−-orbits.
The main technical ingredient in this paper is the positive character-
istic notion of Frobenius splitting. Frobenius splitting is a powerful tool
which has been proved to be very useful in obtaining strong geometric
conclusions for e.g. Schubert varieties and closures of B × B-orbits in
equivariant embeddings. In the present paper we obtain two types of
results related to G-Schubert varieties over fields of positive character-
istic. First of all, if we fix an equivariant embedding X of a reductive
group G then we prove that all G-Schubert varieties in X are simul-
taneously compatibly Frobenius split by a Frobenius splitting of X .
Secondly, concentrating on a single G-Schubert variety X, in a smooth
projective and toroidal embedding X , we prove that this admits a
stable Frobenius splitting along an ample divisor. Statements of this
form put strong conditions on the intertwined behavior of cohomology
groups of line bundles on X and its G-Schubert varieties. As this is re-
lated to geometric properties it therefore seems natural to expect that
G-Schubert varieties should have nice singularities. It therefore comes
as a complete surprise that G-Schubert varieties, in general, are not
even normal. We only provide a single example of this phenomenon
(in the wonderful compactification of a group of type G2), but expect
that this absence of normality is the general picture.
In obtaining the Frobenius splitting result mentioned above, we have
developed some general theory of how to construct Frobenius splitting
of varieties of the form G×PX (see Section 4.2 for the definition). This
part of the paper is influenced by the theory of B-canonical Frobenius
splitting as discussed in [B-K, Chap.4]; in particular the proof of [B-K,
Prop.4.1.17]. The presentation we provide is more general and makes
it possible to extract even better result from the ideas of B-canonical
Frobenius splittings. This theory is presented in Chapter 5 in a general-
ity which is more than necessary for obtaining the described Frobenius
splitting results for G-Schubert varieties. However, we hope that this
theory could be useful elsewhere and we certainly consider it to be of
independent interest.
This paper is organized in the following way. In Section 2 we intro-
duce notation, and in Section 3 we briefly define Frobenius splitting
and explain its fundamental ideas. Section 4 is devoted to some results
on linearized sheaves which should all be well known. In Section 5 we
study the Frobenius splitting of varieties of the form G ×P X for a
variety X with an action by a parabolic subgroup P . The main idea
is to decompose the Frobenius morphism on G×P X into maps associ-
ated to the Frobenius morphism on the base G/P and the fiber X of the
natural morphism G×P X → G/P . In Section 6 we relate B-canonical
Frobenius splittings to the material in Section 5. Section 7 contains
applications of Section 5 to general G × G-varieties. In section 8 we
define the G-stable pieces and G-Schubert varieties. In Section 9 we
apply the material of the previous sections to the class of equivariant
embeddings and obtain Frobenius splitting results for G-Schubert vari-
eties. Section 10 contains results related to cohomology of line bundles
on G-Schubert varieties. Section 11 contains an example of a non-
normal G-Schubert variety. Finally Section 12 contains generalizations
and variations of the previous sections.
We would like to thank the referee for a careful reading of this paper
and for numerous suggestions concerning the presentation.
4 XUHUA HE AND JESPER FUNCH THOMSEN
2. Notation
We will work over a fixed algebraically closed field k. The charac-
teristic of k will depend on the application. By a variety we mean a
reduced and separated scheme of finite type over k. In particular, we
allow a variety to have several irreducible components.
2.1. Group setup. We letG denote a connected linear algebraic group
over k. We fix a Borel subgroup B and a maximal torus T ⊂ B. The
notation P is used for a parabolic subgroup of G containing B. The
set of T -characters is denoted by X∗(T ) and we identify this set with
the set X∗(B) of B-characters.
2.2. Reductive case. In many cases we will specialize to the case
where G is reductive. In this case we will also use the following no-
tation : the set of roots determined by T is denoted by R ⊆ X∗(T )
while the set of positive roots determined by (B, T ) is denoted by R+.
The simple roots are denoted by α1, . . . , αl, and we let ∆ = {1, . . . , l}
denote the associated index set. The simple reflection associated to the
simple root αi is then denoted by si. The Weyl group W = NG(T )/T is
generated by the simple reflections si, for i ∈ ∆. The length of w ∈ W
will be denoted by l(w). For J ⊂ ∆, let WJ denote the subgroup of
W generated by the simple reflection associated with the elements in
J , and let W J (resp. JW ) denote the set of minimal length coset rep-
resentatives for W/WJ (resp. WJ\W ). The element in W of maximal
length will be denoted by w0, while w
0 is used for the same kind of
element in WJ . For any w ∈ W , we let ẇ denote a representative of w
in NG(T ). For J ⊂ ∆, let PJ ⊃ B denote the corresponding standard
parabolic subgroup and P−J ⊃ B
− denote its opposite parabolic. Let
LJ = PJ ∩P
J be the common Levi subgroup of PJ and P
J containing
T . Let UJ (resp. U
J ) denote the unipotent radical of PJ (resp. P
When J = ∅ we also use the notation U and U− for UJ and U
J respec-
tively. When G is semisimple and simply connected we may associate
a fundamental character ωi to each simple root αi. The sum of the
fundamental characters is then denoted by ρ. Then ρ also equals half
the sum of the positive roots.
3. The relative Frobenius morphism
In this section we collect some results related to the Frobenius mor-
phism and to the concept of Frobenius splitting. Compared to other
presentations on the same subject, this presentation differs only in its
emphasis on the set HomOX′
(FX)∗OX ,OX′
(to be defined below) and
not just the set of Frobenius splittings. Thus, the obtained results are
only small variations of already known results as can be found in e.g.
[B-K].
3.1. The Frobenius morphism. By definition a variety X comes
with an associated morphism
pX : X → Spec(k),
of schemes. Assume that the field k has positive characteristic p > 0.
Then the Frobenius morphism on Spec(k) is the morphism of schemes
Fk : Spec(k) → Spec(k),
which on the level of coordinate rings is defined by a 7→ ap. As k
is assumed to be algebraically closed the morphism Fk is actually an
isomorphism and we let F−1k denote the inverse morphism. Composing
pX with F
k we obtain a new variety
p′X : X → Spec(k),
with underlying scheme X . In the following we suppress the morphism
pX from the notation and simply use X as the notation for the variety
defined by pX . The variety defined by p
X is then denoted by X
The relative Frobenius morphism on X is then the morphism of
varieties :
FX : X → X
which as a morphism of schemes is the identity map on the level of
points and where the associated map of sheaves
X : OX′ → (FX)∗OX ,
is the p-th power map. A key property of the Frobenius morphism is
the relation
(1) (FX)
′ ≃ Lp
which is satisfied for every line bundle L on X (here L′ denotes the
corresponding line bundle on X ′).
3.2. Frobenius splitting. A variety X is said to be Frobenius split if
the OX′-linear map of sheaves :
X : OX′ → (FX)∗OX ,
has a section; i.e. if there exists an element
s ∈ HomOX′
(FX)∗OX ,OX′
such that the composition s ◦F
X is the identity endomorphism of OX′ .
The section s will be called a Frobenius splitting of X .
6 XUHUA HE AND JESPER FUNCH THOMSEN
3.3. Compatibility with line bundles and closed subvarieties.
Fix a line bundle L on X and a closed subvariety Y in X with sheaf
of ideals IY . Let Y
′ denote the closed subvariety of X ′ associated to Y
with sheaf of ideals denoted by IY ′ . The kernel of the natural morphism
HomOX′
(FX)∗L,OX′
→ HomOX′
(FX)∗(L⊗ IY ),OY ′
induced by the inclusion L ⊗ IY ⊂ L and the projection OX′ → OY ′ ,
will be denoted by EndLF (X, Y ). The associated space of global sections
will be denoted by EndLF (X, Y ). When Y = X we simply denote
EndLF (X, Y ) (resp. End
F (X, Y )) by End
F (X) (resp. End
F (X)). The
sheaf EndLF (X, Y ) is a subsheaf of End
F (X) consisting of the elements
compatible with Y . Moreover, there is a natural morphism
EndLF (X, Y )|Y → End
F (Y ),
where the notation |Y means restriction to Y .
If Y1, Y2, . . . , Ym is a collection of closed subvarieties of X then the
notation EndLF (X, Y1, . . . , Ym) (or sometimes End
F (X, {Yi}
i=1)) will de-
note the intersection of the subsheaves EndLF (X, Yi) for i = 1, . . . , m.
The set of global sections of the sheaf EndLF (X, Y1, . . . , Ym) will be de-
noted by EndLF (X, Y1, . . . , Ym).
When L = OX we remove L from all of the above notation. In
particular, the vectorspace EndF (X) denotes the set of morphisms from
(FX)∗OX to OX′ and thus contains the set of Frobenius splittings of X .
A Frobenius splitting s of X contained in EndF (X, {Yi}i) is said to be
compatible with the subvarieties Y1, . . . , Ym. When s is compatible in
this sense it induces a Frobenius splitting of each Yi for i = 1 . . . , m.
In this case we also say that s compatibly Frobenius splits Y1, . . . , Ym.
In concrete terms, this is equivalent to
(FX)∗IYi
⊂ IY ′i .
for all i.
Lemma 3.1. Let Y and Z denote closed subvarieties in X and let s
denote a global section of EndLF (X,Z, Y ).
(1) s ∈ EndLF (X, Y1) for every irreducible component Y1 of Y .
(2) If the scheme theoretic intersection Z ∩ Y is reduced then s is
contained in EndLF (X, Y ∩ Z).
Proof. Let Y1 denote an irreducible component of Y and let
J = s
(FX)∗(IY1 ⊗ L)
⊂ OX′ .
Let U denote the open complement (in X ′) of the irreducible compo-
nents of Y ′ which are different from Y ′1 . Then IY ′1 coincides with IY ′
on U and consequently J|U ⊂ (IY ′)|U as s is compatible with Y . In
particular, J|U ⊂ (IY ′1 )|U . We claim that this implies that J ⊂ IY ′1 :
let V denote an open subset of X ′ and let f be a section of J over
V . As J is a subsheaf of OX′ , we may consider f as a function on V ,
and it suffices to prove that f vanishes on Y ′1 ∩ V . If Y
1 ∩ V is empty
then this is clear. Otherwise, U ∩ V ∩ Y ′1 is a dense subset of Y
1 and it
suffices to prove that f vanishes on this set. But this follows from the
inclusion J|U ⊂ (IY ′1 )|U . As a consequence s is compatible with Y1. The
second claim follows as the sheaf of ideals of the intersection Z ∩ Y is
IY + IZ . �
The condition that Z ∩ Y is reduced, in Lemma 3.1, only ensures
that Z ∩ Y is a variety. When L = OX and s is a Frobenius splitting
this is always satisfied [B-K, Prop.1.2.1].
3.4. The evaluation map. Let k[X ′] denote the space of global reg-
ular functions on X ′. Evaluating an element s : (FX)∗OX → OX′ of
EndF (X) at the constant global function 1 on X defines an element in
k[X ′] which we denote by evX(s). This defines a morphism
evX : EndF (X) → k[X
with the property that evX(s) = 1 if and only if s is a Frobenius
splitting of X .
3.5. Frobenius D-splittings. Consider an effective Cartier divisor D
on X , and let σD denote the associated global section of the associated
line bundle OX(D). A Frobenius splitting s of X is said to be a Frobe-
nius D-splitting if s factorizes as
s : (FX)∗OX
(FX)∗σD
−−−−−→ (FX)∗OX(D)
−→ OX′ ,
for some element sD in End
OX(D)
. We furthermore say that the
Frobenius D-splitting s is compatible with a subvariety Y if sD is com-
patible with Y . The following result assures that, in this case, the
compatibility with closed subvarieties agrees with the usual definition
[R, Defn.1.2].
Lemma 3.2. Assume that s defines a Frobenius D-splitting of X.
Then sD is compatible with Y if and only if (i) s compatibly Frobe-
nius splits Y and (ii) the support of D does not contain any irreducible
components of Y .
Proof. The if part of the statement follows from [R, Prop.1.4]. So
assume that sD is compatible with Y . Then sD induces a morphism
sD : (FY )∗OX(D)|Y → OY ′ ,
satisfying sD((σD)|Y ) is the constant function 1 on Y
′. As a conse-
quence (σD)|Y does not vanish on any of the irreducible components
of Y . This proves part (ii) of the statement. Part (i) is clearly satis-
fied. �
8 XUHUA HE AND JESPER FUNCH THOMSEN
It follows that if s is compatible with Y and, moreover, defines a
Frobenius D-splitting of X then D ∩ Y makes sense as an effective
Cartier divisor on Y and, in this case, s induces a Frobenius D ∩ Y -
splitting of Y .
3.6. Stable Frobenius splittings along divisors. Let X(0) = X
and define recursively X(n) = (X(n−1))′ for n ≥ 1. Composing the
Frobenius morphisms on X(i) for i = 0, . . . , n, we obtain a morphism
X : X → X
with an associated map of sheaves
♯ : OX(n) → (F
X )∗OX .
Let, as in Section 3.5, D denote an effective Cartier divisor on X with
associated canonical section σD of OX(D). We say that X admits a
stable Frobenius splitting along D if there exists a positive integer n
and an element
s ∈ HomO
X )∗OX(D),OX(n)
such that the composed map
OX(n)
−−−−→ (F
X )∗OX
−−−−−−→ (F
X )∗OX(D)
−→ OX(n) ,
is the identity map on OX(n) . The element s is called a stable Frobenius
splitting of X along D. When Y is a closed subvariety of X we say
that the stable Frobenius splitting s is compatible with Y if
X )∗(IY ⊗ OX(D))
⊂ IY (n).
Notice that this condition necessarily implies that the support of D
does not contain any of the irreducible components of Y (cf. proof
of Lemma 3.2). Notice also that if X admits a Frobenius D-splitting
which is compatible with Y then X admits a stable Frobenius splitting
along D which is compatible with Y . The following is well known (see
e.g. [T, Lem.4.4])
Lemma 3.3. Let D1 and D2 denote effective Cartier divisors on X and
let Y denote a closed subvariety of X. Then X admits stable Frobenius
splittings along D1 and D2 which are compatible with Y if and only if
X admits a stable Frobenius splitting along D1+D2 which is compatible
with Y .
The following result explains one of the main applications of (stable)
Frobenius splitting. Remember that a line bundle L is nef if L⊗M is
ample whenever M is ample.
Proposition 3.4. Assume that X admits a stable Frobenius splitting
along an effective Cartier divisor D. Then there exists a positive integer
n such that for each line bundle L on X we have an inclusion of abelian
groups
Hi(X,L) ⊂ Hi(X,Lp
⊗ OX(D)).
In particular, if D is ample and L is nef, then Hi(X,L) = 0 for i > 0.
Moreover, if the stable Frobenius splitting of X is compatible with a
subvariety Y , D is ample and L is nef then the restriction morphism
H0(X,L) → H0(Y,L),
is surjective.
Proof. Argue as in the proof [R, Prop.1.13(i)]. �
3.7. Duality for FX. By duality (see [Har2, Ex.III.6.10]) for the finite
morphism FX we may to each quasi-coherent OX′-module F associate
an OX -module denoted by (FX)
!F and satisfying
(FX)∗(FX)
F = HomOX′
(FX)∗OX ,F
Actually, as FX is the identity on the level of points we may define
!F as the sheaf of abelian groups
HomOX′
(FX)∗OX ,F
with OX -module structure defined by
(g · φ)(f) = φ(gf),
for g, f ∈ OX and φ ∈ HomOX′
(FX)∗OX ,F
. When F = OX we
will also use the notation End!F (X) for (FX)
!OX . This sheaf is par-
ticularly nice when X is smooth as (FX)
!OX then coincides with the
line bundle ω
X , where ωX denotes the dualizing sheaf of X (see e.g.
[B-K, Sect.1.3]). If Y1, Y2, . . . , Ym is a collection of closed subvarieties
of X then End!F (X, Y1, . . . , Ym) (or End
F (X, {Yi}
i=1)) will denote the
subsheaf of End!F (X) consisting of the elements mapping the sheaf of
ideals IYi to IY ′i for all i = 1, . . . , m. We say that End
F (X, {Yi}
i=1) is
the subsheaf of elements compatible with Y1, . . . , Ym.
More generally, duality for FX implies that we have a natural iden-
tification
(FX)∗HomOX
G, (FX)
≃ HomOX′
(FX)∗G,F
whenever G (resp. F) is a quasicoherent sheaf on X (resp. X ′). This
leads to the identification
HomOX
G, (FX)
≃ HomOX′
(FX)∗G,F
where a morphism η : G → (FX)
!F is identified with the composed
morphism
η′ : (FX)∗G
(FX)∗η
−−−−→ (FX)∗(FX)
F ≃ HomOX′
(FX)∗OX ,F
Here the latter map is the natural evaluation map at the element 1
in OX . From now on we will specialize to the case where F = OX′
10 XUHUA HE AND JESPER FUNCH THOMSEN
and G equals a line bundle L on X . In this case, an element in
HomOX
L,End!F (X)
may also be considered as a global section of
the sheaf End!F (X)⊗ L
−1. For later use we emphasize
Lemma 3.5. Let η be an element in HomOX
L,End!F (X)
and let
η′ denote the corresponding element in HomOX′
(FX)∗L,OX′
by the
above identification. Then η′ factors through the morphism
(FX)∗L
(FX)∗η
−−−−→ (FX)∗End
F (X).
Moreover, the element η′ is compatible with a collection of closed sub-
varieties Y1, . . . , Ym of X if and only if the image of η is contained in
End!F (X, Y1, . . . , Ym).
Proof. The first part of the statement follows directly from the discus-
sion above. To prove the second statement we may assume that m = 1.
We use the notation Y = Y1. Let σ denote a section of L over an open
subset U of X , and consider s = η(σ) as a map
s : OX(U) → OX′(U
That s is compatible with Y means that s(f) vanishes on Y ′ whenever
f vanishes on Y for a function f on U . Alternatively, the evaluation
of f · s at 1, which coincides with η′(f · σ), should vanish on Y ′. In
particular, the image of η is contained in End!F (X, Y ) if and only if
the restriction of η′ to (FX)∗
IY ⊗ L
maps into IY ′ . This ends the
proof. �
We will also need the following remark
Lemma 3.6. Let D denote a reduced effective Cartier divisor on X
and L denote a line bundle on X. Let M = OX((p − 1)D) ⊗ L and
assume that we have an OX-linear morphism η : M → End
F (X). Let
σD denote the canonical section of OX(D) and consider the map
ηD : L → End
F (X),
induced by σ
D . Then the element
η′D ∈ HomOX′
(FX)∗L,OX′
induced by ηD, is compatible with the support of D. In particular, the
image of ηD is contained in End
F (X,D).
Proof. Notice that η′D is the composition
η′D : (FX)∗L
(FX)∗σ
−−−−−−→ (FX)∗M
−→ OX′ ,
where η′ is the element corresponding to η. Hence, the restriction of
η′D to L⊗ OX(−D) coincides with the map
(FX)∗
L⊗ OX(−D)
) (FX)∗σ
−−−−−→ (FX)∗M
−→ OX′ .
But the restriction of η′ to (cf. (1))
(FX)∗
OX(−pD)⊗M
≃ OX′(−D
′)⊗ (FX)∗M,
maps by linearity into OX′(−D
′). The in particular part follows by
Lemma 3.5. �
3.8. Push-forward operation. Assume that f : X → Z is a mor-
phism of varieties satisfying that the associated map f ♯ : OZ → f∗OX
is an isomorphism. Let f ′ : X ′ → Z ′ denote the associated morphism.
Then f ′∗ induces a morphism
f ′∗EndF (X) → EndF (Z).
If Y ⊂ X is a closed subset then the subsheaf f ′∗EndF (X, Y ) is mapped
to EndF (Z, f(Y )), where f(Y ) denotes the variety associated to the
closure of the image of Y . On the level of global sections this means
that every Frobenius splitting s of X induces a Frobenius splitting f ′∗s
of Z such that when s is compatible with Y then f ′∗s is compatible
with f(Y ). Likewise
Lemma 3.7. With notation as above, let L denote a line bundle on
Z and let s be an element of End
f∗(L)
. Then f ′∗s is an element of
EndLF
. Moreover, if s is compatible with a closed subvariety Y of
X then f ′∗s is compatible with f(Y ).
Proof. This follows easily from the fact that the sheaf of ideals of f(Y )
coincides with f∗IY [B-K, Lem.1.1.8]. �
4. Linearized sheaves
In this section we collect a number of well known facts about lin-
earized sheaves. The chosen presentation follows rather closely the
presentation in [Bri, Sect.2].
Let H denote a linear algebraic group over the field k and let X
denote a H-variety with H-action defined by σ : H × X → X . We
let p2 : H × X → X denote projection on the second coordinate. A
H-linearization of a quasi-coherent sheaf F on X is an OH×X -linear
isomorphism
φ : σ∗F → p∗2F,
satisfying the relation
(2) (µ× 1X)
∗φ = p∗23φ ◦ (1H × σ)
as morphisms of sheaves on H × H × X . Here µ : H × H → H
(resp. p23 : H × H × X → H × X) denotes the multiplication on
H (resp. the projection on the second and third coordinate). Based
on the fact that σ∗OX = p
2OX we see that the sheaf OX admits a
canonical linearization. In the following we will always assume that
OX is equipped with this canonical linearization.
12 XUHUA HE AND JESPER FUNCH THOMSEN
A morphism ψ : F → F′ of H-linearized sheaves is a morphism of
OX -modules commuting with the linearizations φ and φ
′ of F and F′,
i.e. φ′ ◦ σ∗(ψ) = p∗2(ψ) ◦ φ.
Linearized sheaves on X form an abelian category which we denote
by ShH(X).
4.1. Quotients and linearizations. Assume that the quotient q :
X → X/H exists and that q is a locally trivial principal H-bundle.
Then for G ∈ Sh(X/H), q∗G is naturally a H-linearized sheaf on X .
This defines a functor q∗ : Sh(X/H) → ShH(X). On the other hand,
for F ∈ ShH(X), q∗F has a natural action of H . Define a functor
qH∗ : ShH(X) → Sh(
X/H) by qH∗ (F) = (q∗F)
H the subsheaf of H-
invariants of q∗F. It is known that the functor q
∗ : Sh(X/H) → ShH(X)
is an equivalence of categories with inverse functor qH∗ .
In general, if H is a closed normal subgroup of G and X is a G-
variety such that the quotient X/H exists (as above), then X/H is a G/H-
variety and the functor q∗ : ShG/H(X/H) → ShG(X) is an equivalence
of categories with inverse functor qH∗ : ShG(X) → ShG/H(
X/H).
4.2. Induction equivalence. Consider now a connected linear alge-
braic group G and a parabolic subgroup P in G. Let X denote a
P -variety. Then G×X is a G× P -variety by the action
(g, p)(h, x) = (ghp−1, px),
for g, h ∈ G, p ∈ P and x ∈ X . Then the quotient, denoted by
G ×P X , of G × X by P exists and the associated quotient map q :
G×X → G×P X is a locally trivial principal P -bundle. The quotient
of G × X by G also exists and may be identified with the projection
p2 : G×X → X . In particular, we may apply the above consideration
to obtain equivalences of the categories ShP (X), ShG×P (G × X) and
ShG(G×P X). Notice that under this equivalence a P -linearized sheaf
F on X corresponds to the G-linearized sheaf IndGP (F) = (q∗p
P . In
particular, the space of global sections of IndGP (F) equals
IndGP (F)(G×P X) =
p∗2F(G×X)
k[G]⊗k F(X)
= IndGP (F(X)),
where the second equality follows by the Künneth formula. This also
explains the notation IndGP (F). Similarly, starting with a G-linearized
sheaf G on G×P X then the associated P -linearized line bundle on X
equals G′ = ((p2)∗q
∗G)G. However, by [Bri, Lemma 2(1)] the latter also
equals the simpler pull back i∗G by the P -equivariant map
i : X → G×P X,
sending x to q(1, x). In particular, we conclude that the functor i∗ :
ShG(G ×P X) → ShP (X) is an equivalence of categories with inverse
functor IndGP . Notice also that the space of global sections of G is
G-equivariantly isomorphic to
G(G×P X) = Ind
(i∗G)(X)
which follows by (3) above.
4.3. Duality. Assume that the field k has positive characteristic p > 0.
Regard X ′ as a H-variety in the canonical way and let F denote a H-
linearized sheaf on X ′. The sheaf (FX)
!F, defined in Section 3.7, is
then naturally a H-linearized sheaf on X . Moreover, the induced H-
linearization of (FX)∗(FX)
!F coincides with the natural H-linearization
HomOX′
(FX)∗OX ,F
When X is smooth the sheaf (FX)
!OX′ is canonically isomorphic to
X (cf. Section 3.7). We may use this isomorphism to define a H-
linearization of ω
X . Alternatively we may consider the natural H-
linearization of the dualizing sheaf ΩX of X and use this to define a
H-linearization of ω
X . It may be checked that the two stated ways
of defining a H-linearization of ω
X coincide.
5. Frobenius splitting of G×P X
Let G denote a connected linear algebraic group over an algebraically
closed field k of characteristic p > 0. Let P denote a parabolic subgroup
of G and let X denote a P -variety. In this section we want to consider
Frobenius splittings of the quotient Z = G×PX of G×X by P . We let
π : Z → G/P denote the morphism induced by the projection of G×X
on the first coordinate. When g ∈ G and x ∈ X we use the notation
[g, x] to denote the element in Z represented by (g, x).
5.1. Decomposing the Frobenius morphism. The Frobenius mor-
phism FZ admits a decomposition FZ = Fb ◦ Ff where Fb (resp. Ff )
is related to the Frobenius morphism on the base (resp. fiber) of π.
More precisely, define Ẑ and the morphisms π̂ and Fb as part of the
fiber product diagram
(4) Ẑ
Fb //
// (G/P)′
A local calculation shows that we may identify Ẑ with the quotient
G ×P X
′, where the P -action on the Frobenius twist X ′ of X is the
natural one. With this identification π̂ : G ×P X
′ → G/P is just the
14 XUHUA HE AND JESPER FUNCH THOMSEN
map [g, x′] 7→ gP . It also follows that the natural morphism (induced
by the Frobenius morphism on X)
Ff : G×P X → G×P X
makes the following diagram commutative
Fb //
// (G/P)′
5.2. Let M denote a P -linearized line bundle on X and let MZ =
IndGP (M) denote the associated G-linearized line bundle on Z. The
main aim of this section is to construct global sections of the sheaf
F (Z) = HomOZ′
(FZ)∗MZ ,OZ′
To this end we fix a P -character λ and let L denote the associated line
bundle on G/P (cf. Section 4). The pull back π̂∗L of L to Ẑ is then
denoted by LẐ . We then define the following sheaves
F (Z)f := HomOẐ
(Ff)∗MZ ,OẐ
F (Z)b := HomOZ′
(Fb)∗LẐ ,OZ′
with spaces of global sections denoted by End
F (Z)f and End
F (Z)b.
Notice that whenM is substituted with the P -linearized twistM(−λ) :=
M⊗ k−λ then
M(−λ)Z = MZ ⊗ π
∗(L−1) = MZ ⊗ (Ff)
and thus by the projection formula
(5) End
M(−λ)Z
F (Z)f = HomOẐ
(Ff)∗MZ ,LẐ
Sections of End
F (Z) are then constructed as compositions of global
sections of the sheaves End
M(−λ)Z
F (Z)f and End
F (Z)b. More precisely,
v ∈ HomO
(Ff)∗MZ ,LẐ
u ∈ HomOZ′
(Fb)∗LẐ ,OZ′
are global sections of the latter sheaves, then the composition u◦(Fb)∗v
defines a global section of End
F (Z).
5.3. An equivariant setup. We now give equivariant descriptions of
the sheaves End
F (Z)f and End
F (Z)b.
5.3.1. A description of End
F (Z)f . Now End
F (Z)f is a G-linearized
sheaf on Ẑ = G×P X
′. Let Y denote a P -stable subvariety of X and
let ZY = G ×P Y denote the associated subvariety of Z with sheaf of
ideals IZY ⊂ OZ . Let ẐY denote the subvariety G ×P Y
′ of Ẑ. Then
there is a natural morphism of G-linearized sheaves
F (Z)f → HomOẐ
(Ff )∗(MZ ⊗ IZY ),OẐY
induced by the inclusion IZY ⊂ OZ and the projection OẐ → OẐY . We
let End
F (Z,ZY )f denote the kernel of the above map and arrive at a
left exact sequence of G-linearized sheaves
0 → EndMZF (Z,ZY )f → End
F (Z)f → HomOẐ
(Ff)∗(MZ⊗IZY ),OẐY
In particular, the space of global sections of End
F (Z,ZY )f is identified
with the set of elements in End
F (Z)f which map (Ff )∗(MZ ⊗ IZY ) to
⊂ OẐ . Using the observations in Section 4.2 we can give another
description of the space of global sections of End
F (Z,ZY )f . Let i
X ′ → G×P X
′ denote the morphism i′(x′) = [1, x′]. Then the functor
i′ is exact on the category of G-linearized sheaves. We want to apply
this fact on the left exact sequence (6) above : notice first that
(i′)∗End
F (Z)f = HomOX′
(i′)∗(Ff )∗MZ ,OX′
where, moreover, (i′)∗(Ff)∗MZ = (FX)∗M. Thus (i
′)∗End
F (Z)f =
EndMF (X). Similarly,
(i′)∗HomO
(Ff)∗(MZ ⊗ IZY ),OẐY
= HomOX′ ((FX)∗(M⊗ IY ),OY ′).
In particular, we see that the P -linearized sheaf on X ′ corresponding to
the G-linearized sheaf End
F (Z,ZY )f equals the kernel of the natural
EndMF (X) → HomOX′ ((FX)∗(M⊗ IY ),OY ′),
i.e. it equals EndMF (X, Y ). By Section 4.2 the space of global sections
F (Z,ZY )f of End
F (Z,ZY )f is then G-equivariantly isomorphic
IndGP
EndMF (X, Y )).
Applying the above conclusions to the sheaf M(−λ) we find:
Proposition 5.1. There exists a G-equivariant isomorphism
M(−λ)Z
F (Z)f ≃ Ind
EndMF (X)⊗ kλ)
such that when Y is a closed P -stable subvariety of X then the subset of
elements of End
M(−λ)Z
F (Z)f which map (Ff)∗(MZ⊗IZY ) to (IẐY ⊗LẐ) ⊂
LẐ (cf. equation (5)) is identified with
M(−λ)Z
F (Z,ZY )f ≃ Ind
EndMF (X, Y )⊗ kλ).
16 XUHUA HE AND JESPER FUNCH THOMSEN
5.3.2. A description of End
F (Z)b. As π
′ in the fibre-diagram (4) is
flat the natural morphism (π′)∗(FG/P )∗L → (Fb)∗π̂
∗L is an isomor-
phism ([Har2, Prop.III.9.3]). Thus there is a natural isomorphism of
G-linearized sheaves
F (Z)b ≃ (π
′)∗HomO(G/P )′
(FG/P )∗L,O(G/P)′
= (π′)∗EndLF (
G/P).
Let V denote a closed subvariety of G/P . Then EndLF (
G/P , V ) is the
kernel of the natural map
EndLF (
G/P) → HomO(G/P )′
(FG/P )∗(IV ⊗ L),OOV ′
In particular, (π′)∗
EndLF (
G/P , V )
maps into the kernel of the induced
morphism
(7) End
F (Z)b → (π
′)∗HomO
(G/P )′
(FG/P )∗(IV ⊗ L),OOV ′
Let q : G→ G/P denote the quotient map. Then π̂−1(V ) identifies with
the quotient q−1(V )×P X
′. Moreover, as π′ is locally trivial it follows
that π̂∗(IV ) = Iq−1(V )×PX′. In particular, the sheaf
(π′)∗HomO(G/P )′
(FG/P )∗(IV ⊗ L),OOV ′
is isomorphic to
HomOZ′
(Fb)∗(Iq−1(V )×PX′ ⊗ LẐ),O(q−1(V )×PX)′
Thus we see that the kernel of (7) is the subsheaf End
F (Z, π
−1(V ))b
of elements which map (Fb)∗(Iq−1(V )×PX′ ⊗ LẐ) to I(q−1(V )×PX)′ . The
global sections of this subsheaf is denote by End
F (Z, π
−1(V ))b. In
conclusion
Proposition 5.2. The map π′ induces a G-equivariant morphism
(π′)∗ : EndLF (
G/P) → End
F (Z)b.
Moreover, when V is a closed subvariety of G/P then (π′)∗ maps the
subset EndLF (
G/P , V ) into End
F (Z, q
−1(V )×P X)b.
The following is also useful.
Lemma 5.3. Let Y denote a closed P -stable subvariety of X and fix
notation as above. Then each element of End
F (Z)b maps (Fb)∗(IẐY ⊗
LẐ) to I(ZY )′.
Proof. It suffices to show that the natural morphism
HomOZ′
(Fb)∗LẐ ,OZ′
→ HomOZ′
(Fb)∗(IẐY ⊗ LẐ),O(ZY )′
is zero. By linearity, this will follow if the natural morphism
I(ZY )′ ⊗ (Fb)∗LẐ → (Fb)∗(IẐY ⊗ LẐ),
is an isomorphism, which can be checked by a local calculation. �
5.4. Conclusions. By Proposition 5.1 an element v in the vectorspace
IndGP
EndMF (X) ⊗ kλ
defines an element in End
M(−λ)Z
F (Z)f . More-
over, by Proposition 5.2, an element u ∈ EndLF (
G/P) defines an element
(π′)∗(u) in End
F (Z)b. Thus by the discussion in Section 5.2 we obtain
a G-equivariant map
M,λ : End
G/P)⊗ IndGP
EndMF (X)⊗ kλ
→ End
F (G×P X),
defined by
M,λ(u⊗ v) = (π
′)∗u ◦ (Fb)∗v.
We can now prove
Theorem 5.4. Let X denote a P -variety and M denote a P -linearized
line bundle on X. Let L denote the equivariant line bundle on G/P
associated to the P -character λ. Then the G-equivariant map Φ1
defined above, satisfies
(1) When Y is a P -stable closed subvariety of X then the restriction
of Φ1
M,λ to the subspace :
EndLF (
G/P)⊗ IndGP
EndMF (X, Y )⊗ kλ
maps to End
F (G×P X,G×P Y ).
(2) When V denotes a closed subvariety of G/P then the restriction
of Φ1
M,λ to the subspace
EndLF (
G/P , V )⊗ IndGP
EndMF (X)⊗ kλ
maps to End
F (G ×P X, q
−1(V ) ×P X), where q : G → G/P
denotes the quotient map.
Proof. The first statement follows from Proposition 5.1 and Lemma
5.3. The second statement follows from Proposition 5.2 and Lemma
5.5 below. �
Lemma 5.5. Let V denote a closed subset of G/P . Then every element
of End
M(−λ)Z
F (Z)f will map (Ff)∗(MZ ⊗ Iπ−1(V )) to I(π̂)−1(V ) ⊗ LẐ .
Proof. It suffices to prove that the natural morphism
I(π̂)−1(V ) ⊗ (Ff)∗MZ → (Ff)∗
Iπ−1(V ) ⊗MZ
is an isomorphism, which can be checked by a local calculation. �
5.5. Identify IndGP
with the space of global sections of MZ (cf.
Equation (3)). Then we can define a G-equivariant morphism
(8) End
F (G×P X)⊗ Ind
→ EndF (G×P X),
by mapping s⊗σ, for σ a global section of MZ and s : (FZ)∗MZ → OZ′ ,
to the element
(FZ)∗OZ
(FZ )∗σ
−−−−→ (FZ)∗MZ
−→ OZ′ ,
18 XUHUA HE AND JESPER FUNCH THOMSEN
in EndF (G×PX). Combining Φ
M,λ with the morphism in (8) we obtain
a G-equivariant map
ΦM,λ : End
G/P)⊗ IndGP
EndMF (X)⊗kλ
⊗ IndGP
→ EndF (Z),
where an element u⊗ v⊗ σ in the domain is mapped to the composed
(9) (FZ)∗OZ
(FZ)∗σ
−−−−→ (FZ)∗MZ
(Fb)∗v
−−−→ (Fb)∗LẐ
(π′)∗u
−−−→ OZ′.
Notice that by Lemma 3.5 the map u ∈ EndLF (
G/P) factors as
(10) (FG/P )∗L
(FG/P )∗u
−−−−−→ (FG/P )∗ω
→ O(G/P)′ ,
where u! is some global section of the line bundle Ľ := ω
⊗L−1 as-
sociated to u (cf. Section 3.7), and the rightmost map is the evaluation
map with domain (FG/P )∗ω
= EndF (G/P). It follows that we may
extend (9) into a commutative diagram
(FZ)∗OZ
(FZ)∗σ //
(Fb)∗π̂
∗(u!)
(FZ)∗MZ
(Fb)∗v //
(Fb)∗π̂
∗(u!)
(Fb)∗LẐ
(π′)∗u
(Fb)∗π̂
∗(u!)
(FZ)∗π
∗Ľ // (FZ)∗(MZ ⊗ π
∗Ľ) //// (Fb)∗(π̂
rrrrrrrrrrr
where all the vertical maps are induced by multiplication by π̂∗(u!).
Likewise the lower horizontal maps are induced from the upper hori-
zontal maps by multiplication with π̂∗(u!). The triangle on the right is
induced from (10) by pull-back to Z ′.
Theorem 5.6. Let X denote a P -variety and M denote a P -linearized
line bundle on X. Let L denote the equivariant line bundle on G/P
associated to the P -character λ. Then the G-equivariant map ΦM,λ,
defined above, satisfies
(1) When Y is a P -stable closed subvariety of X then the restriction
of ΦM,λ to the subspace :
EndLF (
G/P)⊗ IndGP
EndMF (X, Y )⊗ kλ
⊗ IndGP
maps to EndF (G×P X,G×P Y ).
(2) When V denotes a closed subvariety of G/P then the restriction
of ΦM,λ to the subspace :
EndLF (
G/P , V )⊗ IndGP
EndMF (X)⊗ kλ
⊗ IndGP
maps to EndF (G ×P X, q
−1(V ) ×P X), where q : G → G/P
denotes the quotient map.
Moreover, let u ∈ EndLF (
G/P), v ∈ IndGP
EndMF (X) ⊗ kλ
and σ ∈
IndGP
. Then the element ΦM,λ(u⊗ v ⊗ σ) factorizes both as
(FZ)∗OZ
(FZ)∗σ
−−−−→ (FZ)∗MZ
−→ OZ′,
and as
(FZ)∗OZ
(FZ)∗(σ⊗π
−−−−−−−−→ (FZ)∗(MZ ⊗ π
−→ OZ′,
where s1 and s2 satisfies
i) If v is contained in IndGP
EndMF (X, Y ) ⊗ kλ
then s1 and s2 are
compatible with G×P Y .
ii) If u is contained in EndLF (
G/P , V ) then s1 is compatible with q
−1(V )×P
Proof. Part (1) and (2) follows directly from Theorem 5.4 and the defi-
nition of ΦM,λ. The existence of s1 and s2 follows by the diagram (11).
Finally the claims about the compatibility of s1 and s2 follows from
Theorem 5.4 and Lemma 5.3. �
5.6. We will now describe when an element in the image of ΦM,λ defines
a Frobenius splitting of Z. For this we consider the composed map
evZ◦ΦM,λ. Recall that an element s ∈ EndF (Z) is a Frobenius splitting
of Z if and only if evZ(s) is the constant function 1 on Z
Let u ∈ EndLF (
G/P), v ∈ IndGP
EndMF (X)⊗kλ
and σ ∈ IndGP
By Equation (9) the image of u⊗v⊗σ under evZ ◦ΦM,λ coincides with
the global section of OZ′ determined by the composed map
(12) OZ′
−→ (FZ)∗OZ
(FZ )∗σ
−−−−→ (FZ)∗MZ
(Fb)∗v
−−−→ (Fb)∗LẐ
(π′)∗u
−−−→ OZ′ .
We may divide this composition into two parts. The first part
−→ (FZ)∗OZ
(FZ)∗σ
−−−−→ (FZ)∗MZ
(Fb)∗v
−−−→ (Fb)∗LẐ
is defined by σ and v and defines a global section of LẐ . The corre-
sponding map
M,λ : Ind
EndMF (X)⊗ kλ
⊗ IndGP
→ IndGP
k[X ′]⊗ kλ
is the map induced by the morphism
(13) EndMF (X)⊗M(X) → k[X
mapping s : (FX)∗M → OX′ and τ a global section of M, to s(τ).
Notice that we here identify IndGP
k[X ′]⊗ kλ
with the space of global
sections of LẐ (cf. Equation (3)). The second part takes a global
section τ̃ of LẐ and an element u in End
G/P) to the global section of
OZ′ defined by
−→ (Fb)∗OẐ
(Fb)∗τ̃
−−−→ (Fb)∗LẐ
(π′)∗u
−−−→ OZ′.
20 XUHUA HE AND JESPER FUNCH THOMSEN
The corresponding map is
Φλ : End
G/P)⊗ IndGP
k[X ′]⊗ kλ
→ k[Z ′],
which maps u⊗ τ̃ , to ((π′)∗u)(τ̃) (cf. Proposition 5.2). The restriction
of Φλ :
(14) φλ : End
G/P)⊗ IndGP
is the map corresponding to Φλ in case X is the one point space Spec(k)
(in which case k[X ′] is just k). In combination this defines us a com-
mutative diagram
EndLF (
G/P)⊗ IndGP
EndMF (X)⊗ kλ
⊗ IndGP
Id⊗Φ2
ΦM,λ // EndF (Z)
EndLF (
G/P)⊗ IndGP
k[X ′]⊗ kλ
) Φλ // k[Z ′]
EndLF (
G/P)⊗ IndGP
φλ // k
evG/P
33ggggggggggggggggggggggggggggggg
wheremλ is the natural map which makes the lower part of the diagram
commutative. Notice that when k[X ′] = k, e.g. if X ′ is a complete and
irreducible variety, then φλ and Φλ coincides. Let χ denote the P -
character associated to the canonical G-linearization of ω−1G/P (cf. Sec-
tion 4.3). Then as noted earlier (Section 5.5) the G-module EndLF (
coincides with the space of global sections of Ľ = ω
⊗L−1 and thus
coincides with
(16) EndLF (
G/P) = IndGP
(p− 1)χ− λ
where we abuse notation and write (p− 1)χ− λ for the 1-dimensional
P -representation associated with the character (p− 1)χ−λ. It follows
that mλ is the natural multiplication map
(17) mλ : Ind
(p− 1)χ− λ
⊗ IndGP
→ IndGP
(p− 1)χ
which is surjective if the domain is nonzero, i.e. if L and ω
⊗ L−1
are effective line bundles on G/P [R-R, Thm.3].
The commutativity of the diagram (15) then implies:
Proposition 5.7. Let Ξ denote an element in the domain of ΦM,λ,
and assume that the image (Id⊗Φ2
M,λ)(Ξ) is contained in the subspace
EndLF (
G/P)⊗IndGP
(cf. diagram (15)). Then ΦM,λ(Ξ) is a Frobenius
splitting of Z if and only if φλ((Id ⊗ Φ
M,λ)(Ξ)) equals the constant 1.
In particular, if EndLF (
G/P) ⊗ IndGP
is nonzero and IndGP
contained in the image of Φ2
M,λ, then Z admits a Frobenius splitting.
Proof. The first part of the proof is just a restatement of the fact that
the diagram (15) is commutative. The second part follows by the sur-
jectivity of mλ and the fact that G/P admits a Frobenius splitting.
Corollary 5.8. Assume that X is irreducible and complete. If both
IndGP
and IndGP
(p − 1)χ − λ
are nonzero and Φ2
M,λ is surjective,
then Z admits a Frobenius splitting.
5.7. In many concrete situation the existence of a P -invariant ele-
ment in EndMF (X) ⊗ kλ is given. Notice that this is equivalent to a
G-invariant element v in IndGP
EndMF (X) ⊗ kλ
and thus ΦM,λ defines
a G-equivariant map
(18) EndLF (
G/P)⊗ IndGP
→ EndF (Z),
u⊗ σ 7→ ΦM,λ(u⊗ v ⊗ σ).
Similarly Φ2
M,λ defines a G-equivariant morphism
(19) IndGP
→ IndGP
k[X ′]⊗ kλ
which makes the diagram
(20) EndLF (
G/P)⊗ IndGP
// EndF (Z)
EndLF (
G/P)⊗ IndGP
k[X ′]⊗ kλ
) Φλ // k[Z ′]
commutative. We also note
Corollary 5.9. Assume that X is irreducible and complete and let v
denote a P -invariant element of EndMF (X)⊗ kλ. If the induced map
M,λ)|v⊗IndGP (M(X))
: IndGP
→ IndGP
is surjective then Z admits a Frobenius splitting. In particular, if
IndGP
is an irreducible G-representation then for Z to be Frobenius
split it suffices that the latter map is nonzero.
Proof. Apply Corollary 5.8. �
6. B-Canonical Frobenius splittings
In this section we continue the study of the Frobenius splitting prop-
erties of Z = G ×P X . The notation is kept as in Section 5 but we
restrict ourselves to the case where G is a connected, semisimple and
simply connected linear algebraic group. Moreover, we fix P = B,
M = OX and λ = −(p − 1)ρ. Recall that, in this setup, the dualizing
sheaf ωG/B is the G-linearized sheaf associated to the B-character 2ρ.
22 XUHUA HE AND JESPER FUNCH THOMSEN
Thus, with the notation in Section 5.6, we have χ = −2ρ. Recall also
the G-equivariant identity (see (16))
(21) EndLF (
G/B) ≃ IndGB((p− 1)χ− λ) = Ind
B(λ) = Ind
B((1− p)ρ).
The latter G-module is called the Steinberg module of G and will be
denoted by St. The Steinberg module is a simple and selfdual G-
module. A B-canonical Frobenius splitting ofX is then a B-equivariant
(22) θ : St⊗ k(p−1)ρ → EndF (X),
containing a Frobenius splitting in its image. Notice that a B-canonical
Frobenius splitting of X is not a Frobenius splitting as defined in Sec-
tion 3.2. However, there exists a unique nonzero lowest weight vector
v− of St such that θ(v−) is a Frobenius splitting in the sense of Sec-
tion 3.2. Moreover, as St is a simple G-module the map θ is uniquely
determined by θ(v−), and we may thus identify θ with θ(v−). In this
way θ(v−) will also be called a B-canonical Frobenius splitting of X .
The importance of B-canonical Frobenius splittings was first ob-
served by O. Mathieu in connection with good filtrations of G-modules.
We refer to [B-K, Chapter 4] for a general reference on B-canonical
Frobenius splittings.
6.1. Consider a B-canonical Frobenius splitting as in (22). By Frobe-
nius reciprocity this defines a map
St → IndGB
EndF (X)⊗ kλ
and as IndGB
k[X ]
contains k we may consider the inducedG-equivariant
morphism
θ̃ : St → IndGB
EndF (X)⊗ kλ
⊗ IndGB
k[X ]
Composing θ̃ with the map Φ2
M,λ of Section 5.6 we end up with a map
M,λ ◦ θ̃ : St → Ind
k[X ′]⊗ kλ
We claim
Lemma 6.1. The composed map Φ2
M,λ ◦ θ̃ is an isomorphism on its
image IndGB
Proof. We first prove that the image of Φ2
M,λ◦θ̃ is contained in Ind
For this let EndF (X)c denote the inverse image of k ⊂ k[X
′] under
the evaluation map evX . It suffices to prove that the image of θ is
contained in EndF (X)c. Notice that EndF (X)c is a B-submodule of
EndF (X) containing the set of Frobenius splittings of X . In particular,
the image of the lowest weight space of St under θ is contained in
EndF (X)c. Moreover, as St is an irreducible G-module it is generated
by the lowest weight space as a B-module. Thus, the image of θ will
be contained in the B-module EndF (X)c.
Now Φ2
M,λ ◦ θ̃ is a map from St to Ind
B(λ) = St. Thus, by Frobenius
reciprocity, it suffices to prove that Φ2
M,λ ◦ θ̃ is nonzero which is the
case as θ contains a Frobenius splitting in its image. �
Using Lemma 6.1 we can now combine the diagram (15) with the
map Φ2
M,λ ◦ θ̃ and obtain a commutative and G-equivariant diagram
(23) St⊗ St
≃ Id⊗(Φ2
Θ // EndF (G×B X)
evZ // k[Z ′]
EndLF (
G/B)⊗ IndGB
φλ // k
evG/B
88qqqqqqqqqqqqqq
where Θ is the map induced by θ̃ and ΦM,λ. By Proposition 5.7 it
follows that Θ(Ξ), for Ξ in St⊗ St, is a Frobenius splitting of Z if and
only if the image of Ξ under φλ and Id ⊗ (Φ
M,λ ◦ θ̃) equals 1. The
latter map from St⊗St to k will be denoted by φ. By construction φ is
G-equivariant. Moreover, mλ is surjective and evG/B is nonzero (as G/B
admits a Frobenius splitting) and thus φ is nonzero. As St is a simple
G-module it follows that
(24) φ : St⊗ St → k,
defines a nondegenerate G-invariant bilinear form on St. By Frobenius
reciprocity such a form is uniquely determined up to a nonzero con-
stant. In particular, this provides a very useful way to construct lots
of Frobenius splittings of Z.
Corollary 6.2. Let θ : St⊗ k(p−1)ρ → EndF (X) denote a B-canonical
Frobenius splitting of X. Then the induced morphism (defined above)
Θ : St⊗ St → EndF (G×B X),
satisfies the following
(1) The image Θ(ν) of an element ν in St⊗ St defines a Frobenius
splitting of G×BX up to a nonzero constant if and only if φ(ν)
is nonzero.
(2) If the image of θ is contained in EndF (X, Y ) for a B-stable
closed subvariety Y of X, then the image of Θ is contained in
EndF (G×B X,G×B Y ).
(3) Let v denote an element of St = EndLF (
G/B) which is compatible
with a closed subvariety V of G/B. For any element v′ ∈ St we
Θ(v ⊗ v′) ∈ EndF (G×B X, q
−1(V )×B X),
with q : G→ G/B denoting the quotient map.
24 XUHUA HE AND JESPER FUNCH THOMSEN
(4) Any element of the form Θ(v ⊗ v′) factorizes as
(FZ)∗OZ
(FZ )∗π
−−−−−→ (FZ)∗π
−→ OZ′ ,
where Z = G×B X and L is the line bundle on G/B associated
to the B-character (1 − p)ρ. Moreover, if the image of θ is
contained in EndF (X, Y ) then s is compatible with G×B Y .
Proof. All statements follows directly from Theorem 5.6 and the con-
siderations above. �
The first part (1) and (2) of the above result is well known (see e.g.
[B-K, Ex. 4.1.E(4)]). However, the second part (3) and (4) seems to
be new.
6.2. B-canonical Frobenius splitting when G is not semisim-
ple. Although Corollary 6.2 is only stated for connected, semisimple
and simply connected groups it also applies in other cases : assume
that G is a connected linear algebraic group containing a connected
semisimple subgroup H such that the induced map H/H∩B → G/B is
an isomorphism. E.g. this is satisfied for any parabolic subgroup of a
reductive connected linear algebraic group. Let qsc : Hsc → H denote
a simply connected cover of H . Then X admits an action of the par-
abolic subgroup Bsc := q
sc (B ∩ H) of Hsc. Furthermore, the natural
morphism
Hsc ×Bsc X → G×B X,
is then an isomorphism. We then say that X admits a B-canonical
Frobenius splitting if X , as a Bsc-variety, admits a Bsc-canonical Frobe-
nius splitting. In this case we may apply Corollary 6.2 to obtain Frobe-
nius splitting properties of G×B X .
6.3. Restriction to Levi subgroups. Return to the situation where
G is connected, semisimple and simply connected. Let J be a subset of
the set of simple roots ∆ and let GJ denote the commutator subgroup
of LJ . Then GJ is a connected, semisimple and simply connected linear
algebraic group with Borel subgroup BJ = GJ ∩B and maximal torus
TJ = T ∩ GJ . We let StJ denote the associated Steinberg module.
Notice that StJ = Ind
((1− p)ρJ ) where ρJ denotes the restriction of
ρ to BJ . The following should be well known but we do not know a
good reference.
Lemma 6.3. There exists a GJ -equivariant morphism
StJ → St,
such that the B−J -invariant line of StJ maps surjectively to the B
invariant line of St. In particular, if X is a G-variety admitting a B-
canonical Frobenius splitting then X admits a BJ -canonical Frobenius
splitting as a GJ -variety.
Proof. Let M denote the T -stable complement to the B-stable line in
St. Then M is B−-invariant and thus also B−J -invariant. The trans-
late ẇJ0M is then invariant under BJ and we obtain a BJ -equivariant
morphism
St → St/(ẇJ0M) ≃ k(1−p)ρJ .
By Frobenius reciprocity this defines a GJ -equivariant map St → StJ
such that the B-stable line of St maps onto the BJ -stable line of StJ .
Now apply the selfduality of StJ and St to obtain the desired map.
This proves the first part of the statement.
The second part follows easily by composing the obtained morphism
StJ → St with the B-canonical Frobenius splitting
St → EndF (X)⊗ k(1−p)ρ,
of X and noticing that the restriction of ρ to BJ is ρJ .
7. Applications to G×G-varieties
In this section we consider a linear algebraic group G satisfying the
conditions of Section 6.2, i.e. we assume that G contains a closed
connected semisimple subgroup H such that H/H∩B → G/B is an iso-
morphism. We also let Hsc denote the simply connected version of H
and let Bsc denote the associated Borel subgroup.
7.1. A well known result. Consider for a moment (i.e. in this sub-
section) the case where G is semisimple and simply connected. Re-
member that the G-linearized line bundle on G/B associated to the B-
character 2ρ coincides with the dualizing sheaf ωG/B. Let L denote the
line bundle on G/B associated to the B-character (1−p)ρ and recall from
Section 6 the notation St = IndGB((1−p)ρ) for the Steinberg module. As
the Steinberg module is a selfdual G-module we may fix a G-invariant
nonzero element v∆ in the tensorproduct St⊗ St. We may think of v∆
as a global section of the line bundle L⊠ L on (G/B)2 = G/B × G/B.
Identify G/B × G/B with G×B G/B by the isomorphism
G×B G/B → G/B × G/B,
[g, hB] 7→ (gB, ghB),
and let D denote the subvariety of G/B × G/B corresponding to G ×B
∂(G/B), where ∂(G/B) denotes the union of the codimension 1 Schubert
varieties in G/B. Then, by [B-K, proof of Thm.2.3.8], the zero scheme
of v∆ equals (p− 1)D. Consider then the natural morphism :
η : (L⊠ L)⊗ (L⊠ L) → ω
(G/B)2
= End!F ((
G/B)2)
and define
ηD : (L⊠ L) → End
G/B)2),
26 XUHUA HE AND JESPER FUNCH THOMSEN
as in Lemma 3.6, using the identification L ⊠ L = O(G/B)2
(p − 1)D
Then by Lemma 3.6 the image of ηD is contained in End
G/B)2, D)
and thus the associated element
η′D ∈ HomO((G/B)2)′
(F(G/B)2)∗(L⊠ L),O((G/B)2)′
is compatible with D. It follows
Lemma 7.1. The element in
EndL⊠LF ((
G/B)2) ≃ St⊠ St
defined by v∆ is compatible with the diagonal diag(G/B) in G/B × G/B.
Proof. We have to prove that η′D, defined above, is compatible with the
diagonal diag(G/B). As η′D is compatible with D it suffices to show that
EndL⊠LF ((
G/B)2, D) is contained in EndL⊠LF ((
G/B)2, diag(G/B)
. This fol-
lows by an application of Lemma 3.1 and an argument as at the end of
the proof of [B-K, Thm.2.3.1]. �
7.2. We return to the setup as in the beginning of this section. We
want to apply the results of the preceding sections to the case when
the group equals G×G. So let X denote a B ×B-variety and assume
that X admits a Bsc ×Bsc-canonical Frobenius splitting defined by
θ : (St⊠ St)⊗ (k(p−1)ρ ⊠ k(p−1)ρ) → EndF (X),
which is compatible with certain B×B-stable subvarieties X1, . . . , Xm,
i.e. the image of θ is contained in EndF (X,Xi) for all i. Then
Theorem 7.2. The variety (G × G) ×(B×B) X admits a diag(Bsc)-
canonical Frobenius splitting which compatibly Frobenius splits the sub-
varieties diag(G)×diag(B) X and (G×G)×(B×B) Xi for all i.
Proof. It suffices to consider the case where G = Hsc (cf. discussion
in Section 6.2). By Corollary 6.2 there exists a G × G-equivariant
morphism
Θ : (St⊠ St)⊗ (St⊠ St) → EndF ((G×G)×(B×B) X),
satisfying certain compatibility conditions. Let v∆ ∈ St ⊠ St be a
nonzero diag(G)-invariant element and let v ∈ St ⊠ St be arbitrary.
Then by Corollary 6.2 and Lemma 7.1 the element Θ
v∆ ⊗ v
is com-
patible with diag(G) ×diag(B) X and (G × G) ×(B×B) Xi for all i. In
particular, if we define the diag(G)-equivariant morphism
Θ∆ : St⊗ St → EndF ((G×G)×(B×B) X),
by Θ∆(v) = Θ
v∆ ⊗ v
, then every element in the image of Θ∆ is
compatible with diag(G) ×diag(B) X and (G × G) ×(B×B) Xi for all i.
Consider k(p−1)ρ as the highest weight line in St. Then the restriction
of Θ∆ to St⊗ k(p−1)ρ defines a diag(B)-canonical Frobenius splitting of
(G×G)×(B×B) X with the desired properties. �
Notice that by the general machinery of canonical Frobenius split-
tings (see e.g. [B-K, Prop.4.1.17]) the existence of a Frobenius splitting
of diag(G)×diag(B) X follows if X admits a diag(Bsc)-canonical Frobe-
nius splitting. In the above setup X only admits a Bsc ×Bsc-canonical
Frobenius splitting which is less restrictive. However, in contrast to the
situation when X admits a diag(Bsc)-canonical Frobenius splitting, the
present Frobenius splitting is not necessarily compatible with subvari-
eties of the form BẇB ×B X , with w denoting an element of the Weyl
group and BẇB denoting the closure of BẇB in G.
8. G-Schubert varieties in equivariant Embeddings
From now on, unless otherwise stated, we assume that G is a con-
nected reductive group.
8.1. Equivariant embeddings. Consider G as a G × G-variety by
left and right translation. An equivariant embedding X of G is then
a normal irreducible G × G-variety containing an open dense subset
which is G × G-equivariantly isomorphic to G. In particular, we may
identify G with an open subset of X , and the complement X \ G is
then called the boundary. As G is an affine variety the boundary is of
pure codimension 1 in X [Har, Prop.3.1]. Any equivariant embedding
of G is a spherical variety (with respect to the induced B × B-action)
and thus X contains finitely may B ×B-orbits.
8.2. Wonderful compactifications. When G = Gad is of adjoint
type there exists a distinguished equivariant embedding X of G which
is called the wonderful compactification (see e.g. [B-K, 6.1]).
The boundary X \ G is a union of irreducible divisors Xj , j ∈ ∆,
which intersect transversely. For a subset J ⊂ ∆, we denote the inter-
section ∩j∈JXj by XJ . As a (G×G)-variety, XJ is isomorphic to the
variety (G × G) ×P−
×P∆\J
Y, where Y denotes the wonderful com-
pactification of the group of adjoint type associated to L∆\J . Here the
×P∆\J -action on Y is defined by the quotient maps P∆\J → L∆\J
and P−
→ L∆\J . In particular, X∆ is G×G-equivariantly isomorphic
to the variety G/B− × G/B.
8.3. Toroidal embeddings. Let Gad denote the group of adjoint
type associated to G, and let X denote the wonderful compactifica-
tion of Gad. An embedding X of the reductive group G is then called
toroidal if the canonical map G→ Gad admits an extension X → X.
8.4. G-Schubert varieties. By a G-Schubert variety in an equivari-
ant embedding X we will mean a subvariety of the form diag(G) · V ,
for some B × B-orbit closure V . Notice that diag(G) · V is the image
of diag(G)×diag(B) V under the proper map
diag(G)×diag(B) X → X,
28 XUHUA HE AND JESPER FUNCH THOMSEN
[g, x] 7→ g · x,
and thus G-Schubert varieties are closed diag(G)-stable subvarieties of
If G = Gad and X = X is the wonderful compactification then a G-
Schubert variety in X∆ is diag(G)-equivariantly isomorphic to a variety
of the form G×BX(w), where X(w) denotes a Schubert variety in G/B.
In particular, this explains the name G-Schubert varieties as this is the
name used for varieties of the form G×B X(w).
In the rest of this section, we will relate G-Schubert varieties to
closures of so-called G-stable pieces. Our primary interest are G-stable
pieces in wonderful compactifications but below we will also describe
the toroidal case in general.
8.5. G-stable pieces in the wonderful compactification. LetG =
Gad denote a group of adjoint type and let X denote its wonderful com-
pactification. Let J ⊂ ∆ and identify XJ with (G× G)×P−
×P∆\J
as in Section 8.2. Using this identification it easily follows that there
exists a unique element in XJ which is invariant under U
J × UJ and
diag(LJ). We denote this element by hJ and note that as an element
of (G×G)×P−
×P∆\J
Y it equals [(e, e), eJ ], where e (resp. eJ) denotes
the identity element of G (resp. the adjoint group associated to L∆\J ).
For w ∈ W∆\J , we then let
XJ,w = diag(G)(Bw, 1) · hJ ,
and call XJ,w a G-stable piece of X. A G-stable piece is a locally closed
subset of X and by [L, section 12] and [He, section 2], we can use them
to decompose X as follows
w∈W∆\J
XJ,w.
Moreover, by the proof of [He2, Theorem 4.5], any G-Schubert variety
is a finite union of G-stable pieces. In particular, we may think of
G-Schubert varieties as closures of G-stable pieces.
8.6. G-stable pieces in arbitrary toroidal embeddings. We fix a
toroidal embedding X of G. The irreducible components of the bound-
ary X \G will be denoted by X1, . . . , Xn. For each G×G-orbit closure
Y in X we then associate the set
KY = {i ∈ {1, . . . , n} | Y ⊂ Xi},
where by definition KY = ∅ when Y = X . Then by [B-K, Prop.6.2.3],
Y = ∩i∈KYXi. Moreover, we define
I = {KY ⊂ {1, . . . , n} | Y a G×G-orbit closure in X },
and write XK := ∩i∈KXi for K ∈ I. Then (XK)K∈I are the set of
closures of G×G-orbits in X . Let now πX : X → X denote the given
extension of G → Gad. Then the closure of πX(XK) equals XP (K) for
some unique subset P (K) of ∆. This defines a map P : I → P(∆),
where P(∆) denotes the set of subsets of ∆.
As in [H-T2, 5.4], for K ∈ I we may choose a base point hK in the
open G×G-orbit of XK which maps to hP (K). By [H-T2, Proposition
5.3], XK is then naturally isomorphic to (G×G)×P−
×P∆\J
L∆\J · hK ,
where J = P (K) and L∆\J · hK is a toroidal embedding of a quotient
(L∆\J )/H by some subgroup H of the center of L∆\J .
For K ∈ I and w ∈ W∆\p(K), we then define
XK,w = diag(G)(Bw, 1) · hK ,
and call XK,w a G-stable piece of X . One can then show, in the same
way as in [He2, 4.3], that
w∈W∆\P (K)
XK,w.
Also similar to the proof of [He2, Theorem 4.5], for any B × B-orbit
closure V in X , the G-Schubert variety diag(G) · V is a finite union
of G-stable pieces. In particular, G-Schubert varieties are closures of
G-stable pieces.
9. Frobenius splitting of G-Schubert varieties
In this section, we assume that X is an equivariant embedding of G.
Let Gsc denote a simply connected cover of the semisimple commutator
subgroup (G,G) of G. We fix a Borel subgroup Bsc of Gsc which is
compatible with the Borel subgroup B in G. Similarly we fix a maximal
torus Tsc ⊂ Bsc.
Let X1, . . . , Xn denote the boundary divisors of X . The closure
within X of the B × B-orbit Bsjw0B ⊂ G will be denoted by Dj .
Then Dj is of codimension 1 in X . The translate (w0, w0)Dj of Dj will
be denoted by D̃j.
By earlier work we know
Theorem 9.1. [H-T2, Prop.7.1] The equivariant embedding X admits
a Bsc × Bsc-canonical Frobenius splitting which compatibly Frobenius
splits the closure of every B ×B-orbit.
As a direct consequence of Theorem 7.2 we then obtain
Corollary 9.2. The variety (G × G) ×(B×B) X admits a diag(Bsc)-
canonical Frobenius splitting which is compatible with all subvarieties
of the form (G×G)×(B×B) Y and diag(G)×diag(B)Y , for a B×B-orbit
closure Y in X.
Proposition 9.3. The equivariant embedding X admits a diag(Bsc)-
canonical Frobenius splitting which compatibly splits all G-Schubert va-
rieties in X.
30 XUHUA HE AND JESPER FUNCH THOMSEN
Proof. By Corollary 9.2 the variety Z = diag(G) ×diag(B) X admits
a diag(Bsc)-canonical Frobenius splitting which is compatible with all
subvarieties of the form diag(G)×diag(B) Y , with Y denoting a B ×B-
orbit closure in X . As X is a diag(G)-stable we may identify Z with
G/B ×X using the isomorphism
G×B X → G/B ×X,
[g, x] 7→ (gB, gx).
In particular, we see that the morphism
π : Z = diag(G)×diag(B) X → X,
[g, x] 7→ g · x,
is projective and that π∗(OZ) = OX . As a consequence (see Section 3.8)
the diag(Bsc)-canonical Frobenius splitting of Z induces a diag(Bsc)-
canonical Frobenius splitting of X which is compatible with all subva-
rieties of the form
π(diag(G)×diag(B) Y ) = diag(G) · Y,
i.e. with all the G-Schubert varieties in X . This ends the proof. �
As a direct consequence of Proposition 9.3, we conclude the following
vanishing result (see [B-K, Theorem 1.2.8]).
Corollary 9.4. Let X denote a projective equivariant embedding of G.
Let X denote a G-Schubert variety in X and let L denote an ample line
bundle on X. Then
Hi(X,L) = 0, i > 0.
Moreover, if X̃ ⊂ X is another G-Schubert variety, then the restriction
H0(X,L) → H0(X̃,L),
is surjective.
Later (i.e. Cor. 10.5) we will generalize the vanishing part of this
result to nef line bundle.
9.1. F-splittings along ample divisors. In this subsection we as-
sume that X is toroidal. The following structural properties of toroidal
embeddings can all be found in [B-K, Sect.6.2]. Let X0 denote the com-
plement in X of the union of the subsets BsiB− for i ∈ ∆. If we let T̄
denote the closure of T in X , then X0 admits a decomposition defined
by the following isomorphism
(25) U × U− × (T̄ ∩X0) → X0, (x, y, z) 7→ (x, y) · z.
Moreover, every G×G-orbit in X intersects (T̄ ∩X0) in a unique orbit
under the left action of T . Notice here that as T is commutative the
T × T -orbits and the (left) T -orbit in T will coincide.
Lemma 9.5. Let X denote a projective toroidal equivariant embedding
of G and let Y denote a G × G-orbit closure in X. Let K denote the
subset of {1, . . . , n} consisting of those j such that Y is contained in
the boundary component Xj. Then
Y ∩ (
j /∈K
(1, w0)Di),
has pure codimension 1 in Y and contains the support of an ample
effective Cartier divisor on Y .
Proof. Let XK = ∪j /∈KXj . We claim that Y \X
K coincides with the
open G × G-orbit Y0 of Y . Clearly Y0 is contained in Y \ X
K . On
the other hand, let U be a G×G-orbit in Y \XK . Then Xj contains
U if and only if j /∈ K. But every G × G-orbit closure in X is the
intersection of those Xj which contain it [B-K, Prop.6.2.3]. It follows
that the closure of Y0 and U coincide and thus U = Y0.
As X is normal we may choose a G × G-linearized very ample line
bundle L on X . Then H0(Y,L) is a finite dimensional (nonzero) rep-
resentation of G ×G, and it thus contains a nonzero element v which
is B × B−-invariant up to constants. The support of v is then the
union of B × B−-invariant divisors on Y . As Y0 ∩ (T̄ ∩X0) is a single
T × T -orbit it follows that
Y0 ∩X0 ≃ U × U
− × (Y0 ∩ (T̄ ∩X0)),
is an affine variety and a single B×B−-orbit. In particular, the support
of v is contained in
Y \ (Y0 ∩X0) = Y ∩ (X
(1, w0)Di).
This shows the second part of the statement. The first part follows as
Y0 ∩X0 is affine [Har, Prop.3.1]. �
Let now X denote a smooth projective toroidal embedding of G. As
the line bundles OX(Di) and OX(D̃i) are isomorphic it follows by [B-K,
Prop.6.2.6] that
(26) ω−1X ≃ OX
(Di + D̃i) +
Recall that a X is normal and G is semisimple and simply connected,
any line bundle on X will admit a unique G2sc = Gsc×Gsc-linearization.
In particular, if we let τi denote the canonical section of the line bundle
OX(Di), then we may consider τi as a B
sc = Bsc×Bsc-eigenvector of the
space of global sections of OX(Di). As in the proof of [B-K, Prop.6.1.11]
we find that the associated weight of τi equals ωi ⊠ −w0ωi, where ωi
denotes the i-th fundamental weight. Similarly, we may consider the
canonical section σj of OX(Xj) as a G
sc-invariant element.
32 XUHUA HE AND JESPER FUNCH THOMSEN
Let V denote a B ×B-orbit closure in X . As V is B ×B-stable the
subset Y = (G×G) ·V is closed in X . Thus we may consider Y as the
smallest G×G-invariant subvariety of X containing V . Now define K
as in Lemma 9.5 and let M denote the line bundle
M = OX
(p− 1)(
D̃i +
j /∈K
By Equation (26) and Lemma 3.6 it then follows that multiplication
with τ
i , for i ∈ ∆, and σ
j , for j ∈ K, defines a morphism of
B2sc-linearized line bundles
M → End!F
X, {Di, Xj}i∈∆,j∈K
⊗ kλ⊠λ,
where λ = (1 − p)ρ. By [H-T2, Prop.6.5] and Lemma 3.1 any element
in End!F (X) which is compatible with the closed subvarieties Di, i ∈ ∆,
and Xj, j ∈ K, is also compatible with V and Y . In particular, we
have defined a B2sc-equivariant map
(27) η : M → End!F (X, Y, V
⊗ kλ⊠λ,
which, by Lemma 3.5, is the same as a B2sc-invariant element η
EndMF
X, Y, V
⊗ kλ⊠λ. In particular, this defines us an element
(28) v ∈ Ind
EndMF
X, Y, V
⊗ kλ⊠λ
which is G2sc-invariant. We are then ready to use the ideas explained
in Section 5.7. First we use (18) to construct a morphism
(29) EndL⊠LF
(Gsc/Bsc)
⊗M(X) → EndF
G2sc ×B2sc X
(u, σ) 7→ ΦM,λ⊠λ(u⊗ v ⊗ σ),
where L is the Gsc-linearized line bundle on Gsc/Bsc associated to the
character λ = (1− p)ρ. Notice that we here have used that M(X) is a
G2sc-module.
Lemma 9.6. There exists a G2sc-equivariant map
(30) St⊠ St → M(X),
which maps the B−sc×B
sc-invariant line in St⊠St to a nonzero multiple
of the global section
j /∈K
j ∈ M(X),
where τ̃i denotes the canonical section of OX(D̃i).
Proof. As OX(D̃i) and OX(Di) are isomorphic as line bundles we may
consider the element
j /∈K
as a global section ofM. Then σ is a B2sc-eigenvector inM(X) of weight
(p − 1)ρ ⊠ (p − 1)ρ. In particular, σ induces a Bsc × Bsc-equivariant
k(p−1)ρ ⊠ k(p−1)ρ → M(X).
Applying Frobenius reciprocity and the selfduality of the Steinberg
module St, this defines the desired map
St⊠ St → M(X),
with the stated properties. �
Combining the map (29) with the map (30) in Lemma 9.6 we obtain
a G2sc-equivariant map
(31) Θ : EndL⊠LF
(Gsc/Bsc)
St⊠ St
→ EndF
G2sc ×B2sc X
We will now study when the map (31) describes a Frobenius splitting
of G2sc ×B2sc X . Consider the G
sc-equivariant map
(32) M(X) → St⊠ St,
σ 7→ Φ2
M,λ⊠λ(v ⊗ σ),
defined as the map (19) in Section 5.7. We claim
Lemma 9.7. The composition of the map (30) in Lemma 9.6 and the
map in (32) is an isomorphism on St⊠ St.
Proof. By Frobenius reciprocity it suffices to show that the described
composed map is nonzero. In particular, it suffices to show that
M,λ⊠λ(v ⊗ σ̃) 6= 0,
where σ̃ denotes the global section of M defined in Lemma 9.6. For
this we use the fact that the global section
(τiτ̃i)
X defines a Frobenius splitting of X (see e.g. [B-K, proof of
Thm.6.2.7]). As a consequence η(σ̃) is a Frobenius splitting ofX , where
η is the map defined in (27). Equivalently , the natural G2sc-equivariant
morphism
EndMF (X)⊗M(X) → k[X
′] = k,
defined in (13), will map η′ ⊗ σ̃ to 1. This induces a commutative
diagram
(33) Ind
EndMF
⊗ kλ⊠λ
⊗M(X)
M,λ⊠λ
++VVV
St⊗ St
EndMF (X)⊗ kλ⊠λ ⊗M(X)
// kλ⊠λ
34 XUHUA HE AND JESPER FUNCH THOMSEN
where the image of v⊗ σ̃ under the diagonal map is nonzero. This ends
the proof. �
Proposition 9.8. Let Θ denote the map defined in (31). The image
Θ(ν) of an element ν defines, up to a nonzero constant, a Frobenius
splitting of G2sc ×B2sc X if and only if the image of ν under the map
(34) φλ⊠λ : End
(Gsc/Bsc)
St⊠ St
defined in Section 5.6, is nonzero.
Proof. Apply Proposition 5.7 and Lemma 9.7.
With the identification EndL⊠LF
(Gsc/Bsc)
≃ St⊠ St the map φλ⊠λ,
defined in (34), must necessarily (up to a nonzero constant) be the
G2sc-invariant form on St⊠ St mentioned in Section 6.1. Let v∆ denote
the diag(G)-invariant element in EndL⊠LF
(Gsc/Bsc)2
defined in Section
7.1. Then the diag(G)-equivariant map
St⊗ St → k,
ν 7→ φλ⊠λ(v∆ ⊗ ν),
is nonzero and thus it must coincide (up to a nonzero constant) with
the Gsc-invariant form φ on St defined in (24).
Proposition 9.9. Fix notation as above and let D denote the effective
Cartier divisor
(p− 1)
(1, w0)Di +
j /∈K
on X. Then X admits a Frobenius D-splitting which is compatible with
the subvariety Y and the G-Schubert variety diag(G) · V .
Proof. Consider the diag(G)-equivariant morphism
Θ∆ : St⊠ St → EndF
G2sc ×B2sc X
ν 7→ Θ(v∆ ⊗ ν),
where Θ is the map in (31). By Lemma 9.8 the image Θ∆(ν) of an
element ν ∈ St⊗ St is a Frobenius splitting, up to a nonzero constant,
if and only if φ(ν) is nonzero. Here φ is the the map defined in (24).
Let v+ (resp. v−) denote a nonzero B (resp. B
−)-eigenvector of St
and let ν = v+ ⊗ v−. After possibly multiplying v+ with a constant
we may assume that s = Θ∆(ν) defines a Frobenius splitting of Z =
G2sc ×B2sc X . As v is compatible with Y and V (cf. (28)) it follows by
Theorem 5.6 and Lemma 7.1 that s factorizes as
(35) s : (FZ)∗OZ
(FZ )∗σ
−−−−→ (FZ)∗MZ
−→ OZ′,
where s1 is compatible with the subvarieties G
sc×B2sc V , G
sc×B2sc Y and
diag(Gsc)×diag(Bsc) X . Here MZ is the G
sc-linearized line bundle on Z
associated with the B2sc-linearized line bundle M on X as explained in
Section 5.2, and σ is the global section of MZ defined as the image of ν
under the map (30) in Lemma 9.6. Notice that as M is a G2sc-linearized
line bundle on the G2sc-variety X we may identify the global sections of
M and MZ . Actually , as X is a G
sc-variety the morphism
G2sc ×B2sc X →
Gsc/Bsc × Gsc/Bsc ×X,
[(g1, g2), x] 7→ (g1B, g2B, (g1, g2) · x),
is an isomorphism. Moreover, under this isomorphism, the line bundle
MZ is just the pull back of M under projection pX on the third coor-
dinate. Thus, by Lemma 9.6 it follows that σ is the pull back from X
of the effective Cartier divisor
D = (p− 1)
(1, w0)Di +
j /∈K
Applying the functor (pX)∗ to (35) we obtain the Frobenius D-splitting
(pX)∗s : (FX)∗OX
(FX)∗σD
−−−−−→ (FX)∗O(D)
(pX)∗s1
−−−−→ OX′
ofX where (pX)∗s1 is compatible with the subvarieties pX(G
sc×B2scY ) =
Y and pX(diag(Gsc)×diag(Bsc) V ) = diag(G) · V (by Lemma 3.7). This
ends the proof. �
Corollary 9.10. Let X denote a G-Schubert variety in a smooth pro-
jective toroidal embedding of a reductive group G. Then X admits a
stable Frobenius splitting along an ample divisor.
Proof. Apply Proposition 9.9, Lemma 9.5 and Lemma 3.3. �
10. Cohomology of line bundles
The main aim of this section is to obtain a generalizing the vanishing
part of Corollary 9.4 to nef line bundles. The concept of a rational
morphism is here central and for this we use [B-K, Sect.3.3] as a general
reference. First we recall :
Definition 10.1. A morphism f : Y → Z of varieties is a called a ra-
tional morphism if the induced map f ♯ : OZ → f∗OY is an isomorphism
and Rif∗OY = 0, i > 0.
The following criterion for a morphism to be rational will be very
useful ([R, Lem.2.11]).
Lemma 10.2. Let f : Y → Z denote a projective morphism of ir-
reducible varieties and let Ŷ denote a closed irreducible subvariety of
Y . Consider the image Ẑ = f(Ŷ ) as a closed subvariety of Z. Let L
denote an ample line bundle on Z and assume
(1) f ♯ : OZ → f∗OY is an isomorphism.
(2) Hi(Y, f ∗Ln) = Hi(Ŷ , f ∗Ln) = 0, for i > 0 and n≫ 0.
36 XUHUA HE AND JESPER FUNCH THOMSEN
(3) The restriction map H0(Y, f ∗Ln) → H0(Ŷ , f ∗Ln) is surjective
for n≫ 0.
Then the induced map f̂ : Ŷ → Ẑ is a rational morphism.
10.1. Toric variety. An equivariant embedding Z of the (reductive)
group T is called a toric variety (wrt. T ). Notice that, as T is commu-
tative, we may consider the T ×T -action on Z as just a T -action. The
following result should be well known but, as we do not know a good
reference, we include a proof.
Lemma 10.3. Let f : Y → Z denote a projective surjective morphism
of equivariant embeddings of T . Let T · z denote a T -orbit in Z and let
T · y denote a T -orbit in f−1(T · z) of minimal dimension. Then the
map T · y → T · z, induced by f , is an isomorphism.
Proof. Let T · z and T · y denote the closures of T · z and T · y in Z
and Y respectively. Then the induced map
f̂ : T · y → T · z,
is a projective morphism. Moreover, by the minimality assumption
on T · y, the inverse image f̂−1(T · z) equals T · y. In particular, the
induced morphism : T · y → T · z is projective. But any T -orbit in a
toric variety (wrt. to T ) is isomorphic to a torus T1 satisfying that the
cokernel of the induced map of character groups X∗(T1) → X
∗(T ) is a
free abelian group ([Ful, Sect.3.1]). In particular, the varieties T ·y and
T · z are tori and the cokernel of the induced map of character groups
X∗(T · z) → X∗(T · y) is a free abelian group. But T · y → T · z is
an affine projective morphism and thus it must be a finite morphism.
Thus the cokernel of X∗(T · z) → X∗(T · y) is a finite group and, as it
is already a free group, it must be trivial. This ends the proof as tori
are determined by their character groups. �
Lemma 10.4. Let X denote a projective embedding of a reductive
group G and let Y denote a G × G-orbit closure of X. Then there
exists a smooth toroidal embedding X̂ of G, a projective G-equivariant
morphism f : X̂ → X and a G×G-orbit closure Ŷ in X̂ such that the
induced morphism f : Ŷ → Y is a rational morphism.
Proof. Assume first that X is toroidal. By [B-K, Prop.6.2.5] there ex-
ists a smooth toroidal embedding X̂ of G with a projective morphism
f : X̂ → X . Let X0 denote the open subset of X introduced in the
beginning of Section 9.1, and let X̂0 denote the corresponding sub-
set of X̂ . Then the inverse image f−1(X0) coincides with X̂0 [B-K,
Prop.6.2.3(i)]. Let T (resp. T̂ ) denote the closure of T in X (resp. X̂).
Then T and T̂ are toric varieties [B-K, Prop.6.2.3], and the induced
map f : T̂ → T is a projective morphism of toric varieties. Thus also
the induced map
X̂0 ∩ T̂ → X0 ∩ T ,
is a projective morphism of toric varieties. As mentioned in Section 9.1
every G×G-orbit in X will intersect X0 ∩ T in a unique T -orbit. We
let T · x denote the open T -orbit in the intersection of Y with X0 ∩ T .
By Lemma 10.3 we may find a T -orbit T · x̂ in X̂0 ∩ T̂ which by f
is isomorphic to T · x, and we then define Ŷ to be the closure of the
G×G-orbit through x̂. By the isomorphism (25) we then conclude that
f induces a projective birational morphism Ŷ → Y . By [H-T2, Cor.8.4]
the orbit closure Y is normal and thus, by Zariski’s main theorem, we
conclude f∗OŶ = OY . By Lemma 10.2 (used on the morphism Ŷ → Y
and the closed non-proper subvariety Ŷ of Ŷ ) it now suffices to prove
Hi(Ŷ , f ∗L) = 0, i > 0,
for a very ample line bundle L on Y . This follows from [H-T2, Prop.7.2]
and ends the proof in the case when X is toroidal.
Consider now an arbitrary projective equivariant embedding X of
G. Let X̂ denote the normalization of the closure of the image of the
natural G×G-equivariant embedding
G→ X ×X,
where X denotes the wonderful compactification of Gad. Then X̂ is a
toroidal embedding of G with an induced projective equivariant mor-
phism f : X̂ → X . Let Ŷ denote any G×G-orbit closure in X̂. Then
f : Ŷ → f(Ŷ ) is a rational morphism [H-T2, Lem.8.3]. In particular,
we may find a G × G-orbit closure Ŷ of X̂ with an induced rational
morphism f : Ŷ → Y . Finally we may apply the first part of the proof
to Ŷ and X̂ and use that a composition of rational morphisms is again
a rational morphism. �
Corollary 10.5. Let X denote a projective embedding of a reductive
group G and let X denote a G-Schubert variety in X. Let Y = (G×G)·
X denote the minimal G × G-orbit closure of X containing X. When
L is a nef line bundle on X then
Hi(X,L) = 0, i > 0.
Moreover, when L is a nef line bundle on Y then the restriction mor-
phism
H0(Y,L) → H0(X,L),
is surjective.
Proof. Assume first that X is smooth and toroidal. Then by Propo-
sition 9.9, Lemma 9.5 and Lemma 3.3 the variety Y admits a stable
Frobenius splitting along an ample divisor which is compatibly with X.
Thus the statement follows in this case by Proposition 3.4.
38 XUHUA HE AND JESPER FUNCH THOMSEN
Let now X denote an arbitrary projective equivariant embedding of
G. Choose, using Lemma 10.4, a smooth projective toroidal embedding
X̂ with a projective equivariant morphism f : X̂ → X onto X , and a
G × G-orbit closure Ŷ in X̂ with an induced rational morphism onto
Y . Let V denote a B×B-orbit closure in Y such that X = diag(G) ·V .
As Y is the minimal G × G-orbit closure containing X it follows that
V will intersect the open G×G-orbit of Y . In particular, there exists
a B×B-orbit closure V̂ in X̂ which intersects the open G×G-orbit of
Ŷ and which maps onto V . In particular,
X̂ := diag(G) · V̂ ,
is a G-Schubert variety in X̂ which by f maps onto X. Moreover, Ŷ is
the minimal G×G-orbit closure containing X̂.
We claim that the induced morphism X̂ → X is a rational morphism.
To prove this we apply Lemma 10.2 to the rational morphism f : Ŷ →
Y . Choose an ample line bundle M on Y . Then it suffices to prove
(36) Hi(Ŷ , f ∗Mn) = Hi(X̂, f ∗Mn) = 0, i > 0, n > 0,
and that the restriction map
(37) H0(Ŷ , f ∗Mn) → H0(X̂, f ∗Mn),
is surjective for n > 0. But Mn is an ample, and thus nef, line bundle
on Y and therefore the pull back f ∗Mn is a nef line bundle on Ŷ ([Laz,
Ex. 1.4.4]). As X̂ is smooth and toroidal, the conclusion of the first
part of this proof then shows that conditions (36) and (37) are satisfied.
Now both X̂ → X and Ŷ → Y are rational morphisms. In particular,
we have identifications
Hi(Ŷ , f ∗L) ≃ Hi(Y,L), i ≥ 0,
Hi(X̂, f ∗L) ≃ Hi(X,L), i ≥ 0,
for any line bundle L on Y or, in the second equation, on X . When L is
a nef line bundle the pull back f ∗L is also nef ([Laz, Ex. 1.4.4]). Thus
as we have already completed the proof of the statement for smooth
toroidal embeddings, in particular for X̂ , this now ends the proof. �
By the proof of the above result we also find that any G-Schubert
variety X in a projective equivariant embedding of G, will admit a G-
equivariant rational morphism f : X̂ → X by a G-Schubert variety X̂
of some smooth projective toroidal embedding of G.
Remark 10.6. When X = X is the wonderful compactification of a
group G of adjoint type and L is a nef line bundle on X, then the
restriction morphism
H0(X,L) → H0(Y,L),
to any closed G×G-stable irreducible subvariety Y of X is surjective.
In particular, also the restriction morphism
H0(X,L) → H0(X,L),
to any G-Schubert variety X is surjective by the above result. We do
not know if the latter is true for arbitrary equivariant embeddings.
11. Normality questions
The obtained Frobenius splitting properties of G-Schubert varieties
in Section 9 and the cohomology vanishing results in Corollary 10.5
should be expected to have strong implications on the geometry of
these varieties. However, in this section we provide an example of a G-
Schubert variety in the wonderful compactification of a group of type
G2 which is not even normal. In fact, it seems that there are plenty of
such examples.
11.1. Some general theory. We keep the notations as in Section
8.5. For J ⊂ ∆ and w ∈ W∆\J , we let XJ,w denote the closure of XJ,w
in X. Let
K = max{K ′ ⊂ ∆ \ J ;wK ′ ⊂ K ′}.
By [He2, Prop. 1.12], we have a diag(G)-equivariant isomorphism
diag(G)×diag(PK) (PKẇ, PK)hJ ≃ XJ,w
induced by the inclusion of (PKẇ, PK)hJ in X. Let V denote the
closure of (PKẇ, PK)hJ within X. Then V is the closure of a B × B-
orbit and we find that the induced map
(38) f : diag(G)×diag(PK) V → XJ,w,
is a birational and projective morphism. Thus, by Zariski’s Main The-
orem, a necessary condition for XJ,w to be normal is that the fibers of
f are connected. Actually, in positive characteristic, connectedness of
the fibers is also sufficient forXJ,w to be normal. This follows asXJ,w is
Frobenius split (Prop. 9.3) and thus weakly normal [B-K, Prop.1.2.5].
11.2. An example of a non-normal closure. Let now, further-
more, G be a group of type G2. Let α1 denote the short simple root
and α2 denote the long simple root. The associated simple reflections
are denoted by s1 and s2. Let J = {α2} and w = s1s2 ∈ W
∆\J . In this
case K = ∅ and we obtain a birational map
f : diag(G)×diag(B) V ≃ XJ,w
where V is the closure of (Bẇ,B)hJ . By [Sp, Prop. 2.4], the part of
V which intersect the open G×G-orbit of XJ equals
(Bẇ′, B)hJ ∪
ws1≤w′
(Bẇ′, Bṡ1)hJ .
40 XUHUA HE AND JESPER FUNCH THOMSEN
In particular, x := (v̇, 1)hJ is an element of V , where v = s2s1s2. We
claim that the fiber of f over x is not connected. To see this let y
denote a point in the fiber over x. Then we may find g ∈ G and x̃ ∈ V
such that
y = [g, x̃].
By (39), x̃ = (bẇ′, b′)hJ for some b ∈ B, b
′ ∈ P∆\J and w
′ ≥ w. Then
(gbẇ′, gb′)hJ = (v̇, 1)hJ .
It follows that (v̇−1gbẇ′, gb′) lies in the stabilizer of hJ . In particular,
gb′ ∈ P∆\J and thus also g ∈ P∆\J . If g ∈ B then y = [1, x]. So assume
that g = u1(t)ṡ1 where u1 is the root homomorphism associated to α1.
Assume that t 6= 0. Then we may find b1 ∈ B and s ∈ k such that
g = u−1(s)b1 where u−1 is the root homomorphism associated to −α1.
x̃ = (g−1, g−1)(v̇, 1)hJ
= (b−11 u−1(−s)v̇, g
−1)hJ
= (b−11 v̇, g
−1)hJ
∈ (Bv̇, Bṡ1)hJ
where the third equality follows as v̇−1u−1(−s)v̇ is contained in the
unipotent radical of P−
. But (Bv̇, Bṡ1)hJ has empty intersection
with V (by (39)) which contradicts the assumption that t 6= 0. It
follows that the only possibilities for y are [1, x] and [ṡ1, (ṡ
1 v̇, ṡ
1 )hJ ].
As (ṡ−11 v̇, ṡ
1 ) is contained in V (by (39)) we conclude that the fiber
of f over x consists of 2 points; in particular the fiber is not connected
and thus XJ,w is not normal.
Remark 11.1. It seems likely that normalizations of G-Schubert vari-
eties should have nice singularities : If we let ZJ,w denote the normal-
ization of the closure of XJ,w, then the map (38) induces a birational
and projective morphism
f̃ : diag(G)×diag(PK) V → ZJ,w.
We expect that f̃ can be used to obtain global F -regularity of ZJ,w
(see [S] for an introduction to global F -regularity). In fact, by the
results in [H-T2] the B×B-orbit closure V is globally F -regular. Thus
diag(G)×diag(PK) V is locally strongly F -regular, and as
f̃∗Odiag(G)×diag(PK )V
= OZJ,w ,
it seems likely that ZJ,w is also locally strongly F -regular. Moreover,
similarly to Corollary 9.10 one may conclude that ZJ,w admits a stable
Frobenius splitting along an ample divisor. Thus ZJ,w is globally F -
regular if it is locally strongly F -regular. At the moment we do not
know if ZJ,w is locally strongly F -regular.
12. Generalizations
Fix notation as in Section 2. An admissible triple of G × G is by
definition a triple C = (J1, J2, θδ) consisting of J1, J2 ⊂ ∆, a bijection
δ : J1 → J2 and an isomorphism θδ : LJ1 → LJ2 that maps T to T
and the root subgroup Uαi to the root subgroup Uαδ(i) for i ∈ J1. To
each admissible triple C = (J1, J2, θδ), we associate the subgroup RC of
G×G defined by
RC = {(p, q) : p ∈ PJ1, q ∈ PJ2, θδ(πJ1(p)) = πJ2(q)},
where πJ : PJ → LJ , for a subset J ⊂ ∆, denotes the natural quotient
Let X denote an equivariant embedding of the reductive group G.
A RC-Schubert variety of X is then a subset of the form RC · V for
some B × B-orbit closure V in X . When G = Gad is a group of
adjoint type and X = X is the associated wonderful compactification
the set of RC-Schubert varieties coincides with closures of the set of RC-
stable pieces. By definition [L-Y, section 7], a RC-stable piece in the
wonderful compactification X of Gad is a subvariety of the form RC ·Y ,
where Y = (Bv1, Bv2) · hJ for some J ⊂ ∆, v1 ∈ W
J and v2 ∈
(notation as in Section 8.5). Notice that when J1 = J2 = ∆ and θδ
is the identity map then a RC-stable piece is the same as a G-stable
piece. On the other hand, when J1 = J2 = ∅, then a RC-stable piece
is the same as a B × B-orbit. Moreover, any RC-Schubert variety is a
finite union of RC-stable pieces [L-Y, Section 7].
The following is a generalization of Proposition 9.3 and Proposition
Proposition 12.1. Let C = (J1, J2, θδ) denote an admissible triple of
G×G and let X denote an equivariant embedding of G. Then X admits
a Frobenius splitting which compatible splits all RC-Schubert varieties
in X. If, moreover, X is a smooth, projective and toroidal embedding
and Y = XK = (G × G) · V , for some B × B-orbit closure V in X,
then X admits a Frobenius splitting along the Cartier divisor
D = (p− 1)
(wJ10 , 1)D̃i +
j /∈K
which is compatibly with Y and RC · V .
Proof. As the proof is similar to the proof of Proposition 9.3 and Propo-
sition 9.9 we only sketch the proof. In the following GJ , for a subset
J ⊂ ∆, denotes the commutator of the Levi subgroup in Gsc associated
to J . The Borel subgroup GJ ∩Bsc of GJ is denoted by BJ . Define XC
to be the G2J1-variety which as a variety is X but where the action is
twisted by the morphism
GJ1 ×GJ1
−−−→ GJ1 ×GJ2.
42 XUHUA HE AND JESPER FUNCH THOMSEN
Then the BJ1 × BJ2-canonical Frobenius splitting of X defined by
Theorem 9.1 and Lemma 6.3 induces a B2J1-canonical Frobenius split-
ting of XC. In particular, all subvarieties of XC which corresponds
to B × B-orbit closures in X will be compatibly Frobenius split by
this canonical Frobenius splitting. Now apply an argument as in the
proof of Proposition 9.3 and use the identification of RC · V ⊂ X with
diag(GJ1) · V ⊂ XC. This ends the proof of the first statement.
Assume now that X is a smooth, projective and toroidal embedding
and consider the B2sc-equivariant morphism
η : M → End!F (X, Y, V )⊗ k(1−p)ρ⊠(1−p)ρ,
defined in (27). Let YC and VC be defined similar to XC. Then η induces
a B2J1-equivariant morphism
ηC : M → End
F (XC, YC, VC)⊗ k(1−p)ρJ1⊠(1−p)ρJ1 .
Similar to the definition of v in (28) we obtain from ηC an element
vC ∈ Ind
EndF (XC, YC, VC)⊗ k(1−p)ρJ1⊠(1−p)ρJ1
and from this a G2J1-equivariant morphism
(40) End
(GJ1/BJ1)
⊗M(XC) → EndF
G2J1 ×B2J1
similar to (29). Here LJ1 is the line bundle on
GJ1/BJ1 associated to the
character (1 − p)ρJ1. Combining Lemma 6.3 and Lemma 9.6 we also
obtain a map
(41) StJ1 ⊠ StJ1 → M(XC),
with properties similar to the ones described in Lemma 9.6. As in (32)
we may also use vC to construct a morphism
M(XC) → StJ1 ⊠ StJ1 ,
such that the composition with (41) is an isomorphism on StJ1 ⊠ StJ1 .
Finally we may construct
ΘC : End
(GJ1/BJ1)
⊗ (StJ1 ⊠ StJ1) → EndF
G2J1 ×B2J1
similar to (31). In particular, a statement equivalent to Proposition
9.8 is satisfied for ΘC. Let v
+ (resp. v
− ) denote a highest (resp.
lowest) weight vector in StJ1 and let v
∆ denote the diag(GJ1)-invariant
element of End
(GJ1/BJ1)
. Imitating the proof of Proposition
9.9 we then find that ΘC(v
∆ ⊗ (v
+ ⊗ v
− )) is a Frobenius splitting of
G2J1×B2J1
XC (up to a nonzero constant). Moreover, the push forward of
this Frobenius splitting to X has the desired properties. We only have
to note that the effective Cartier associated to the image of vJ1+ ⊗ v
under the map (41) equals
D = (p− 1)
(wJ10 , 1)D̃i +
j /∈K
This ends the proof. �
We may also argue as in Corollary 10.5 to obtain
Corollary 12.2. Let X denote a projective embedding of a reductive
group G and let V denote the closure of a B × B-orbit in X. Let
Y = (G×G) · V and XC = RC · V . When L is a nef line bundle on XC
Hi(XC,L) = 0, i > 0.
Moreover, when L is a nef line bundle on Y then the restriction mor-
phism
H0(Y,L) → H0(XC,L),
is surjective.
Remark 12.3. In the case where k = C and X is the wonderful com-
pactification, the subvarieties (wJ10 , 1)D̃i, Xj and all the RC-Schubert
varieties are Poisson subvarieties with respect to the Poisson structure
on X corresponding to the splitting
Lie(G)⊕ Lie(G) = l1 ⊕ l2,
where l1 = Lie(RC) and l2 is a certain subalgebra of Ad(w
0 )Lie(B
Lie(B−). See [L-Y2, 4.5].
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Department of Mathematics, Stony Brook University, Stony Brook,
NY 11794, USA
E-mail address : hugo@math.sunysb.edu
Institut for matematiske fag, Aarhus Universitet, 8000 Århus C,
Denmark
E-mail address : funch@imf.au.dk
