Abelian variety and spin representation∗
Shigeru MUKAI
Abelian varieties and K3 surfaces1 have bigger symmetry than their automorphisms. We have learned this from the study of vector bundles on them. Thisphenomenon is similar to the action of metaplectic group on the function space
L2 (Rn ) and to sphere geometry of Lie. In this article, extending the Fourier
functor deﬁned in [M2], we shall show that a unitary group2 U (X × X̂) actson the derived category Dbc (X) of an abelian variety modulo shift of complex(Theorem 1.14). Moreover, using the spin representations (§2), we shall showthat a double covering group U Spin(X × X̂) has a ﬁner action. The Chern
character map and Riemann-Roch theorem are equivariant on this group action(§3). The action will be constructed by semi-homogeneous vector bundles (ortheir universal family), in place of the Poincaré line bundle (§4). In §5, weshow that the Lie group U (X × X̂)R is of Hermitian type and that the group
U (X × X̂) of autoequivalences acts on the tube domain
DX = NS(X)R +√−1(ample cone)

associated with the (formally real) Jordan algebra NS(X)R of Néron-Severigroup. (This might be a sign of a mirror symmetry for Abelian varieties if
there’s any.)
Notation• g denotes the dimension of an abelian variety X.
• X̂ is the dual abelian variety of X, that is, the neutral component Pic 0 X
ˆof the Picard group Pic X. The double dual X̂ is canonically isomorphic
to X.
• For a homomorphism of abelian varieties φ : X −→ Y , φ̂ : Ŷ −→ X̂ is its
transpose, which is the pull-back φ∗ : Pic Y −→ Pic X by deﬁnition.
• P is a normalized Poincaré bundle, that is, universal bundle on X × X̂such that both P|X×0 and P|0×X̂ are trivial.
0 This is a translation of the author’s article in Proceedings of the symposium ”Hodge theory
and Algebraic Geometry”, Sapporo, 1994, pp. 110-135. He is grateful to the University of
Warwick for various supports where this translation was made in stimulating atomosphere
during his stay in 1998. All footnotes were added in this occasion.
1 See [6], [7] and [8].
2 In the original article this group was called orthogonal and denoted by O(X × X̂).

1

• Aut X denotes the group {φ : X −→ X|φ(0) = 0} of automorphisms of X
as Abelian variety.
• For a point a ∈ X, Ta : X −→ X is the translation x → x + a, which is
an automorphism of X as variety.
• For a morphism π : X −→ Y of schemes, Rπ∗ : Dbc (X) −→ Dbc (Y ) is thederived functor of the functor π∗ : (OX − mod) −→ (OY − mod) taking
the direct image of sheaves by π.
• χ(E, F ) is the alternating sum i (−1)i dim Ext iO (E, F ) for a pair of coherent sheaves E and F on X.

1

Fourier transformation and Fourier functor

We recall the basic back-ground. Let G be a ﬁnite abelian group and G∗ thegroup of characters χ. A function on G is expanded to a linear combinationcχ χ of characters in the unique way and we get a function(1.1)

fˆ : G∗ −→ C,χ → cχ =1 
f (g)χ(g)
|G|
g∈G

∗

on G . This is the simplest example of the Fourier transformation. Note that
this is an isometry of two inner product spaces Map(G, C) and Map(G∗ , C).
The most famous one is obtained, at least formally, by replacing the pair(G, G∗ ) of groups with the pair (V, V ∨ ) of a real vector space V and its dual.
The Fourier transformation is deﬁned by

√
(1.2)
fˆ(α) =
f (x) exp(2π −1 < x, α >)dx
Vfor f ∈ L2 (V ). As is well-known, this gives an isometry of two Hilbert spaces
L2 (V ) and L2 (V ∨ ).
These are special cases of the expansion of a function by special functions(characters in the above case). We make an analogy of this in Algebraic Geometry. Namely we consider the expansion of a sheaf by a family of special sheaves.
As coeﬃcients, we obtain a sheaf on another variety, the moduli space. The
best sample3 is the Fourier functor, which expands a sheaf on an abelian variety
by the family of line bundles.
We denote the line bundle (or more precisely its isomorphism class) corresponding to a point α ∈ X̂ = Pic 0 X by Pα . The Fourier transform F̂ of a sheaf
F on X is the sheaf, or precisely speaking a complex of sheaves, on X̂ obtainedin the following way regarding the cohomology H • (X, F ⊗ Pα ) as the coeﬃcientor multiplicity of P−α .
3 Another sample is a K3 surface. See references in the footnote of the ﬁrst page. Beilinson’s
spectral sequence is also an expansion of a sheaf. His functor gives an equivalence of the derived
categories of sheaves on Pn and modules over the exterior algebra.

2

Deﬁnition 1.3 Let P be the normalized Poincaré bundle on X × X̂. For a
coherent sheaf F on X, its Fourier transform F̂ is the element∗
RπX̂,∗ (πXF ⊗ P)of the derived category Dbc (X̂), where πX and πX̂ are the projections of thedirect product X × X̂ onto the ﬁrst and second factors, respectively.
Remark 1.4 Dbc (X̂) is the triangulated category consisting of certain equivalence classes, called quasi-isomorphism, of complexesK• :· · · −→ K n−1 −→ K n −→ K n+1 −→ · · ·of quasi-coherent sheaves K n of OX -modules such that the cohomology groupH i (K • ) is coherent for every i and zero except a ﬁnite number of i ([H]). This
category is considered as a (cohomological) completion of the category of coherent sheaves.4
The following is the main result of [M2].
Theorem 1.5

ˆˆ = (−1X )∗ [g],where −1X is the isomorphism x → −x of X and [g] is the g-shift operator of acomplex. In particular,ˆ is an equivalence between the derived categories Dbc (X)and Dbc (X̂).
This theorem has a several variants. Recall that the Grothendiek K-group
K(X) is the abelian group whose generators are the isomorphism classes [F ]of coherent sheaves F on X and relations are [F1 ] − [F2 ] + [F3 ] for all exact
sequences
0 −→ F1 −→ F2 −→ F3 −→ 0
on X. We obtain the well-deﬁned homomorphismˆ: K(X) −→ K(X̂) by

∗
(1.6)
[F ] →(−1)i [Ri πX̂,∗ (πXF ⊗ P)]iand the duality ˆˆ= (−1)g (−1X )∗ . We have similar duality ˆ for the Chow group•
CH (X) ⊗ Q and singular cohomology group H ∗ (X, Z) by using the Chern
characters of of the Poincaré bundle P. We have the commutative diagram:(1.7)

Dbc (X) −→ K(X)↓↓
Dbc (X̂) −→ K(X̂)

Chern char.

−→−→CH • (X)Q↓CH • (X̂)Q

H ∗ (X, Q)↓−→ H ∗ (X̂, Q)−→

4 Compare with the fact that L2 (Rn ) is a completion of the space of usual functions.

3

where the downward arrows are all Fourier isomorphisms. The most right one
is simple. Its degree i part is just the isomorphism
(1.8)

i


H 1 (X, Q) −→2g−i


H 1 (X̂, Q)of the complementary exterior products for the pair of mutually dual vector
spaces H 1 (X, Q) and H 1 (X̂, Q).
The group action on the derived category Dbc (X), which we are going todiscuss, has grown out from the following:
Observation 1.9 ([M2], p.163) Let (X, L) be a principally polarized abelianvariety. We identify X and its dual X̂ by the isomorphism φL : X −→ X̂ (see(4.3)). If we neglect the shift of complexes, then the relation between the two
equivalences
F → F̂ , (Fourier transformation)
andF → F ⊗ L,

(basic twist)




0 1
1 1
b
of Dc (X) is the same as the two matrices
and. In other−1 0
0 1
words, the modular group SL(2, Z) acts on Dbc (X) (modulo shift).
If we look for group actions on the category Dbc (X) for general abelian varieties, then we immediately ﬁnd the following two actions:(1.10) If φ is an automorphism of X as variety, then φ∗ : Dbc (X) −→ Dbc (X) is
an autoequivalence. Therefore the automorphism group of X acts on Dbc (X).
(1.11) If L is a line bundle on X, then F → F ⊗ L is an autoequivalence.
Therefore the Picard group of X acts on Dbc (X).
The automorphism group of X as variety is generated by all the translations Ta , a ∈ X, and the automorphisms as Abelian variety. Hence both
X and the dual abelian variety X̂ act on Dbc (X). In the sequel we consider actions and autoequivalences modulo these actions. Two discrete groups,that is, the automorphism group Aut X of X and the Néron-Severi group
NS(X) = Pic X/Pic 0 X ⊂ H2 (X, Z) still act on Dbc (X) by (1.10) and (1.11),
respectively.
Problem 1.12 Does Dbc (X) has autoequuivalences other than the semi-direct
product (Aut X) · NS(X)?


The answer is yes by virtue of the Fourier equivalence Dbc (X) −→ Dbc (X̂).
In fact, the Picard group Pic X̂ of the dual abelian variety X̂ acts on Dbc (X̂).
Though the action of Pic 0 X̂  X is translation Ta∗ , the action of the NéronSeveri group NS(X̂) does not belong to (Aut X) · NS(X). A better answer to thequestion is this:4

Deﬁnition 1.13 We denote an endomorphism φ of the product abelian variety
X × X̂ in the matrix form:

 

a b
End (X)
Hom (X, X̂)∈= End (X × X̂).
c d
Hom (X̂, X)
End (X̂)Then the unitary group 5 U (X × X̂) associated with an abelian variety X is thegroup consisting of all automorphisms φ which satisfy


 

a b
1X 0
dˆ −b̂
=
.
c d
0 1X̂
−ĉ â
Theorem 1.14 The group U (X × X̂) acts on the derived category Dbc (X) ofan abelian variety X modulo shift of complex (and modulo the actions of X andX̂).
The functors induced from L ∈ NS(X), ϕ ∈ Aut X and M ∈ NS(X̂) are
included in this action and the following matrices correspond to them:

 



1 φL
1 0
ϕ
0
,
,
.
φM 1
0 1
0 ϕ̂−1
Example 1.15 (1) If (X, L) is a principally polarized abelian variety, thenthe group U (X × X̂) contains


aX
bφL
a, b, c, d ∈ Z, ad − bc = 1  SL(2, Z)
cφ−1
dX̂
L
as its subgroup. They coincide when End (X)  Z.
(2) When End (X) = Z, U (X × X̂) is isomorphic to the Hecke group
a b
Γ0 (N ) =c ≡ mod N ⊂ SL(2, Z),c dwhere N is the smallest natural number which annihilates the kernel of thegenerator φ : X −→ X̂ of Hom (X, X̂)  Z.
(3) If X is the product of g copies of the same elliptic curve E (or thesame abelian variety), then U (X × X̂) contains a subgroup isomorphic to thesymplectic group Sp(2g, Z). They coincide when End (X)  Z.


0 I2g
We endow H 1 (X×X̂) = H 1 (X)⊕H 1 (X)∨ with the inner product.
I2g 0Then the group U (X × X̂) coincides with the group of Hodge isometries preserving this inner product.
We improve the theorem using the spin representation in §3. (U (X × X̂) has
no natural action on K(X) or CH(X)Q .)5 If we tensor Q, then this group is the SL(2) over the algebra End (X) with the Rosati
Qinvolution  . See §5.

52Metaplectic representation and spin representation

First we review these representations a little bit. We consider the endomorphism
algebra of the polynomial ring k[x1 , . . . , xn ] as vector space. The multiplications
by xi and partial derivations ∂/∂xi , i = 1, 2, . . . , n, form a 2n-dimensional
subspace, which we denote by V . The subalgebra W generated by them iscalled the Weyl algebra. The commutator [A, B] = AB − BA induces a nondegenerate skew-inner product on V and makes W a Lie algebra.6 Moreover,the subspace spanned byxi xj ,

xi

∂
1
+ δij
∂xj
2

and

∂2
∂xi ∂xj

is a Lie subalgebra of W and isomorphic to the symplectic Lie algebra sp(V ) 
sp(2n) of V . Hence the polynomial ring k[x1 , . . . , xn ] is a representation ofsp(2n). The metaplectic representation is the lift of this Lie algebra representation to a unitary representation of a Lie group in the case k = R.7 This action
does not lift to Sp(2n, R) but to its double covering denoted by M p(2n, R).
Moreover, the space is no more the polynomial ring but itscompletion L2 (Rn ).
This story goes similarly when the variables x1 , · · · , xn anti-commutes. We•
replace the polynomial ring with the exterior algebra(x1 , . . . , xn ) and consider its endomorphism algebra
End k

•


(x1 , . . . , xn )as vector space. This algebra is generated by n multiplications xi and n partial
derivations8 ∂/∂xi . Let V be the subspace sppened by these 2n generators. The
anti-commutator [A, B]+ = A◦B +B ◦A induces an inner product on this vector•(x1 , . . . , xn )space V .9 The Cliﬀord algebra is the pair of this algebra End k
and this generating vector space V . We denote it by C . The vector subspacespanned by∂1∂2
xi ∧ xj , xi ∧+ δij and∂xj
2∂xi ∂xj
is a Lie subalgebra 10 isomorphic to the orthogonal Lie algebra so(V )  so(2n)•(x1 , . . . , xn ) is a representation of so(2n). The (group-theoretic)of V . Hence
6 The (2n + 1)-dimensional vector space V ⊕ C · 1 is also a Lie subalgebra. This is called a
Heisenberg algebra. sp(V ) is the normalizer of this in W .
7 Metaplectic representation is called the Weil representation in arithmetic context. This
plays a crucial role in the theory of modular functions and theta functions. See [11], [12], [5]and [10]. Metaplectic representation appears in Frenel optics also. See[3].
8 In literatures, the symbol x ?u is used in place of ∂u/∂x .
i
i
9 The (2n + 1)-dimensional space V ⊕ C · 1 is closed under anti-commutator and normalizedby so(V ). This is called a Heisenberg superalgebra.
10 The anti-commutator [ , ]+ and the commutator [ , ] are the same on the even part ofalgebras.

6

spin representation is a lift of this Lie algebra representation to a representation of an algebraic group. The even part and odd parts are both irreducible
representations and called half spin representations. Similar to the symplectic
case, the Lie algebra representation does not lift to SO(2n) but its double cover,Spin(2n). The construction is as follows:
The even invertible elements g ∈ C such that gV g −1 = V form a group
under multiplication. This is called the special Cliﬀord group and denoted by
CSpin(2n). This acts on V preserving the inner product and we obtain theexact sequence(2.1)

1 −→ Gm −→ CSpin(2n) −→ SO(V ) −→ 1.

The spinor group is the subgroup of this Cliﬀord group consisting of those gwith spinor norm11 1. Hence we have the exact sequence(2.2)

1 −→ Spin(2n) −→ CSpin(2n)sp. norm−→

Gm −→ 1.

Combining the two exact sequences, we have
(2.3)

1 −→ {±1} −→ Spin(2n) −→ SO(V ) −→ 1,

which shows Spin(2n) is a double over of SO(2n).

3Action of U Spin(X × X̂) on the derived category

As Theorem 1.14 shows the derived category Dbc (X) of an abelian variety X hasa bigger symmetry than its automorphism and Picard group. This is similar
to the Hamiltonian formalism in classical and quantum mechanics and also to
the contact transformation of Lie in the classical theory of partial diﬀerential
equations. We recall the former a little bit (cf. [A]). Let
m

d2 2x
= F2
dt2

in Rn

be the equation of motion of the location 2x = (x1 , . . . , xn ) of particles. This istransformed to the canonical equations
(3.1)

dxi
∂H
=
,
dt
∂pi

∂H
dpi
=−
dt
∂xi

i
on the phase space Rn ⊕ Rn by introducing the new variables pi = m dx
dt , where
H = H(2x, p2) is the Hamiltonian function. The symmetry of the symplectic
group Sp(2n) thus obtained is a great advantage of this formalism.

11 There is a unique anti-automorphism ∗ of C which is identity on V . If g ∈ CSpin(2n),then gg ∗ = N (g) · id. This N (g) is called the spinor norm.

7

When we pass to the quantum mechanics, the problem changes from the
motion of particles in Rn to the (wave) functions on the phase space. But thesymmetry of Sp(2n) is still vital. It survives as the action of metaplectic groupM p(2n) reviewed in the previous section. We are now making an analogy ofthis for Abelian varieties. The dictionary is this:(3.2)
Rnfunction f (x)multiplication f (x)g(x)integral transformation K(x, y)f (y)dy
Hilbert space L2 (Rn )polynomial ring C[x1 , . . . , xn ]Heisenberg group1 −→ C∗ −→ H −→ Rn ⊕ Rn −→ 0
2:1
M p(2n) −→ Sp(n, R)

Abelian variety X
sheaf F
tensor product F ⊗ G
integral functor π∗ (K ⊗ τ ∗ F )derived category Dbc (X)•exterior algebra(x1 , . . . , xn )odd Heisenberg group12
2:1

U Spin(X × X̂) −→ U (X × X̂)Restricting the exact sequence (2.3), we have the exact sequence
1 −→ {±1} −→ Spin(4g, Z) −→ O(H1 (X × X̂), Z) −→ 1∪
U (X × X̂).
Deﬁnition 3.3
X̂).

U Spin(X × X̂) is the inverse image in Spin(4g, Z) of U (X ×Theorem 3.4 The group U Spin(X × X̂) acts on the derived category Dbc (X)modulo even shift of complexes. The nontrivial element z of the kernel of
2:1
U Spin(X × X̂) −→ U (X × X̂) shifts complexes by one under this action, i.e.,z
K • → K • [1].
The group U Spin(X × X̂) also acts on K(X), CH(X) and H • (X) and theaction of z is the multiplication by −1. Note that the even partH ev (X) =ev


H 1 (X)is the half spin representation of Spin(4g), by deﬁnition.
Theorem 3.5 The Chern character homomorphism ch : Dbc (X) −→ H ev (X)is equivariant with respect to the action of U Spin(X × X̂) ⊂ Spin(4g). Moreover,χ(E, F ) = β(ch(E), ch(F ))12 The exact sequence 1 −→ C∗ −→ P ∗ −→ X × X̂ −→ 0 was put here in the original article
instead.

8

holds for a pair of coherent sheaves on X, where β is the invariant bilinear formon the half spin representation.
The second half is the equivariance of the Riemann-Roch formula with respect to the action of U Spin(X × X̂). The bilinear form β is called the fundamental polar form in [Ca, §102]. This is symmetric or anti-symmetric according
as g is even or odd. 13

4

Semi-homogeneous sheaf

The Fourier functor for sheaves is an integral functor whose kernel (sheaf) is
the Poincaré line bundle P. We prove Theorems 1.14 and 3.4 by replacing P
with the universal family of semi-homogeneous sheaves. 14
Let E be a coherent sheaf on an abelian variety. We deﬁne a subgroup of
X × X̂ by(4.1)
Φ(X) = {(X, α)|Tx∗ E  E ⊗ Pα }.
This is also a subvariety15 . We denote its neutral component by Φ0 (E).
Deﬁnition 4.2 A coherent sheaf E on an abelian variety X is semi-homogeneous
if dim Φ(E) = dim X holds.
If E is a (holomorphic) vector bundle, then the ﬁrst projection Φ(E) −→ X
is ﬁnite. Hence the above deﬁnition is compatible with the one in [M1], that is, avector bundle E is semi-homogeneous if for every x ∈ X there exists P ∈ Pic 0 X
such that Tx∗ E  E ⊗ P . A line bundle L is always semi-homogeneous and thesubgroup Φ(L) is the graph of the homomorphism(4.3)

φL : X −→ Pic 0 X = X̂,x → Tx∗ L ⊗ L−1

associated with L. Another typical example of semi-homogeneous sheaf is asheaf which has ﬁnite support. In this case, the connected component Φ0 (E) is
0 × X̂. The category of semi-homogeneous sheaves is closed under two operations, the pull-backs π∗ and the direct images π ∗ by isogenies π. It is also closed
under Fourier transformation. For example, the category of artinian sheaves
and that of homogeneous vector bundles are interchanged by the Fourier functor ([M2, (2.9)]).
We look at the Abelian subvariety Φ0 (E) more closely. The homomorphism(4.3) deﬁned for a line bundle L is characterized by the symmetric property
13 Note that χ(E, F ) = (−1)g χ(F, E) holds by Hirzebruch-Riemann-Roch theorem or by
Serre duality.
14 The basic reference of this section is [M1]. The proof of results is similar to the case of
semi-homogeneous vector bundles, or reduces easily to that case. Note that if F is a semihomogeneous sheaf, then the Fourier transform of F ⊗ L is a semi-homogeneous vector bundle
for a suﬃciently ample line bundle L.
15 More precisely, a natural scheme structure on Φ(E) is deﬁned as in [M1] using the technique of [Mum §10].

9

φ̂L = φL among all homomorphisms from X to X̂. So the image NS(X) of
Pic X in the exact sequence(4.4)

0 −→ Pic 0 X −→ Pic X
L−→ Hom (X, X̂)→φL

coincides with the subgroup {φ̂ = φ} in Hom (X, X̂). When we regard the linear
map H1 (φL ) : H1 (X) −→ H1 (X̂) as a tensor w ∈ H 1 (X) ⊗ H 1 (X), φ̂L = φL
is equivalent to the anti-symmetricity of w. This tensor w is nothing but the2
Chern class c1 (L) ∈ H 2 (X) =H 1 (X) of L. The anti-symmetricity is also
equivalent to saying that the graph of H1 (X)−→ H1 (X̂) is totally isotropic
0 I2g
with respect to the inner product.
I2g 0
Φ0 (E) ⊂ X ×X̂, or more 
precisely, the subspace H1 (Φ0 (E))
0 I2g
is totally isotropic with respect to. In particular, we have dim Φ(E)
I2g 0
≤ g.
Proposition 4.5

Remark 4.6 Let ψ : Z −→ Ẑ be a homomorphism from an abelian variety
to its dual. We can deﬁne a homomorphism f : Y −→ Z be isotropic withrespect to ψ by the property fˆ ◦ φ ◦ f = 0. Then the totally isotropicness inthe proposition is equivalent to that of the natural inclusion homomorphism
Y <→ X × X̂ with respect to the homomorphism


0
1X
J :=
: X × X̂ −→ X̂ × X = dual of X × X̂.
−1X̂ 0(It will be natural to call this homomorphism J skew-polarization since it satisﬁes Jˆ = −J.)
Recall that a maximally totally isotropic subspace, or a Lagrangian, determines a vector in the spin representation, which is unique up to constantmultiplication ([Ch], [Ca]). We call it the spinor coordinate. In our situation, E
is semi-homogeneous and H1 (Φ0 (E)) ⊂ H1 (X) ⊕ H1 (X̂) is a Lagrangian. Hence
we obtain an element of the cohomology group H ∗ (X) as spinor coordinate.The
following is important:(4.7) For a semi-homogeneous sheaf E its Chern character ch(E) is the spinor
coordinate of H1 (Φ0 (E)).
(4.8) The Chern character of semi-homogeneous vector bundle E is equal toc1 (E)H 1 (X),
)∈
r(E)
ev

r(E) exp(

where r(E) is the rank of E. For a general semi-homogeneous sheaf E, theChern character ch(E) is equal to [Y ] ∧ exp w for suitable w ∈ H 1,1 (X, Q),
where [Y ] is a connected component of the support of E.
10

A semi-homogeneous sheaf E has a ﬁltration
E = En ⊃ En−1 ⊃ · · · ⊃ E2 ⊃ E1 ⊃ E0 = 0such that all successive quotients Ei /Ei−1 are simple, semi-homogeneous andhave the same Abelian subvariety, i.e. Φ0 (Ei /Ei−1 ) = Φ0 (E). (A semihomogeneous vector bundle E is semi-stable and the above ﬁltration coincides
with the JHS-ﬁltration.) Therefore, simple ones are important. They are classiﬁed in the following way: (A sheaf E is simple if End O (E) = C.)
Theorem 4.9 (1) If E is simple and semi-homogeneous, then Φ(E) is connected.
(2) (Riemann-Roch) For a pair of simple semi-homogeneous sheaves E1 andE2 , we have|Φ(E1 ) ∩ Φ(E2 )|2
χ(E1 , E2 ) =0according as the intersection Φ(E1 ) ∩ Φ(E2 ) is ﬁnite or not. In particular, wehaveχ(E)2 = |Φ(E) ∩ X × 0| and r(E)2 = |Φ(E) ∩ 0 × X̂|for a semi-homogeneous sheaf E.
(3) Let E1 and E2 be as above. Then Φ(E1 ) and Φ(E2 ) coincide if and only
if there exist a ∈ X and α ∈ X̂ such that E1  Ta∗ E2 ⊗ Pα . When both E1 andE2 are vector bundles, then these are also equivalent to the condition
c1 (E1 )c1 (E2 )=r(E1 )
r(E2 )

in H 1 (X, Q).

(4) For a Lagrangian Abelian subvariety Y of X × X̂, there exists a simple
semi-homogeneous sheaf E on X such that Φ(E) = Y .
Remark 4.10 For a semi-homogeneous vector bundle E, the Abelian subvariety Φ0 (E) coincides with the image of the homomorphism(rX , φL ) : X −→ X × X̂
where r is the rank of X and L is determinant det E. Hence the semi-homogeneous
vector bundles modulo deformation are parameterized by NS(X) ⊗ Q by thecorrespondence E → φdet E /r(E). All semi-homogeneous sheaves are parameterized by the rational points of the projective line over the Jordan algebra
NS(X), which is the natural compactiﬁcation of NS(X).
All deformations of a simple semi-homogeneous sheaf E is isomorphic toTa∗ E ⊗ Pα for some a and α and the moduli space M (E) is isomorphic to thequotient Abelian variety (X × X̂)/Φ(E). Now assume that there exists anothersemi-homogeneous sheaf F with χ(E, F ) = ±1. By (2) of the theorem, X × X̂is the direct product Φ(E) × Φ(F ). Moreover on the product X × M (E), there∗
exists a universal family E. The direct image RπM,∗ (πXF ∨ ⊗ E) is a line bundle
on M (E) (modulo shift). We normalize E so that this line bundle is trivial.
11

Proposition 4.11

The integral functor∗
RπX,∗ (E ⊗ πM?) : Dbc (M (E)) −→ Dbc (X)is an equivalence of categories and send the sky-scraper sheaf k(0) to E and thestructure sheaf OX to F (modulo shift).
Using Theorem 4.9, especially (4) of it, we can eliminate E and F from theabove statement.
Proposition 4.12 Assume that an isomorphism

 

a b
Hom (Y, X) Hom (Y, X̂)ω=∈= Hom (Y × Ŷ , X × X̂)c d
Hom (Ŷ , X) Hom (Ŷ , X̂)satisﬁes


a b
c d



dˆ −b̂
−ĉ â




=

1X
0

0
1X̂


.

Then there exists an integral functorΩ : Dbc (Y ) −→ Dbc (X)whose kernel is a universal family of semi-homogeneous sheaves and such that∗Ω(Ty∗ G ⊗ Pα )  Ta(y)+c(α)Ω(G) ⊗ Pb(y)+d(α)

holds 16 for every G ∈ Dbc (Y ), y ∈ Y and α ∈ X̂.
Note that the assumption of theorem implies that the images of Y × 0 and
0 × Ŷ are both totally isotropic subvarieties of X × X̂. Theorem 1.14 is the
special case Y = X of this proposition.
Remark 4.13 In the notation of §2, there exists a standard isomorphism:: :from the exterior algebra
isomorphism is••V −→ CV to the Cliﬀord algebra ([SMJ]). In our case the: : : H ∗ (X × X̂) −→ C.

The equivalence of categories associated withω̃ ∈ U Spin(X × X̂) ⊂ C,
a lift of ω, is the composite of two functors: one is the integral functor whose
kernel is the semi-homogeneous sheaf F on X × X̂ with : ch(F ⊗ P ±1 ) := ω̃and the other is the Fourier functor. This has a similarity with the formalismof Fourier integral operators in the theory of partial diﬀerential equations ([D]).
16 This functor Ω may be called semi-homogeneous by this property. Another possible namewill be a spinor functor.

125

Action on a tube domain

The unitary group U (X × X̂) of an abelian variety X is the set of integral pointsof an algebraic group. The Lie algebra17 of this algebraic group is

 

a b
dˆ −b̂
(5.1)
u(X × X̂) =+= 0 ⊂ End (X × X̂)c d−ĉ â
and decomposed into three parts:u(X × X̂)
(5.2)



ϕ
φM

φL
−ϕ̂



NS(X) ⊕End (X)⊕ NS(X̂)



.
↔

(L,

ϕ,

M)

In this section we study the structure of this Lie algebra u(X × X̂) over Q.
We ﬁx an ample line bundle L on X and let φL : X −→ X̂ be the homomorphism in (4.3). Since this is an isogeny, φ−1
L is deﬁned as element of
Hom (X̂, X)Q . The map
End (X)Q " a → a = φ−1
L aφL ∈ End (X)Q
is called the Rosati involution. The positivity Tr(aa ) > 0 is famous ([Mum,§21]). By (4.4) the Néron-Severi group NS(X)Q is isomorphic to the subspace{a = a}. This subspace is closed under the product a ◦ b = (ab + ba)/2. Hence
NS(X)Q becomes a Jordan algebra whose unit is φL . This Jordan algebra isformally real by the above mentioned positivity ([Mum, ibid.]).
Now we consider the element


1
0
φL
(5.3)
HL =
−φ−1
0
2
L
of the Lie algebra u(X × X̂)Q . By easy computation, u(X × X̂)Q decomposes
into the direct sum of 0-eigenspace


a
φL b
(5.4)
k=
a, b ∈ End (X)Q , a + a = 0, b = b−bφL −âand (−1)-eigenspace

(5.5)
p=

a
bφ−1
L

φL b
−â


a, b ∈ End (X)Q ,

a=a,

b=b

of (ad HL )2 . By the positivity of Rosati involution , the Lie algebra kR is ofcompact type. Moreover, ad HL induces a complex structure on p.
17 a) This Lie algebra acts on the cohomology group H ∗ (X). The subaction of

a
cφL̂

bφL
−a



a, b, c ∈ R

is the sl(2)-action of the Lefschetz decomposition (see [2]).
b) This kind of Lie algebras is studied for general varieties in [4].

13Proposition 5.6 The Lie algebra u(X × X̂) is of Hermitian type and HL is
its H-element (see [S, Chap. II] for the terminology.)
In particular, the Lie group U (X × X̂)R acts transitively on a boundedsymmetric domain. In our case the domain is the tube domain√
DX = NS(X)R + −1B ⊂ NS(X) ⊗ C,
where B is the set of positive elements of NS(X)R , that is, the ample cone.
So the discrete group U (X × X̂), studied in the previous sections, acts thisdomain DX discontinuously. It will be interesting to study how the category
relates with the quotient variety U (X × X̂)\DX , Shimura variety, Hodge group,
Hodge structures, the Kuga-Satake Abelian varieties of K3 surfaces, etc.. 18 But
we do not pursue them here.
Remark 5.7 When L is a principal polarization, then the H-element of (5.3)
corresponds to the Fourier transformation ˆ.
Example 5.8 Let X be the product E × · · · × E of elliptic curve as in (3) of(1.15). If End E = Z, the Lie algebra u(X × X̂)R is the symplectic Lie algebrasp(2g, R) and k is the unitary group u(g). Hence the tube domain DX is theSiegel upper half space of degree g. When E has a complex multiplication, thenu(X × X̂)R is isomorphic to the general linear Lie algebra gl(2g, R) and k is asubalgebra isomorphic to u(g) ⊕ u(g). The domain DX is of dimension g 2 andof type Ig,g .
October 3, 1994

References[A]

Arnold, V.I.: Mathematical Methods in Classical Mechanics, Springer
Verlag, 1978.

[Ca]

Cartan, E.: The Theory of Spinors, Hermann, 1966.

[Ch]

Chevalley, C.: The Algebraic Theory of Spinors, Columbia Univ. Press,
1954.

[D]

Duistermaat, J.J.: Fourier integral operators, Courant Inst., 1973.

[H]

Hartshorne, R.: Residues and duality, Lecture Notes in Math. 20(1966).

[M1]

Mukai, S.: Semi-homogeneous vector bundles on an abelian variety. J.
Math. Kyoto. Univ., 18 (1978), 239–272.

[M2]—— : Duality between D(X) and D(X̂) with its application to Picard
sheaves, Nagoya Math. J., 81(1981), 153–175.

18 Cf. [9], [1] and [13].

14

[Mu]

Mumford, D.: Abelian Varieties, Oxford Univ. Press, 1970.

[S]

Satake, I.: Algebraic Structure of Symmetric Domains, Iwanami Shoten
and Princeton Univ. Press, 1980.

[SMJ] Sato, M. Miwa, T. and Jimbo, M.: Holomorphic quantum ﬁeld I, Publ.
Res. Inst. Math. Sci., Kyoto Univ., 14(1978), 223-267.
Added in translation[1] Deligne, P.: La conjecture de Weil pour les surfaces K3, Inv. math.
15(1972), 206-226.
[2] Griﬃths, P.A. and Harris, J.: Principles of Algebraic Geometry, WileyInterscience, New York, 1978.
[3] Guillemin, V. and Stenberg, S.: Symplectic Techniques in Mathematical
Physics, Cambridge Univ. Press.
[4] Looijenga, E. and Lunts, V.A.: A Lie algebra attached to a projective
variety, Invent. Math. 129(1997), 361-412.
[5] Lion, G. and Vergne, M. : The Weil Representation, Maslov Index and
Theta Series, Birkhäuser, 1978.
[6] Mukai, S.: On classiﬁcation of vector bundles on an abelian surface, Res.
Inst. Math. Sci. Kōkyūroku 409(1980), 103-127 (in japanese).
[7] —— :On the moduli space of bundles on K3 surfaces, I, in ‘Vector Bundles on Algebraic Varieties ’, Tata Institute of Fundamental Research,
Bombay, 1987, pp. 341–413.
[8] —— : Duality of polarized K3 surfaces, to appear in Proceedings ofEuroconference on Algebraic Geometry, (K. Hulek and M. Reid eds.),
1998.
[9] Mumford, D.: Families of abelian varieties, Proc. Symp. in Pure Math.
9(1966), 347-351.
[10] —— , Nori, M. and Norman, P.: Tata Lectures on Theta III, Progressin Math. 97, Birkhäuser, 1991, Boston.
[11] Weil, A.: Sur certain groupes d’operateurs unitaires, Acta Math.
111(1964), 143 211.
[12] —— : Sul la formula de Siegel dans l théorie de groupes classique, Acta
Math. 113(1965), 1-87.
[13] —— : Abelian varieties and the Hodge ring, Collected works, III, pp.
421-429.
Added after translation19
19 I thank Prof. A. Tyurin who taught the author this reference.

15[14] Orlov, D.: On equivalence of derived categories of coherent sheaves on
abelian varieties: e-print, alg-geom/9712017.
Graduate School of Mathematics
Nagoya University
Furō-chō, Chikusa-ku
Nagoya, 464-01, Japane-mail: mukai@math.nagoya-u.ac.jp

Current address (until 6th April 1998)
Mathematics Institute
University of Warwick
Coventry CV4 7AL England
mukai@maths.warwick.ac.uk

16