Stratiﬁcations on generic torus orbit closures

Hitoshi Yamanaka (OCAMI)
Toric Topology 2019, Okayama University of Science‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌

Contents
1. Generic torus orbit closures (GTOCs) in T -varieties
2. The case of Schubert varieties of type A (Lee-Masuda’s work)
3. Main results
4. Sketch of proof‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌

1.

Generic torus orbit closures (GTOCs) in T -varieties‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌• X : irreducible complex variety• T = (C∗ )r (r ≥ 1) : algebraic torus of rank r• T acts on X algebraically• X T : the set of T -ﬁxed points in X .
We assume that |X T | < ∞Deﬁnition
A point x ∈ X is called T-generic point if X T ⊂ T · x
The condition
XT ⊂ T · x
is seemingly very extremal.
−→ Is this condition really a “generic condition” ?
‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌Examples• Consider the action C∗ ↷ CP 2 given byt · [x0 : x1 : x2 ] := [x0 : tx1 : t 2 x2 ].
There is no T -generic point.
• Consider the action C∗ ↷ C2 given byt · (x, y ) := (tx, t −1 y ).
The origin (0, 0) is a unique T -generic point.

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Lemma (Y.)
Assume that X is T -equivariantly embedded to a T -variety Xsatisfying the following conditions:• X has a T -invariant open covering {Up }p∈X T s.t. each Up isT -isomorphic to some rational complex T -representation Vp .
• The weights α1 , . . . , αn : T → C∗ of Vp satisfy
hv, αi i > 0 (i = 1, . . . , n)for some v ∈ Hom(T , C∗ ) ⊗Z R ∼= Rn .
• X is covered by {Up }p∈X T .
Then the set of T -generic points of X is given by ∩p∈X T (X ∩ Up ).
In particular T -generic points form a Zariski dense subset in X .

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Remark• A point x ∈ X is called regular point if
dim T · x = max{dim T · y |y ∈ X }.
Regular points also form a non-empty Zariski open set.
• But “regular” and “T -generic” are not equivalent in general:
Let
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1 1 1
X := GL3 (C)/B, g := 0 1 1 , x := gB.
1 0 1
Then x is regular (∵ T · x is a free orbit) but not T -generic.

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Example• G : complex reductive algebraic group• B : Borel subgroup, B − : opposite Borel subgroup• W : Weyl group• Xw = BwB/B, X u = B − uB/B : (opposite) Schubert variety• Xwu := Xw ∩ X u : Richardson variety
X = Xwu satisﬁes the assumptions in above Lemma (X = G /B).
−→ Xwu has T -generic points.
Since Xw = Xwid , Schubert variety Xw has T -generic points.

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2.

The case of Schubert varieties of type A (Lee-Masuda’s work)‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌

In the rest of this talk we focus on Schubert variety Xw of type A.
(G = GLn (C), B = (upper triangular Borel))In the paper
Generic torus orbit closures in Schubert varieties, J. of Comb.
Series. A, 2019,
Lee-Masuda investigate GTOCs Yw in Xw .
Every torus orbit closure in Fl(Cn ) is normal by Carrell-Kurth,Thus Yw is a normal toric variety. Yw can be singular.
Lee-Masuda obtain:• Explicit description of the fan of Yw .
• Combinatorial criterion for the smoothness of Yw• Some interesting conjectures which indicate algebro-geometricsimilarity between Xw and Yw .
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Notation
Let u, w ∈ Sn , u ≤ w and tu(i),u(j) u be the transpositionu = · · · u(i) · · · u(j) · · · =⇒ tu(i),u(j) u = · · · u(j) · · · u(i) · · ·ew (u) be the set of pairs (u(i), u(j)) s.t.
• Let E(1) 1 ≤ i < j ≤ n(2) tu(i),u(j) u ≤ w(3) |ℓ(tu(i),u(j) u) − ℓ(u)| = 1

ew (u) is said to be indecomposable if there is no• (a, b) ∈ Esequenceew (u)(a1 , b1 ), (a2 , b2 ), . . . , (ak , bk ) ∈ E
s.t. k ≥ 2, a = a1 , b1 = a2 , . . . , bk−1 = ak , bk = b.
ew (u))• Ew (u) := (the set of indecomposables in E‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌

Description of dual cones
Note thatYwT = XwT = {u ∈ Sn |u ≤ w }.
We denote by Cw (u) the maximal cone corresponding to u ∈ YwT .
Theorem (Lee-Masuda)
The set of primitive edge vectors of the dual cone
Dw (u) := (Cw (u))∨
is given by{eb − ea |(a, b) ∈ Ew (u)}.

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Remark(1) GTOCs in Xw are mutually isomorphic as T -varieties(this justiﬁes the notation “Yw ”).
This kind of “invariance” is also true for GTOCs in ﬂag Bottmanifolds (Lee-Suh’s work. Explicit description of fans).
(2) Question:
Does the invariance also hold in the setting of Section 1 ?
If the invariance holds for X , it follows thatevery T -generic point in X is a regular point in X .

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3.

Main result‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌

Theorem (Y., a conjecture of Lee-Masuda)
The Poincaré polynomial of Yw is given byXt aw (u)Pt (Yw ) =u∈Sn ,u≤w

whereaw (u) := |{(a, b) ∈ Ew (u)|a < b}|‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌

4.

Sketch of proof‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌

Step 1: Local description of Yw
Recall that Dw (u) is the dual cone of the maximal cone Cw (u)corresponding to u ∈ YwT .
Key Lemma
The monoid Dw (u) ∩ Zn of lattice points in Dw (u) is generated byprimitive edge vectors.
−→ The aﬃne variety
Yw (u) := Max(C[Dw (u)∩Zn ]) (T -inv. open nbd around u ∈ Yw )can be realized as an algebraic set in CEw (u) .

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Step 2: Deﬁning equations
One can show that the deﬁning equations ofYw (u)are given byY(⊂ CEw (u) )x(a,b) =(a,b)∈LY

x(a,b)(a,b)∈R

where• L, R ⊂ Ew (u)XX•(eb − ea ) =(eb − ea )(a,b)∈L(a,b)∈R

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Step 3: Computing BB-cellThanks to the deﬁning equations, one can analyze the BB-cell
Yw+ (u) with respect to the C∗ -action on Yw (u) induced from theregular cocharacterC∗ → T , t 7→ (t, t 2 , . . . , t n ) :Theorem (Y.)(1) Yw+ (u) ∼= CEw (u)<(2) {Yw+ (u)}u∈YwT is an aﬃne paving of Yw .
Theorem in Section 3 follows from above theorem.

‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌Summary• Concept of a T -generic point in a T -variety:(1) T -generic points form a Zariski dense subset under some mildconditions(2) Question on uniqueness of T -isomorphic types of GTOCs in aT -variety• GTOCs in the Schubert variety Xw of type A(1) Lee-Masuda’s conjecture on the Poincaré polynomials ofGTOCs in Xw(2) Paving theorem‌

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Summary• Concept of a T -generic point in a T -variety:(1) T -generic points form a Zariski dense subset under some mildconditions(2) Question on uniqueness of T -isomorphic types of GTOCs in aT -variety• GTOCs in the Schubert variety Xw of type A(1) Lee-Masuda’s conjecture on the Poincaré polynomials ofGTOCs in Xw(2) Paving theorem

Thank you for your attention !

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