CHARACTERISTIC CLASSES

OF SINGULAR MANIFOLDS

WITH TORUS ACTION

Andrzej Weber

University of Warsaw

Liverpool, June 2012

Plan

Equivariant cohomology for torus action

Localization Theorem

• calculus of symmetric rational functions

Method of computation of equivariant characteristic classes

• Chern (-Schwartz-MacPherson) class

• Todd class, Hirzebruch L-class, χy genus

Positivity conjectures

Equivariant cohomology

Torus T = (S1)n (or (C∗)n) acts on a manifold M .

Denote by t ' Rn the Lie algebra of T

Equivariant cohomology

H∗T(M) = H∗(

Ω•(M)T ⊗ Sym(t∗) , d)

is a module over

H∗T(pt) = Sym(t∗) = R[t1, t2, . . . tn]

H∗T(M) ' H∗(M)⊗ R[t1, t2, . . . tn]

as R[t1, t2, . . . tn]-modules, provided that M is acompact algebraic manifold and T acts algebraically

Localization

(Almost) everything about equivariant cohomology can beread from some data concentrated at the fixed points.

For example: Euler characteristic Euler(M) = Euler(MT) .

Borel localization theorem

The restriction to the fixed set

H∗T(M)−→H∗T(MT)

is an isomorphism after inverting t∗ − 0.

Atiyah-Bott or Berline-Vergne formula

Assumption about fixed points:MT = p0, p1, . . . pn is discrete.

For p ∈ MT : Define the Euler class ep as the product ofweights of T appearing in the tangent representation.

Integration Formula

For a ∈ H∗T(M) the integral is the sum of fractions∫M

a =∑p∈MT

a|pep,

where∫M

: H∗T(M) −→ H∗−dim(M)T (pt) = R[t1, t2, . . . tn]

i.e. the integral can be expressed by the local data.

Example of computation

M = P2 = P(C3) , T = (C∗)3

MT = p0, p1, p2 fixed points

c1 := c1(O(−1)) Chern class of the tautological bundle

We apply Berline-Vergne formula to compute∫P2 cn

1 :

tn0(t1 − t0)(t2 − t0)

+tn1

(t0 − t1)(t2 − t1)+

tn2(t0 − t2)(t1 − t2)

=

Resz=∞zn

(z − t0)(z − t1)(z − t2)

set z = w−1

= Coefficient of w inw 3−n

(1− w t0)(1− w t1)(1− w t2)

The goal and strategy

Use Localization Theorem to compute some invariants ofT-invariant singular varieties X ⊂ M .

The main interest:

• Chern class

• or Todd class or Hirzebruch L-class, in general χy genus.

These three invariants have generalization for algebraiccomplex singular varieties

We will give an example of computation when

• M is the Grassmannian Gm(Cn)

• X is a Schubert variety.

We compute local contributions in global formulas.

Example: Chern class c(X )

Euler(X ) =

∫Xc(X )

Suppose that X ⊂ M is an invariant subvariety in a T-manifoldwith isolated fixed points:

Euler(X ) =

∫Mc(X ) =

∑p∈XT

c(X )|pep

Suppose the point p ∈ XT is smooth. Then the contribution tothe Euler characteristic is equal to

c(X )|p =∏

(1 + wi ) ·∏

nj ∈ H∗T(pt) = Sym(t∗) ,

• wi are the weights of tangent representation TpX

• nj are the weights of normal representation.

Example: Chern class c(X ), cont.

Suppose that all but one points of X are smooth.

Then one can compute

c(X )|psing = epsing

Euler(X )−∑

p∈XT−psing

c(X )|pep

To compute the local contribution of the Chernclass at singularities one does not even have toknow the definition. It is enough to know it exists!

Example: Grassmannian Grass2(C4), its GKM-graph

Schubert variety of planes in C4 which are 6t lin(e1, e2)

Schubert variety of planes in C4 which are 6t lin(e1, e2)

c(X )|p12 =(5− (t4 − t1)(1 + t2 − t1)(1 + t2 − t3)(1 + t4 − t3)

(t4 − t1)(t2 − t1)(t2 − t3)(t4 − t3)

− (t4 − t2)(1 + t1 − t2)(1 + t1 − t3)(1 + t4 − t3)

(t4 − t2)(t1 − t2)(t1 − t3)(t4 − t3)

− (t3 − t1)(1 + t2 − t1)(1 + t2 − t4)(1 + t3 − t4)

(t3 − t1)(t2 − t1)(t2 − t4)(t3 − t4)

− (t3 − t2)(1 + t1 − t2)(1 + t1 − t4)(1 + t3 − t4)

(t3 − t2)(t1 − t2)(t1 − t4)(t3 − t4)

)multipied by

(t3 − t1)(t3 − t2)(t4 − t1)(t4 − t2)

Schubert variety of planes in C4 which are 6t lin(e1, e2)

The result with t1 = −s1, t2 = −s2:

Inductive computation of local equivariant Chern classes

• smooth point• already computed Chern class• unknown Chern class

This procedure can be applied to compute local Chern class ofthe Schubert variety of codimension one in Grassn(C2n). Thesingularity is the determinant singularity:

A ∈ M(n × n) : det(A) = 0 .

3-spaces in C6 which are 6t lin(e1, e2, e3)

It is more economic to present the result for the Chern classes ofthe complement of the Schubert variety:

c(open cell) = c(Grassmanian)− c(Schubert variety) .

The result is given in monomial basis of symmetric functions ins1, s2, s3 and t4, t5, t6.

4-spaces in C8 which are 6t lin(e1, e2, e3, e4)

The result in monomial basis of symmetric functions in s1, s2, s3, s4and t5, t6, t7, t8

Computation of local equivariant Chern class in Grass4(C8)

Already for Grass4(C8) appears a problem with the size ofthe expressions since dim(Grass4(C8)) = 16 and dim(T ) = 8

In a polynomial of degree 15 in 8 variables there are

245 157 monomials.

The expression is a sums of 79 fractions with factors ti − tjin denominators.

Positivity

The local Chern classes are positive combinationof monomials in ti and sj .

This supports the conjecture of Aluffi andMihalcea that the Chern class (or even equivariantChern class) of a Schubert variety is an effectivecycle.

So far the conjecture is proved for Grassk(Cn),k ≤ 3.

Hirzebruch χy -genus

To compute Chern class we have used the rule

line bundle L 7→ 1 + c1(L)

If we replace that rule by

line bundle L 7→ (1 + y e−t)t

1− e−t,

where t = c1(L), we obtain the χy -genus.

y = −1 Euler characteristic (top Chern class)

y = 1 Hizebruch L-class

y = 0 Todd class

Hirzebruch L-class

Smooth point in a line

(1 + e−t)t

1− e−t

For the complement of a smooth point(1+e−t)t

1−e−t − t = 2e−t t1−e−t

For the complement of the determinantsingularity: set Ti = e−ti and Si = e−si

2n∏i<j

(Si + Sj)∏i<j

(Ti + Tj)∏i

(SiTi)∏i ,j

si + tj1− SiTj

Todd class

Smooth point in a linet

1− e−t

For the complement of a smooth pointt

1−e−t − t = e−t t1−e−t

For the complement of the determinantsingularity: set Ti = e−ti and Si = e−si∏

i

(SiTi)∏i ,j

si + tj1− SiTj

χy -genus: Smooth point in a line

(1 + y e−t)t

1− e−t

The formulas for χy genus seem to be complicated. Afterrestriction to one dimensional torus acting by scalar multiplication:

n = 1 : t(1−T )(y + 1)T

n = 2 : t4

(1−T )4(y + 1)2T 2

(1− y + 4yT + (y2 − y)T 2

) n = 3 : t9

(1−T )9(1 + y)3T 3

((1− 2y + 2y2 − y3)

+(9y − 18y2 + 9y3)T+(45y2 − 45y3)T 2

+(−y − 14y2 + 94y3− 14y4− y5)T 3

+(−45y3 + 45y4)T 4

+(9y3 − 18y4 + 9y5)T 5

+(−y3 + 2y4 − 2y5 + y6)T 6)

χy genus for determinant variety in M(4× 4)

Further directions of work

Develop a calculus of symmetric rational functions

Deduce positivity results for Chern classes

Study equivariant characteristic classes of

• Schubert varieties in homogeneous manifolds G/P ,

• spherical varieties etc.

Understand the structure of expressions for characteristicclasses and its relation to geometry and combinatorics

Thank You