Cohomology theories on projective homogeneous varieties

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1 Cohomology theories on projective homogeneous varieties Baptiste Calmès RAGE conference, Emory, May 2011

2 Goal: Schubert Calculus for all cohomology theories Schubert Calculus? Cohomology theory?

3 (Very) classical example of Schubert (1879) How many lines in R 3 intersect 4 given lines? Answer: often 2, but not always... After projectivization: {lines in P 3 } {planes in A 4 } = Gr(2, 4) {lines intersecting D} subvariety V (D)

4 Modern answer using Chow groups subvariety V [V ] CH(Gr(2, 4)) [V (D 1 )] = [V (D 2 )] = [V (D 3 )] = [V (D 4 )] = [V ] CH(Gr(2, 4)) = Z Z Z Z Z Z Ring structure: [V ] [pt] [V ] 4 = 2[pt]

5 General problem: Ring structure of h(g/b) oriented cohomology split semi-simple linear algebraic group Borel subgroup

6 Why is it useful? In our world: Computing invariants using cohomology theories K. Zainoulline s talk (K-theory and Chow groups) In other worlds: Combinatorics of various objects (Young diagrams) Everywhere: Algebraic groups (or compact Lie groups) are fundamental objects

7 G split semi-simple, simply connected linear T W maximal split torus Weyl group over a field k B Borel subgroup containing T α 1,...,α n s 1,...,s n simple roots simple reflexions I =(i 1,...,i q ) sequence of integers is a reduced decomposition of w W if w = s i1 s iq and it is of minimal length

8 open cell Schubert varieties Xw= B wb/b A l(w) Schubert variety X w = B wb/b Bruhat order X w1 X w2 w 1 w 2 Cellular decomposition CH(G/B) = w W Z[X w ] K(G/B) = w W Z[O Xw ]

9 Oriented cohomology theory (Quillen, Morel-Levine) c 1 (L 1 L 2 )=F (c 1 (L 1 ),c 1 (L 2 )) formal group law a commutative formal group law F on a ring R F R[[x, y]] written x + F y x + F (y + F z)=(x + F y)+ F z x + F 0=0+ F x = x x + F y = y + F x CH : x + y K 0 : x + y xy Ω : x + y a 11 xy +...

10 Formal group law of h Invariant theory under the Weyl group of G Schubert Calculus in h(g/b) Equivariant cohomology

11 Schubert calculus Find good (geometric) generators of h(g/b) as a module over h(pt) Compute multiplication tables using these elements Do this in a reasonable language

12 A theorem of Borel G =SL n 0 σ 1,...,σ n Q[x 1,...,x n ] H (SL n /B, Q) 0 Description for other types (Demazure)? Description for other cohomology theories?

13 Demazure s approach M characters of the torus T L(λ) =λ T G line bundle over G/T characteristic map λ + µ c 1 L(λ + µ) = c1 L(λ) L(µ) c g : S(M) CH(G/B) = c 1 L(λ) + c1 L(µ) symmetric algebra (polynomial ring)

14 differential operators on S(M) i (u) = u s i(u) α i α i s i simple root simple reflexion i (uv) = i (u)v + s i (u) i (v) I =(i 1,...,i q ) I (u) = i1... iq (u) Theorem (Demazure, 73) I and I two reduced decompositions of w W Then I = I

15 Algebraic reconstruction of CH(G/B) D(M) subring of endomorphisms of S(M) generated by the i D(M) the Z module of elements of D(M) composed by the augmentation S(M) Z H(M) the Z dual of D(M) properties of the i yield a ring structure on H(M)

16 c a : S(M) h(m) by double-duality characteristic map (algebraic) Theorem: There is a unique iso CH(G/B) H(M) it is a ring map s. t. c g CH(G/B) S(M) It sends the class c a t [X w ] to H(M) c a w 1(u 0 ) torsion index a part. element

17 Theorem (Demazure 73) S 1 t (M) :=S(M) Z [ 1 t ] S 1 is a free S 1 t (M)W t (M) w (u 0 ) w W module, with basis The char. map is surjective. c a : S 1 t (M) H(M) Z [ 1 t ] Its kernel is the ideal generated by ker() W The c a w (u 0 ) form a basis of its image

18 How to do the same thing for any other cohomology theory? (when the formal group law is not the additive one?)

19 Formal group ring R[[M]] F = R[[x M ]]/ x 0 x λ+µ x λ + F x µ h(pt) char. of T c g : R[[M]] F h(g/t )=h(g/b) x λ c 1 (L(λ)) x λ+µ c 1 L(λ + µ) = c1 L(λ) L(µ) = c 1 L(λ) +F c 1 L(µ)

20 Demazure operators F i (u) = u s i(u) x αi C F i (u) = ux α i + s i (u)x αi x αi x αi F I = F i 1 F i q C F I = Main problem: F I = F I for two reduced dec. of the same element Still possible to build up H(M) F as before Choose a reduced dec. I w for each w W

21 Theorem (C., Petrov, Zainoulline) 2t regular in R R 1 [[M]] F is a free R 1 t t [[M]]W F module, with basis Iw (u 0 ) w W The char. map c a : R 1 t [[M]] F H 1 t (M) F is surjective. Its kernel is the ideal generated by ker() W c a Iw (u 0 ) The form a basis of its image.

22 h oriented cohomology with formal group law F over R = h(pt) There is a unique ring isomorphism s. t. c g h(g/b) R[[M]] F It sends t[z I ] to c a H(M) F c g CI rev(u 0 ) where [Z I ]=(p I ) (1) p I : Z I X w G/B (depends on I ) desingularization of X w Bott-Samelson

23 Formulas for G 2 and for algebraic cobordism Z =1+4a 2 Z (10a 3 10a 1 a 2 )Z 212 (4a 4 +9a 1 a 3 +3a 2 2 9a 2 1a 2 )Z 12 (54a 5 459a 1 a a 2 a 3 108a 2 1a a 1 a a 3 1a 2 )Z 2 ; Z =3Z a 1 Z (13a 2 +2a 2 1)Z 21 +(2a 3 +7a 1 a 2 + a 3 1)Z 1 ; Z = Z a 2 Z 12 +(6a 3 5a 1 a 2 )Z 2 ; Z Z = Z Z a 1 Z a 1 Z 212 +(8a 2 + a 2 1)Z 12 +(8a 2 + a 2 1)Z 21 +(4a 3 +8a 1 a 2 + a 3 1)Z 1 + (10a 3 +6a 1 a 2 + a 3 1)Z 2 +( 4a 4 + a 1 a a a 2 1a 2 + a 4 1)pt; Z Z 1212 = Z Z a 1 Z 12 +3a 1 Z 21 +(8a 2 +4a 2 1)Z 1 + (13a 2 +5a 2 1)Z 2 +(a a 1 a 2 +5a 3 1)pt; Z Z 2121 =2Z a 1 Z 21 +(4a 2 + a 2 1)Z 1 ; Z Z 1212 =2Z a 1 Z 12 +4a 2 Z 2 ; Z Z 2121 = Z Z a 1 Z 12 + a 1 Z 21 +(5a 2 + a 2 1)Z 1 +(8a 2 + a 2 1)Z 2 +(3a 3 +6a 1 a 2 + a 3 1)pt

24 Z Z 121 =3Z 21 +2a 1 Z 1 ; Z Z 212 =2Z 12 + Z 21 +2a 1 Z 1 +3a 1 Z 2 +(4a 2 +3a 2 1)pt; Z Z 121 = Z 12 +2Z 21 +2a 1 Z 1 +2a 1 Z 2 +(4a 2 +2a 2 1)pt; Z Z 212 = Z 12 ; Z =2Z 12 + a 1 Z 2 ; Z =2Z 21 + a 1 Z 1 ; Z 1212 Z 2121 =2Z 12 +2Z 21 +3a 1 Z 1 +4a 1 Z 2 +(4a 2 +4a 2 1)pt; Z Z 12 = Z 1 +3Z 2 +3a 1 pt; Z Z 21 = Z 1 ; Z Z 12 = Z 2 ; Z Z 21 = Z 1 + Z 2 + a 1 pt; Z 1212 Z 121 =2Z 1 +3Z 2 +4a 1 pt; Z 1212 Z 212 = Z 2 ; Z 2121 Z 121 = Z 1 ; Z 2121 Z 212 = Z 1 +2Z 2 +2a 1 pt.

25 Equivariant cohomology G X a linear algebraic group a G-variety (smooth) h G(X) Properties: Pull-backs, push-forwards along equiv. morph. Restriction along morphisms of groups Chern classes for equivariant bundles Various compatibilities...

26 Important property: f : X Y G H-morphism G-torsor h H(X) res π H h G H(X) f h G H(Y ) is an isomorphism. Corollary: if H G ψ f,g,h : h H(X) h G H(Y ) h G(Y/H)

27 Topological filtration: h G(X) =F 0 h G(X) F 1 h G(X) F j h G(X) x F j h G G(X) υ : U X υ (x) =0 X \ U of codim j F j h G(X) =0 j=0

28 Variation on Totaro s method for Chow groups X \ U of codim j h G(X) F j h G(X) h G(U) F j h G(U) Corollary: V linear rep. of G s. t. U G acts freely on with V \ U of codim j h G(X) F j h G(X) p X h G(X U) F j h G(X U) h (X G U) F j h (X G U)

29 Can be used to construct h G from h (equivariant cobordism Deshpande, Krishna, Malagon-Lopez) h G(X) lim j h (X G U j ) F j h (X G U j ) where the U j are...

30 Equivariant cohomology of the point under a group of multiplicative type R graded ring R[[M]] F =lim j M abelian group of finite type R[[M]] F /I j in cat. of graded rings augmentation ideal Theorem: R[[M]] F h D(M)(pt) x λ c 1 (λ) Cartier dual Ex: M = Z n and D(M) =G n m

31 Interpretation of the caracteristic map h T (pt) ψ pt,g,t h G(G/T ) res h (G/T ) R[[M]] F h (G/T )

32 Open problems Universal formulas (Pieri, Monk, etc.) Positivity results What is so special about K-theory and twisting?