(Equivariant) Chern-Schwartz-MacPherson classes
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1 (Equivariant) Chern-Schwartz-MacPherson classes Leonardo Mihalcea (joint with P. Aluffi) November 14, 2015 Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
2 Let X be a compact manifold, T X tangent bundle, with Chern class Gauss-Bonnet Theorem: c(t X ) = 1 + c 1 (T X ) c n (T X ). c n (T X ) [X ] = χ(x ) the topological Euler characteristic of X. Question: What happens if X is singular? Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
3 Constructible functions Let X be an algebraic variety. Constructible functions: F(X ) = { c i 1 Vi : c i Z, V i X constructible }. If f : X Y is a proper map, define a push-forward f : F(X ) F(Y ); f ( 1 V )(y) = χ(f 1 (y) V ). Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
4 Chern-Schwartz-MacPherson classes Theorem (Deligne - Grothendieck Conjecture; MacPherson 74, M. H. Schwartz) There exists a unique natural transformation c : F(X ) H (X ) such that: 1 If X is projective, non-singular, c ( 1 X ) = c(t X ) [X ]. 2 c is functorial with respect to proper push-forwards f : X Y : F(X ) f F(Y ) c H (X ) f c H (Y ) c SM (X ) := c ( 1 X ) is the Chern-Schwartz- MacPherson class. Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
5 Aluffi s method Let X closed, π : Z X be a resolution of singularities, and D X X a divisor such that π 1 (D X ) = D := D 1... D k is simple normal crossing (SNC) and π : Z \ D X \ D X. Then c SM (X \ D X ) = π (c SM (Z \ D)) = π (c SM (Z) c SM (D)) c(t Z ) = π ( (1 + D 1 )(1 + D 2 )... (1 + D k ) [Z]). Goal: Apply this to a (Schubert variety \ boundary divisor) and a Bott-Samelson resolution. Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
6 Lie and Schubert data G - complex simple Lie group and T B G (torus Borel G). E.g. G = SL n (C) and B = upper triangular matrices. W := N G (T )/T - the Weyl group. l : W N - length function. s i - simple reflections; w 0 - longest element in W. G/B - generalized flag manifold; e.g. Fl(n) = {F 1... F n = C n }. X (w) o := BwB/B - Schubert cell. X (w) := BwB/B - Schubert variety. dim C X (w) = l(w); X (w) := X (w) \ X (w) o - boundary divisor. l(w 0 ) = dim G/B. Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
7 Bott-Samelson varieties and CSM classes To any reduced decomposition w = s i1... s ik one can define the Bott-Samelson-(Demazure-Hansen) variety Z(w) inductively as a tower of P 1 -bundles. It comes equipped with proper, birational such that π : Z(w) X (w) π 1 ( X (w)) = D := D 1... D k is a SNC divisor and π : Z(w) \ D X (w) \ X (w). Corollary (Aluffi) c SM (X (w) o c(t Z(w) ) ) = π ( (1 + D 1 )(1 + D 2 )... (1 + D k ) [Z(w)]). Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
8 Examples Recall that H (G/B) = w W Z[X (w)]. Corollary immediately implies: c SM (X (w) o ) = v w c(w; v)[x (v)] = 1 [X (w)] [pt]. 1 G/B = P 1. Then c SM (P 1 ) = c(t P 1) [P 1 ] = [P 1 ] + 2[pt]. 2 c SM [pt] = [pt] thus c SM (A 1 ) = c SM (P 1 )) c SM ([pt]) = [P 1 ] + [pt]. Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
9 Operators on H (G/B) The BGG operator: let P k G minimal parabolic. G/B G/P k G/B pr 2 G/B pr 1 G/B p p G/P k k = (pr 2 ) (pr 1 ) : H (G/B) H 2 (G/B). Right Weyl group action: Let s k W. Since G/B hom G/T, right multiplication induces s k : H (G/B) H (G/B) automorphism. Alternatively, using Chevalley rule s k = id c 1 (L αk ) k. Formulas for left/right W -actions on HT (G/B) found by: Peterson, Knutson, Tymoczko,... Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
10 A Demazure-Lusztig type operator Define T k := k s k. Note: This operator is a specialization of an operator which appears in the study of a degenerate affine Hecke algebra, in relation to the Steinberg variety in T G/B T G/B. Lemma The operators T k satisfy the following properties: 1 (commutativity) E.g. in type A, T i T j = T j T i if i j 2; 2 (braid relations) E.g. in type A: T i T i+1 T i = T i+1 T i T i+1 ; 3 (square) T 2 i = id. 4 (Schubert action): T k ([X (w)]) = { [X (w)] if l(ws k ) < l(w) [X (ws k )] + [X (w)] + α k, β [X (ws k s β )] if l(ws k ) > l(w) where β > 0, β α k and l(ws k s β ) = l(w). Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
11 CSM classes of Schubert cells Theorem (Aluffi-M.) 1 Let w W be a Weyl group element, and X (w) o G/B the Schubert cell. Then T k (c SM (X (w) o )) = c SM (X (ws k ) o ). 2 Let P G be any parabolic subgroup and pr : G/B G/P be the projection. Then pr (c SM (X (w) o )) = c SM (X (ww P ) o ) where W P W the the subgroup generated by the reflections in P. Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
12 Positivity Fl(3) = {F 1 F 2 C 3 } - the flag variety. dim Fl(3) = l(w 0 ) = l(s 1 s 2 s 1 ) = 3. c SM (Fl(3) o ) = [Fl(3)] + [X (s 2 s 1 ] + [X (s 1 s 2 )] + 2[X (s 1 )] + 2[X (s 2 )] + [pt] Conjecture The coefficients c(w; u) > 0 for any u w. J. Huh proved the conjecture in the case G/P = Grassmannian. Positivity has been checked for Fl(n), n 7. We proved the conjecture in some cases: l(w) l(u) 1 or if w has a reduced decomposition into distinct reflections. Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
13 Equivariant case T. Ohmoto defined an equivariant version of MacPherson s transformation: which satisfies functoriality and c T : F T (X ) H T (X ) c T ( 1 X ) = c T (T X ) [X ] T if X is projective, nonsingular. A. Weber proved properties of localizations of CSM classes, and Rimányi-Varchenko used these and Maulik-Okounkov stable envelopes to obtain localization formulas for CSM classes c SM T (X (w) o ) u. Theorem (Aluffi - M.) Let T T k := k s k. Then T T k (c SM T (X (w) o )) = c SM T (X (ws k ) o ). Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
14 The equivariant operator The operator T T k acts almost as T k on Schubert classes: T k ([X (w)]) = { [X (w)] (1 + w(α k ))[X (ws k )] + [X (w)] + α k, β [X (ws k s β )] where branches are as before. The (c(w; u)) matrix for cells in Fl(3) is: α α 1 + α α α 1 + α α α α 1 + α α 1 + α (1 + α 1 )(1 + α 1 + α 2 ) 0 (1 + α 1 )(1 + α 1 + α 2 ) (1 + α 2 )(1 + α 1 + α 2 ) (1 + α 2 )(1 + α 1 + α 2 ) (1 + α 1 )(1 + α 2 )(1 + α 1 + α 2 ) Read on columns! Conjecture (Equivariant positivity) For any u w, the coefficients c(w; u) are polynomials with non-negative coefficients in simple roots α i. Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
15 Further connections Let ι : G/B TG/B be the zero section and let Stab + (w) HT C (T G/B ) be the stable envelope. Changjian Su used the operator Tk T to prove: ι Stab + (w) =1 = ±P.D. c SM T (X (w) o ). Let L C (TG/B ) be the group of Lagrangian cycles. Ginzburg proved that MacPherson s map c factors as F(G/B) L C (T G/B ) Then (C. Su - M., J. Schürmann): c Gi H (G/B) c Gi (Stab + (w)) = ±c SM (X (w) o ). Seung-Jin Lee: in type A, the coefficients c(w; u) coincide with certain specializations in Fomin-Kirillov algebra. Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
16 THANK YOU! Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, / 16
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