EQUIVARIANT CHERN-SCHWARTZ-MACPHERSON CLASSES IN PARTIAL FLAG VARIETIES: INTERPOLATION AND FORMULAE

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1 EQUIVARIANT CHERN-SCHWARTZ-MACPHERSON CLASSES IN PARTIAL FLAG VARIETIES: INTERPOLATION AND FORMULAE R. RIMÁNYI AND A. VARCHENKO Dedicted to Piotr Prgcz on the occsion of his 60th irthdy Astrct. Consider the nturl torus ction on prtil flg mnifold F λ. Let Ω I F λ e n open Schuert vriety, nd let c sm (Ω I ) H T (F λ) e its torus equivrint Chern-Schwrtz- McPherson clss. We show set of interpoltion properties tht uniquely determine c sm (Ω I ), s well s formul, of locliztion type, for c sm (Ω I ). In fct, we proved similr results for clss κ I H T (F λ) in the context of quntum group ctions on the equivrint cohomology groups of prtil flg vrieties. In this note we show tht c SM (Ω I ) = κ I. 1. Introduction An interesting chpter of enumertive geometry is the theory of chrcteristic clsses of singulr vrieties. One of the fundmentl results out Chern-Schwrtz-McPherson (CSM) chrcteristic clsses is their clcultion for degenercy loci in [PP] y Prgcz nd Prusiński. In this short note dedicted to the 60th irthdy of P. Prgcz we prove result out CSM clsses of Schuert cells in prtil flg vrieties. Nmely, we present set of interpoltion properties tht uniquely determine the sought CSM clss. Such interpoltion chrcteristion ws known efore for the leding term of the CSM clss, the fundmentl clss [FR]. A solution of these interpoltion conditions is weight function in the terminology of our erlier works. Weight functions were considered y Trsov nd Vrchenko [TV1, TV2] in the context of q-hypergeometric solutions of qkz differentil equtions, nd turn up in our recent works (with Trsov, Gorounov) in the context of geometric interprettions of Bethe lgers. In fct, our results on the interpoltion conditions nd their solution re present in [RTV1, RTV2, RTV3] for certin cohomology clsses κ I see lso one of the min motivtions [MO, Section 3.3.4]. In the present note we only prove tht κ I is equl to the CSM clss of Schuert cell Ω I. By giving proof of this fct nd y presenting n ccessile description of the interpoltion conditions nd the weight functions (with pproprite convention chnges) we hope to ring the ttention of reserchers in Schuert clculus nd chrcteristic clsses to the quntum group spects of CSM clsses. In Section 8 we extend this result to the cse of Chern-Schwrtz-McPherson clsses of Schuert cells in G/P where P is prolic sugroup of semisimple group G. R.R. ws supported y NSF grnt DMS nd A.V. ws supported in prt y NSF grnt DMS , Simons Foundtion, nd the Mx Plnck Institute in Bonn. 1

2 2 R. RIMÁNYI AND A. VARCHENKO The uthors thnk P. Aluffi, L. M. Fehér, nd T. Ohmoto for useful discussions on the topic. 2. Weight functions 2.1. Definition of weight functions. Let us fix non-negtive integers n nd N, s well s λ = (λ 1,..., λ N ) {0, 1, 2,...} N with λ = N k=1 λ k = n. Consider N-tuples I = (I 1,..., I N ), where I k {1,..., n}, I k = λ k, nd I k I l = for k l. The set of such N-tuples will e denoted y I λ. For I I λ we will use the nottions nd the vriles I (k) = k l=1 I l = {i (k) 1 < i (k) 2 <... < i (k) λ (k) }, λ (k) = I (k) = (1) t = {t (k) : k = 1,..., N 1, = 1,..., λ (k) }, z = {z 1,..., z n }. For I I λ, k = 1,..., N 1, = 1,..., λ (k), = 1,..., λ (k+1) define { 1 + t (k+1) t (k) if i (k+1) < i (k) l (k) I (, ) = t (k+1) if i (k+1) > i (k), with the convention tht t (N) = z. Let the group S (λ) = S λ (1)... S λ (N 1) ct on the set of vriles t in such wy tht elements of S λ (k) permute the lower indexes of t (k) s. Let Sym λ f(t) = σ S f(σ(t)). (λ) Denote e λ (t) = N 1 k=1 λ (k) λ (k) =1 =1 (1 + t(k) Definition 2.1. Define the weight function N 1 λ (k) W I (t, z) = Sym λ nd the modified weight function k=1 =1 λ (k+1) =1 ). λ (k) l (k) I (, ) λ (k) =1 =+1 W I (t, z) = W I (t, z)/e λ (t). 1 + t (k) t (k) k l=1 λ l, The weight functions W I re polynomils (despite the ppernce of t (k) denomintors), while the modified weight functions W I re rtionl functions. fctors in the Exmple 2.2. For N = 2, λ = (1, n 1) we hve k 1 n W ({k},{1,...,n} {k}) = (1 + z i t (1) 1 ) (z i t (1) 1 ). i=1 i=k+1

3 EQUIVARIANT CSM CLASSES IN PARTIAL FLAG VARIETIES 3 3. Comintoril description of weight function in the distinguished oxes of the k th column from top to ottom. Ech such filled tle will define rtionl function s follows. Let u e vrile in the filled tle in one of the columns 1,..., N 1. If v is vrile in the next column, ut ove the position of u then consider the fctor 1 + v u ( type-1 fctor ). If v is vrile in the next column, ut elow the position of u then consider the fctor v u ( type-2 fctor ). If v is vrile in the sme column, ut elow the position of u then consider the fctor (1+v u)/(v u) ( type-3 fctor ). The rule is illustrted in the following figure. In this section we show digrmmtic interprettion of the terms of the weight function. Let I I λ. Consider tle with n rows nd N columns. Numer the rows from top to ottom nd numer the columns from left to right. Certin oxes of this tle will e distinguished, s follows. In the k th column distinguish oxes in the i th row if i I (k). This wy ll the oxes in the lst column will e distinguished since I (N) = {1,..., n}. Now we will define fillings of the tles y putting vrious vriles in the distinguished oxes. First, put the vriles z 1,..., z n into the lst column from top to ottom. Now choose permuttions σ 1 S λ (1),..., σ N 1 S λ (N 1). Put the vriles t (k) σ k (1),..., t(k) σ k (λ (k) ) v u u u v v 1 + v u v u 1 + v u v u type -1 type -2 type -3 For ech vrile u in the tle consider ll these fctors nd multiply them together. This is the term ssocited with the filled tle. One sees tht W I is the sum of terms ssocited with the filled tles corresponding to ll choices σ 1,..., σ N 1. For exmple, W {2},{1},{3} is the sum of two terms ssocited with the filled tles t (2) 1 z 1 t (1) 1 t (2) 2 z 2, z 3 The term corresponding to the first filled tle is t (2) 2 z 1 t (1) 1 t (2) 1 z 2. z 3 (1 + t (2) 1 t (1) 1 )(1 + z 1 t (2) 2 ) (z }{{} 2 t (2) 1 )(z 3 t (2) 1 )(z 3 t (2) 2 ) }{{} type 1 type 2 (1 + t (2) 2 t (2) 1 ) (t (2) 2 t (2) 1 ) }{{} type 3,

4 4 R. RIMÁNYI AND A. VARCHENKO nd the term corresponding to the second filled tle is (1 + t (2) 2 t (1) 1 ) (1 + z 1 t (2) 1 ) (z }{{} 2 t (2) 2 ) (z 3 t (2) 2 ) (z 3 t (2) 1 ) }{{} type 1 type 2 (1 + t (2) 1 t (2) 2 ) (t (2) 1 t (2) 2 ) } {{ } type Prtil flg mnifold, equivrint cohomology, Schuert vrieties Let ɛ 1,..., ɛ n e the stndrd sis in C n. Consider the prtil flg mnifold F λ prmeterizing chins 0 = F 0 F 1... F N = C n, where dim F i /F i 1 = λ i. The stndrd ction of the torus T = (C ) n on C n induces n ction of T on F λ. The fixed points of this ction re the points x I = (F i ) with F k = spn{ɛ i : i I (k) }. Consider vriles Γ i = {γ i,1,..., γ i,λi } for i = 1,..., N, nd let Γ = {Γ 1,..., Γ N }. The group S λ = i S λi cts on Γ y permuting the vriles with the sme first index. The complex coefficient, T equivrint cohomology ring of F λ is presented s (2) H T (F λ ) = C[Γ] S λ C[z] / f(γ) = f(z) for ny f C[z] Sn. Here γ i,j for j = 1,..., λ i re the Chern roots of the undle whose fier is F i /F i 1, nd z re the Chern roots of the torus. Let V i = spn(ɛ 1,..., ɛ i ). For I I λ define the Schuert cell Ω I = {F F λ dim(f p V q ) = #{i I (p) i q} p N q n}. The Schuert cell Ω I is n ffine spce of dimension #{(i, j) {1,..., n} 2 i I, j I, <, i > j}. The point x I is smooth point of the Schuert cell Ω I. The weights of the torus ction on the tngent spce T xi Ω I re z z for >, I k, I l, k < l. The weights of the torus ction on T invrint norml spce N xi Ω I to T xi Ω I in T xi F λ re z z for <, I k, I l, k < l. Hence we hve c(t xi Ω I ) = k<l (1 + z z ), e(n xi Ω I ) = k<l (z z ), I k I l > for the tngent totl Chern clss nd the norml Euler clss of Ω I t x I. I k I l >

5 EQUIVARIANT CSM CLASSES IN PARTIAL FLAG VARIETIES 5 5. Unique clsses defined y interpoltion The restriction of cohomology clss ω H T (F λ) to the fixed point x J will e denoted ω xj. In terms of vriles this mens the sustitution (3) {γ k,i i = 1,..., λ k } {z J k }. For n S (λ) symmetric function f(t, z) in vriles t, z s in (1) let f(γ, z) denote the sustitution {t (k) = 1..., λ (k) } {Γ 1,..., Γ k }. Theorem 5.1. There re unique clsses κ I H T (F λ) stisfying (I) κ I xj is divisile y c(t xj Ω J ); (II) κ I xi = c(t xi Ω I )e(n xi Ω I ); (III) κ I xj hs degree less thn dim F λ = k<l λ kλ l, if I J. Moreover, (4) κ I = [ W I (Γ, z)]. Since W I is rtionl function we need to explin wht we men y (4). This mens tht the sustitution (3) of WI (Γ, z) is ( polynomil, nd is) equl to κ I xj for ll J I λ. It is remrkle tht lthough κ I cn e represented y polynomil in Γ, z vriles (y definition, see (2)) ut we represented it y rtionl function. The first prt of Theorem 5.1 follows from [MO, Thm 3.3.4]. Although (I) is locl version of the xiom in [MO], the proof is essentilly the sme. Both prts of Theorem 5.1 follow from the K-theory nlogue [RTV3, Thm 3.1 nd Thm 6.7], see lso [RTV1, RTV2]. 6. Chern-Schwrtz-McPherson clsses Let the torus T = (C ) n ct on the lgeric mnifold M. For T equivrint constructile function f on M one cn consider its Chern-Schwrtz-McPherson (CSM) clss c SM (f) HT (M). For n invrint suset X M we write c SM (X) for c SM (1 X ) where 1 X is the indictor function of X. The non-equivrint version of this notion ws introduced y McPherson [M], see lso [Sch], nd the equivrint version y Ohmoto [O1], see lso [W, O2]. We refer the reder to these ppers for definitions nd min properties, nmely nturl normliztion, functorility, product, nd locliztion properties. CSM clsses of Schuert cells nd vrieties, their recursion nd positivity properties, re studied in [AM1, H, AM2]. Remrk 6.1. It is customry to consider the CSM clss of vriety X in its (equivrint) homology c SM (X) H T (X) this version does not depend on n mient mnifold M. In the present pper we will not consider this homology CSM clss. We follow [O1] nd [W] y mpping the homology CSM clss c SM (X) first to H T (M) y homology push-forwrd, then pplying (equivrint) Poincré dulity for M, nd considering the resulting cohomology c SM (X) clss living in HT (M).

6 6 R. RIMÁNYI AND A. VARCHENKO The clss c SM (X) is refinement of the notion of (equivrint) fundmentl clss; nmely c SM (X) = [X]+ higher degree terms, where [X] is the (equivrint) fundmentl clss of the vriety X in H T (M). Now we recll the properties of equivrint CSM clsses we will need in the next section. (A) (Linerity.) For equivrint constructile functions f nd g we hve c SM (f + g) = c SM (f) + c SM (g). (B) (Locl model of CSM clsses of vrieties.) Suppose T cts on vector spce V, nd let X V e n invrint smooth suvriety. Then (5) c SM (X) = c(t 0 X)e(V/T 0 X) = (1 + α i ) β j HT (V ) = HT ({pt}). i where α i s re the tngent weights, nd β j s re the norml weights of X. Indeed, when c SM (X) is considered in the (co)homology of X then it is the totl Chern clss of the tngent undle of X [O2, Thm. 3.10]. In our convention (i.e. when c SM (X) is pushed forwrd to the cohomology of the mient spce) we otin (5). (C) Suppose T cts on vector spce V, nd let X e n invrint suvriety which contins n invrint smooth suvriety X 0. Moreover ssume tht n invrint liner suspce W is trnsversl to X nd W T 0 X 0 = V. Then c SM (X) is divisile y c(t 0 X 0 ). Indeed, in our convention Theorem 3.13 (2) of [O2] reds j c SM (X) c(v ) = csm (X W ), c(w ) which implies c SM (X) = c(v/w )c SM (X W ) = c(t 0 X 0 )c SM (X W ). (D) Let X e suvriety of the T -mnifold M with isolted fixed points. Let x e fixed point of M. Then [W, Thm 20] { c SM e(t x M) + lower degree terms if x X (X) x = 0 if x X. A remrkle feture of this fct which we will use elow is tht the top degree prt of c SM (X) x does not depend on the vriety X s long s x elongs to the vriety. 7. CSM clsses coincide with κ clsses Consider the torus ction on F λ nd their Schuert cells Ω I F λ s in Section 4. Also recll the κ I clsses defined xiomticlly in Section 5. Theorem 7.1. We hve c SM (Ω I ) = κ I H T (F λ ). Proof. We need to prove tht the clss c SM (Ω I ) stisfies the xioms defining κ I.

7 EQUIVARIANT CSM CLASSES IN PARTIAL FLAG VARIETIES 7 Let us write the indictor function of the Schuert cell s liner comintion of the indictor functions of Schuert vrieties: (6) 1 ΩI = K I d IK 1 ΩK. Here K I mens Ω K Ω I nd d IK re integer coefficients, d II = 1. By linerity of CSM clsses c SM (Ω I ) = d IK c SM (Ω K ), K I nd hence c SM (Ω I ) xj = K I d IK c SM (Ω K ) xj = J K I d IK c SM (Ω K ) xj. The ehviour of the vrieties Ω K ppering on the RHS ner x J re known in Schuert clculus: they ll contin Ω J, Ω J is smooth t x J nd the suspce W required in (C) ove exists. Therefore ech term is divisile y c(t xj Ω J ). This proves xiom (I). If I = J then the right hnd side is just the term c SM (Ω I ) xi which ccording to the locl model property ove is c(t xi Ω I )e(n xi Ω I ). This proves xiom (II). Now let J I, J I. Using Theorem [W, Thm 20] s reclled ove in (D) we otin c SM (Ω I ) xj = ( ) d IK (e(t xj F λ ) + l.d.t.) = e(t xj F λ ) d IK + l.d.t., J K I J K I where l.d.t. (=lower degree terms) mens terms of degree strictly less thn dim F λ = i<j λ iλ j. However, the sum J K I d IK vnishes due to sustituting x J into the functionl identity (6). We otin tht c SM (Ω I ) xj hs degree strictly less thn dim F λ. This proves xiom (III). Corollry 7.2. The equivrint CSM clsses of Schuert cells of prtil flg mnifolds re defined y the xioms of Theorem 5.1. Moreover c SM (Ω I ) = [ W I (Γ, z)]. Remrk 7.3. The CSM clsses stisfy ovious vnishing properties respecting the Bruht order. Nmely, if Ω J Ω I then c SM (Ω I ) xj = 0. It is remrkle tht one does not need to list this ovious vnishing property mong the xioms, this property is consequence of the xioms. Other remrkle properties of κ I clsses include orthogonlity properties [RTV2, Sec.5], [RTV3, Thm. 6.6], recursion [RTV2, Lemm 3.3],[RTV3, Thm. 6.10], nd R-mtrix properties [RTV2, Cor. 3.8],[RTV3, Section 7]. According to Theorem 7.1 these lso hold for equivrint CSM clsses of Schuert cells. 8. CSM clsses of Schuert cells in generlized flg mnifolds Let G e semisimple lgeric group nd let T B P e mximl torus, Borel sugroup, nd prolic sugroup in G. The B orits Ω I re clled Schuert cells in G/P. They re prmeterized y certin elements of the Weyl group. Ech orit contins T fixed point x I. The first prt of Corollry 7.2 generlizes to the generl flg vriety G/P s follows.

8 8 R. RIMÁNYI AND A. VARCHENKO Theorem 8.1. The T equivrint Chern-Schwrtz-McPherson clsses c SM (Ω I ) re the unique clsses in HT (G/P ) stisfying the properties (I) c SM (Ω I ) xj is divisile y c(t xj Ω J ); (II) c SM (Ω I ) xi = c(t xi Ω I )e(n xi Ω I ); (III) c SM (Ω I ) xj hs degree less thn dim G/P, if I J. Proof. The existence nd uniqueness of clsses stisfying the itemized properties is proved in [MO]. (A guide to the reder: Mulik nd Okounkov prove tht certin clsses re chrcterized y three xioms, see [MO, Section 3.3.4]. Their three xioms for G/P re (II), (III) ove, together with glol version of (I). The glol versus locl versions of (I) does not ffect the rguments. Also, the Mulik-Okounkov clsses re homogeneous, depending on n extr prmeter h. Our non-homogeneous versions re otined y putting h = 1.) The fct tht the c SM (Ω I ) clsses stisfy the properties follows the sme wy s in the specil cse detiled in Section 7. Remrk 8.2. The mp sending fixed points {x I } to the clsses {c SM (Ω I )} is essentilly the stle envelope mp of Mulik-Okounkov corresponding to the pir P G. References [AM1] P. Aluffi, L. C. Mihlce: Chern clsses of Schuert cells nd vrieties; J. Algeric Geom., 18(1):63 100, [AM2] P. Aluffi, L. C. Mihlce: Chern-Schwrtz-McPherson clsses for Schuert cells in flg mnifolds, rxiv: , 2016 [FR] L. Feher, R. Rimnyi: Schur nd Schuert polynomils s Thom polynomils cohomology of moduli spces; Cent. Europen J. Mth. 4 (2003) [H] J. Huh: Positivity of Chern clsses of Schuert cells nd vrieties, Journl of Algeric Geometry, to pper. [M] R. McPherson: Chern clsses for singulr lgeric vrieties, Ann. of Mth. 100 (1974), [MO] D. Mulik, A. Okounkov: Quntum Groups nd Quntum Cohomology; rxiv: , 2012 [O1] T. Ohmoto: Equivrint Chern clsses of singulr lgeric vrieties with group ctions, Mth. Proc. Cmridge Phil. Soc.140 (2006), [O2] T. Ohmoto: Singulrities of mps nd chrcteristic clsses, to pper in Proc. of Rel nd Complex Singulrities 2012 [PP] A. Prusiński nd P. Prgcz, Chern-Schwrtz-McPherson clsses nd the Euler chrcteristic of degenercy loci nd specil divisors, Jour. Amer. Mth. Soc. 8 (1995), no. 4, [RTV1] R. Rimnyi, V. Trsov, A. Vrchenko: Cohomology clsses of conorml undles of Schuert vrieties nd Yngin weight functions; to pper in Mth. Z., 2015 [RTV2] R. Rimnyi, V. Trsov, A. Vrchenko: Prtil flg vrieties, stle envelopes nd weight functions; Quntum Topology 6 (2015) 1-32 [RTV3] R. Rimnyi, V. Trsov, A. Vrchenko: Trigonometric weight functions s K-theoretic stle envelop mps for the cotngent undle of flg vriety; to pper in Geometry nd Physics 2015 [Sch] M. H. Schwrtz, Clsses crctéristiques definies pr une strtifiction dune vriété nlytique complexe, C. R. Acd. Sci. Pris t.260 (1965), , [TV1] V. Trsov, A. Vrchenko: Jckson Integrl Representtions for Solutions to the Quntized Knizhnik- Zmolodchikov Eqution; 1-50, (1993) rxiv:hep-th/ [TV2] V. Trsov, A. Vrchenko, Comintoril formule for nested Bethe vectors, SIGMA 9 (2013), 048, 1 28

9 EQUIVARIANT CSM CLASSES IN PARTIAL FLAG VARIETIES 9 [W] A. Weer: Equivrint Chern clsses nd locliztion theorem, Journl of Singulrities Volume 5 (2012), Deprtment of Mthemtics, University of North Crolin t Chpel Hill, USA E-mil ddress: rimnyi@emil.unc.edu Deprtment of Mthemtics, University of North Crolin t Chpel Hill, USA E-mil ddress: nv@emil.unc.edu