Cohomological properties of certain quiver ag varieties

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1 Algebra Extravaganza Temple University Philadelphia, PA Cohomological properties of certain quiver ag varieties Mee Seong m United States Military Academy July 7, 2017

2 Joint work. 1 This is a joint work in-progress with Brian Allen.

3 Background. Preliminary definitions. Work over C. 2 A quiver Q = (Q0, Q1 ) is a finite directed graph, where Q0 is the set of vertices and Q1 is the set of arrows. 0 Let β = (β1,..., βq0 ) NQ 0, a dimension vector. A representation of a quiver assigns a finite-dimensional vector space to each vertex and a linear map to each arrow. Let Rep(Q, β) := M HomC (W (βta ), W (βha )), a Q1 where W (βi ) is a βi -dimensional complex vectorq space. t has a natural (change-of-basis) group action by Gβ = i Q0 GLβi (C). We say Rep(Q, β) is the representation space of a quiver.

4 Filtered representation space. 3 Let γ (1), γ (2),..., γ (M) = β be a sequence of dimension vectors. Then (i) (i) F Rep(Q, β) := {(Aa )a Q1 : Aa (W (γta )) W (γha ) for all 1 i M, a Q1 }. F Rep(Q, β) is the filtered representation space for Q and β. f Pi GLβi (C) is the group of automorphisms preserving the filtration of vector spaces at vertex i, then we define Y Pβ := Pi, i Q0 which acts as a change-of-basis on F Rep(Q, β). So there is a parabolic group action Pβ on F Rep(Q, β). f we have a complete filtration of vector spaces over each vertex, we will write Bβ instead of Pβ.

5 Universal quiver flag. 4 We denote Flγ (β) := Y Flγi (βi ), i Q0 the product of usual flag varieties on the vector space W (i) parametrizing flags of subspaces (1) 0 Ui (2) Ui (M)... Ui (j) (j) = W (i), where dim(ui ) = γi, i Q0. We denote U := {0 U (1) U (2)... U (M) }, the set of flags of subspaces. The universal quiver flag is defined as (i) (i) FlγQ (β) = {(W, U ) Rep(Q, β) Flγ (β) : W (a)uta Uha for each a Q1, 1 i M}.

6 Projections from the universal quiver flag. 5 ^ β), where we have Also denote the universal quiver flag as Rep(Q, natural projections p1 and p2 into appropriate factors p1 p2 ^ β) Rep(Q, β) Rep(Q, Flγ (β). Properties about morphisms p1 and p2 p1 and p2 are Gβ -equivariant projections, a generic fiber of p1 over W : a product U of flags of vector spaces on U (M) such that WU U (so p1 is projective), a generic fiber of p2 over a flag: a homogeneous vector bundle, i.e., the set of all W such that WU = U (so p2 is flat). The fibers of p1 are called quiver flag varieties, the fibers of p2 are called filtered quiver representations.

7 Quiver G-S resolution & Springer resolution. 6 Fix Q = (Q0, Q1 ), where Q0 = {v0, v11, v21,..., v`11, v12, v22,..., v`22,..., v1i, v2i,..., v`i i,..., v1k, v2k,..., v`kk } and Q1 = {a01, a11,..., a`11, a02, a12,..., a`22,..., a0k, a1k,..., a`kk }, satisfying ai ai ai ai a`i 1 a`i 3 0 i i v`i i v0 for 1 i k. v0 v1i v2i v3i Also let M = 2N. Fix sequences of dimension vectors γ (i) so that γ (2i) = (i, i,..., i) for each 1 i N and γ (2i 1) = (i 1, i 1,..., i 1) for each 1 i N. Note that β = (N, N,..., N). (1)

8 Generalized G-S resolution. 7 ^ β) is called the With Q and β from the previous slide, Rep(Q, generalized Grothendieck-Springer resolution corresponding to a ^ β) complete flag. t will be denoted as Rep(Q, G Spr. Denote bq1 := F Rep(Q, β). Proposition p2 ^ β) The second projection Rep(Q, Flγ (β) is Gβ -equivariant, ^ β). inducing an isomorphism Gβ B bq = Rep(Q, β 1 Proposition p1 A generic fiber of Gβ Bβ bq1 Rep(Q, β) consists of (N!)Q0 points.

9 Generalized Springer resolution. 8 Continue to let M = 2N. Fix γ (i) so that γ (2i) = (i, i,..., i) for each 1 i N and γ (2i 1) = (i 1, i 1,..., i 1, i) for each 1 i N. (2) For the 1-Jordan quiver, ^ β) Rep(Q, = T (GLN (C)/B) = GLN (C) B N, the cotangent bundle of the flag variety (the Springer resoln of the nilpotent cone N ). ^ β) a generalized For the general quiver and satisfying (2), Rep(Q, Springer resolution of the nilpotent cone NQ1 in Rep(Q, β), where NQ1 = {(W (a01 ),..., W (a`kk )) : W (a01 a`kk ) is nilpotent}.

10 Resolution of singularities. 9 Now fix the filtration of vector spaces in (2) w.r.t. the standard basis. Then F Rep(Q, β) is contained in the product of upper triangular matrices. Let nq1 = b (Q1 1) n, (3) where n = gln /b, the set of strictly upper triangular matrices. Proposition T (Gβ /Bβ ) = Gβ Bβ nq1 is a resolution of singularities of NQ1. Corollary φ The morphism T (Gβ /Bβ ) NQ1 is generically finite. φ The morphism T (Gβ /Bβ ) NQ1 is also called generalized Springer desingularization.

11 Diagram of morphisms. 10 Summary We have the diagram of morphisms, where Gβ Bβ nq1, Gβ Bβ bq1 is an inclusion of subbundles:

12 Weights of quiver flag varieties. 11 Q For a fixed Gβ, let Tβ = i Q0 Ti, where Ti is the maximal torus consisting of diagonal matrices in GLβi (C) for each i Q0. Let diag(ti1, ti2,..., tin ) Ti for each i Q0. Denote X (Tβ ) to be the character group of Tβ, where each character i,j X (Tβ ) is defined as i,j (tv0,1,..., tv0,n,..., ti,1,..., ti,n,..., tv`k k,1,..., tv`k,n ) = ti,j k for each i Q0, 1 j N. Then { ij }i Q0,1 j N forms a standard basis for X (Tβ ), which gives an isomorphism of weight lattices X (Tβ ) = Z Q0 N. Dominant weights of X (Tβ ) are defined to be 1 1 k v X + (Tβ ) = {~λ = (λv0, λv1, λv2,..., λ `k ) Z Q0 N : λi = (λi,1 λi,2... λi,n )}.

13 Cohomology of line bundles. 12 Lemma Let ~κ = PN j=0 ( v0,j v`k,j ). Then k K (Gβ Bβ (nq1 C~λ )) ' LGβ Bβ (nq1 C~λ ) (C ~λ ~κ ), (4) where K (Gβ Bβ (nq1 C~λ )) is the canonical line bundle of Gβ Bβ (nq1 C~λ ). Proposition Given ~λ X + (Tβ ), H 0 (Gβ /Bβ, LGβ /Bβ (C ~λ )) = V~λ = Vλ v0 V v 1 V v k, λ 1 λ `k where Vλ i is the dual of the GLN (C)-irreducible representation Vλi with highest weight λi.

14 Higher cohomology. 13 Theorem We have H i (T (Gβ /Bβ ), LT (Gβ /Bβ ) (C ~λ )) = 0 for any ~λ and for all i > 0. Theorem Given a generalized Grothendieck-Springer resolution ^ β) Rep(Q, G Spr, we have ^ β) H i (Rep(Q, (C ~ )) = 0 ^ G Spr, LRep(Q,β) G Spr λ for any ~λ and for all i > 0.

15 Flag Hilbert schemes. 14 Why are flag Hilbert schemes interesting? The adjointness of a monoidal functor from the symmetric monoidal category CohFHilbn (C) of coherent sheaves on the flag Hilbert scheme FHilbn (C) to the non-symmetric monoidal category SBimn of Soergel bimodules produces a 1-1 correspondence between the Euler characteristic of a sheaf on the flag Hilbert scheme with the Hochschild homology of a braid (Gorsky-Negut-Rasmussen, 2016). Flag Hilbert schemes are interesting in their own right: birationality of FHilbn (X), singular locus of FHilbn (X), irreducible components of FHilbn (X), Hilbert function of FHilbn (X).

16 Flag Hilbert schemes. 15 Constructions The flag Hilbert scheme on C or C2 parametrizes full flags of ideals: FHilbn (C) := {n = C[x, y ] : dimc C[x, y ]/i = i, ideals supported on y = 0}, FHilb n 2 (C ) := {n = C[x, y ] : dimc C[x, y ]/i = i}. ADHM description Let B be lower triangular matrices in GLn (C), b = Lie(B), and let n be nilpotent matrices in b. Then FHilbn (C) = {(X, Y, v ) b n Cn : [X, Y ] = 0, X a Y b v span Cn }/B, FHilbn (C2 ) = {(X, Y, v ) b b Cn : [X, Y ] = 0, X a Y b v span Cn }/B.

17 Facts about flag Hilbert schemes. 16 Facts We have FHilbn (C2 ) Hilbn (C2 ), sending (n ) 7 n. We have FHilbn (C2 ) C2n, sending (n ) 7 (x1,..., xn, y1,..., yn ), where (xj, yj ) = supp(j 1 /j ). FHilbn (C2 ) is a closed subscheme of Hilbn (C2 ) Hilbn 1 (C2 ) Hilb1 (C2 ) Hilb0 (C2 ). FHilbn (C), FHilbn (C2 ) are singular for n 0, reducible, their dimensions expected dimensions.

18 Filtered quiver variety. 17 Let B be upper triangular matrices in GLn (C) and let b = Lie(B). dentify T (b Cn ) = b b Cn (Cn ). Consider the moment map µb T (b Cn ) b = gln /n+, where (X, Y, v, w) 7 [X, Y ] + vw. We define n n µ 1 B (0)/B := {(X, Y, v, w) b b C (C ) : [X, Y ] + vw = 0, X a Y b v span Cn }/B. Conjecture n 2 There is a birational map µ 1 B (0)/B 99K FHilb (C ).

19 Thank you. Questions?