Punctual Hilbert and Quot Schemes on Kleinian Singularities

Punctual Hilbert and Quot schemes on Kleinian singularities

Ad´amGyenge´ 1

1Mathematical Institute, University of Oxford

28 February 2020 Outline

1. Hilbert scheme of points on curves and smooth surfaces 2. The orbifold and coarse Hilbert schemes 3. Euler characteristic calculations (joint work with András Némethi and Balázs Szendr˝oi) 4. Quiver variety description (joint work with Alastair Craw, Søren Gammelgaard and Balázs Szendr˝oi) 5. Quot schemes (work in progress) Hilbert scheme of points Let X be a quasiprojective variety over C. Definition (Theorem) n For every n N there is a Hilbert scheme Hilb X ,which parametrizes 0 dimensional subschemes (ideal sheaves) of colength n on X . ∈ ( ) Remark 1. Hilbn X represents a moduli functor. 2. Every Z Hilbn X decomposes as Z Z ,wherethe ( ) j supports Pj supp Zj are mutually disjoint. ∈ ( ) = 3. colength Z, P h0 X , . = j ( ) Z,Pj 4. n colength Z colength Z, P . ( ) = (j O ) j 5. Hilbert-Chow morphism = ( ) = ∑ ( ) n n ⇧ Hilb X S X , Z colength Z, Pj Pj . j ∶ ( ) → ( ) Hilbert scheme of points

Question: From topological/analytical properties of X what can we infere about the topology of Hilbn X ?

( ) Usually better to work with the collection of the Hilbert schemes for all n together.

n n ZX q top Hilb X q ∞ n 0 ( ) = ( ( )) Relative version: For Y =X a (locally) closed subvariety: n n Hilb X , Y Hilb X is the Hilbert scheme of points (set-theoretically) supported⊂ on Y . ( ) ⊂ ( ) Curves C curve over C with singularities pi Csm C i pi smooth part

= n n ZC q Hilb C q ∞ n 0 ( ) = ( ( )) = n0 n0 ni ni Hilb Csm q Hilb C, pi q ∞ n 0 n0 n i

= = +⋅⋅⋅= ( ( n)) n( ( )) ZC q Hilb C, pi q sm ∞ i n 0

= ( ) = ( Z ( q )) C,pi ( )( ) Theorem (Macdonald, Maulik, Oblomkov-Shende)

n n 1. ⇧ Hilb Csm S Csm is an isomorphism 2. Z q 1 C∶sm ( 1 q) →Csm ( ) 3. If p C is a( − planar) singularity, then the topology of the link ( ) =3 L C,0 S (i.e. its embedding type) determines Z C,p q ∈ ( ) ⊆ ( )( ) Smooth surfaces

X surface over C with isolated singularities pi Xsm X i pi smooth part. = Again,

ZX q ZXsm q Z X ,pi q . i ( ) = ( ) ( )( ) Theorem (Fogarty, Ellingsrud-Strømme, G¨ottsche)

n 1. Hilb Xsm is smooth of dimension 2n 2. ⇧ Hilbn X SnX is a resolution of singularities ( )sm sm 3. ∶ ( ) → 1 ZXsm q ∞ n Xsm i 1 1 q ( ) = ( ) 4. In particular, if X Xsm is smooth,= ( − then) up to a suitable power of q, ZX q transforms a modular form. = ( ) An example: the ane plane

Let X C2. n 2 Points= of Hilb C Ideals in C x, y of colength n. Special ideals: monomial ideals partitions. ( ) ↔ [ ] Using the technique of torus localization, we obtain ↔ n Hilb C2 # monomial ideals of colength n # a partition of n ( ( )) = p {n } = { } and so = ( ) n 1 Z 2 q p n q C ∞ n n 0 i 1 1 q ( ) = ( ) = as stated by G¨ottsche’s formula.≥ = − An example: the ane plane (cont’d)

1 n Z 2 q p n q . C ∞ n i 1 1 q n 0 ( ) = = ( ) This is the character formula= for− , the≥ Fock space representation of the Heisenberg algebra. Recall: F

p 1blockadded , F = ∈P ′ q 1blockremoved , ( ) = ∑ p, q Id. ′ ( ) = ∑ Theorem[ ] (Nakajima,= Grojnowski)

2 n 2 H Hilb C H Hilb C . ∞ ∗ n 0 ∗ Moreover, the operators( ( p))and= =q can( be constructed( )) ≅ F geometrically. Hilbert schemes of surface singularities

We study the simplest possible class: rational double points. These can be obtained as quotients: 2 G Sl2 C ﬁnite subgroup, acting on C . C2 G qoutient variety. ⊂ ( ) Deﬁnition Coarse (invariant) Hilbert scheme:

2 G Hilb C G Z C x, y Z is of ﬁnite colength .

( ) = { ◁ [ ] } As before, this decomposes as

2 m 2 Hilb C G Hilb C G . m N

( ) = ∈ ( ) Orbifold Hilbert schemes Orbifold singularities have another type of Hilbert scheme. Deﬁnition Orbifold (equivariant) Hilbert scheme:

2 2 Hilb C G I Hilb C I is G-invariant .

([ ]) = { ∈ ( ) } This stratiﬁes as

2 ⇢ 2 Hilb C G Hilb C G , ⇢ Rep G ([ ]) = ([ ]) where ∈ ( )

⇢ 2 2 0 0 Hilb C G I Hilb C H I H C2 I G ⇢ . Scheme-theoretic([ ]) pushforward= { ∈ ( map:) (O ) = (O ) }

2 2 G G p Hilb C G Hilb C G , I I I C x, y .

∗ ∶ ([ ]) → ( ) = ∩ [ ] Generating series

Let ⇢0,...⇢n be the irreducible representations of G. Deﬁnition (a) Coarse generating series (or coarse partition function):

m 2 m Z 2 q Hilb G q . C G ∞ C m 0 ( ) = ( ) (b) Orbifold generating series= (or orbifold partition function):

Z C2 G q0,...,qn [ ]( ) = Hilbm0⇢0 ... mn⇢n 2 G qm0 ... qmn . ∞ C 0 n m0,...,mn 0 + + = = ([ ]) ⋅ ⋅ Simple singularities

Finite subgroups of Sl2 C An n 1 , Dn n 4 , E6, E7, E8. To the quotient C2 G we can associate its resolution graph: ( ) ∶ ( ≥ ) ( ≥ )

There is also a corresponding Lie algebra. The abelian case

G cyclic subgroup of Sl2 C of order n 1. Generated by = ( ) ! 0 + , 0 ! 1 − where ! is a n 1 -st root= of unity. All representations are one dimensional. ( + ) j They are given by ⇢j ! ,forj 0,...,n .

∶ ∈ { } The abelian case: C 2 ﬁxpoints Monomials transform according∗ to irreducible representations of G: ( )

0⋮ 1

1 2

⋮ ⋮ n 0 n 2 n 1 n 0

0 1 ... n−1 −n 0 1 ...

− We can apply again torus localization to count only the monomial ideals. We get a coloured version of the partition counting problem: m ⇢ 2 Hilb i i C # a coloured partition∑ with m boxes of colour i ( )i = and so { } colj Z C2 G q0,...,qn 1 qj j ( ) [ ] is the coloured generating( function) = of+ partitions . The coarse Hilbert scheme

Points of Hilb C2 G ideals in C x, y G

Again, we can( apply ) torus↔ localization.[ ] Torus ﬁxed points of Hilb C2 G monomial ideals in C x, y G Young diagrams where( all the ) generators are of color 0: 0-generated↔ Young diagrams[ ] ↔ Colength of a monomial ideal I C x, y G = number of 0 boxes

In particular, [ ] n 2 Hilb C # a 0-generated Young diagram with n boxes of colour 0 ( ) = { } Abelian case cont’d C x, y G C xn 1, xy, y n 1 Recall the pushforward+ map:+ [ ] = [ 2 ] 2 G G Hilb C G Hilb C G , I I I C x, y . On the level of diagrams, we extend the diagram of I to the ([ ]) → ( ) = ∩ [ ] smallest 0-generated diagram which covers it. Example Abacus on n 3rulers:toadiagram associate the inﬁnite series of = numbers: , 1, 2,... : 2 1 2 3 0 1 { − − } 1 2 8 7 6 ⋮⋮⋮ 2 0 5 4 3 − − − 0 1 2 0 2 1 0 − − − 123 − − 4 5 6

⋮⋮⋮ Abelian case cont’d C x, y G C xn 1, xy, y n 1 Recall the pushforward+ map:+ [ ] = [ 2 ] 2 G G Hilb C G Hilb C G , I I I C x, y . On the level of diagrams, we extend the diagram of I to the ([ ]) → ( ) = ∩ [ ] smallest 0-generated diagram which covers it. Example 0 Abacus on n 3rulers:toadiagram associate the inﬁnite series of = numbers: , 1, 2,... : 2 0 1 2 3 0 1 { − − } 1 2 0 8 7 6 ⋮⋮⋮ 2 0 0 5 4 3 − − − 0 1 2 0 0 2 1 0 − − − 123 − − 4 5 6

⋮⋮⋮ Abelian case cont’d C x, y G C xn 1, xy, y n 1 Recall the pushforward+ map:+ [ ] = [ 2 ] 2 G G Hilb C G Hilb C G , I I I C x, y . On the level of diagrams, we extend the diagram of I to the ([ ]) → ( ) = ∩ [ ] smallest 0-generated diagram which covers it. Example 0 Abacus on n 3rulers:toadiagram associate the inﬁnite series of 1 = numbers: , 1, 2,... : 2 0 1 2 3 0 1 { − − } 1 2 0 8 7 6 ⋮⋮⋮ 2 0 1 2 0 5 4 3 − − − 0 1 2 0 1 2 0 2 1 0 − − − 123 − − 456

⋮⋮⋮ Abelian case cont’d C x, y G C xn 1, xy, y n 1 Recall the pushforward+ map:+ [ ] = [ 2 ] 2 G G Hilb C G Hilb C G , I I I C x, y . On the level of diagrams, we extend the diagram of I to the ([ ]) → ( ) = ∩ [ ] smallest 0-generated diagram which covers it. Example Abacus on n 3rulers:toadiagram associate the inﬁnite series of 1 = numbers: , 1, 2,... : 2 1 2 3 0 1 { − − } 1 2 8 7 6 ⋮⋮⋮ 2 0 1 2 5 4 3 − − − 0 1 2 0 1 2 2 1 0 − − − 123 − − 456

⋮⋮⋮ Abelian case cont’d So p on the abacus corresponds to pushing all the bead to the right as much as possible. In particular, ∗ abacus conﬁgurations where no 0-generated Young diagrams bead can be moved to the right. Theorem (Gy–N´emethi–Szendr˝oi,←→ 2015)

2⇡i Let ⇠ e n 2 . Under the substitution qi ⇠ i 1,...,n , n + q0 ⇠ q the terms in Z C2 G q0,...,qn corresponding to the preimage=− of a 0-generated diagram =0 add∈ { up to the} term = [ ]( ) corresponding to 0 in ZC2 G q :

qwti µ ( ) qwt0 0 . i n 1 q1 qn ⇠,q0 ⇠ q µ p 0 i ( ) ( ) − − ∗ =⋅⋅⋅= = = = As a consequence,∈ ( ) Z 2 q Z 2 q0,...,qn . C G C G n q1 qn ⇠,q0 ⇠ q [ ] − ( ) = ( ) =⋅⋅⋅= = = Nonabelian cases

In type D a similar but more complicated combinatorics works. Diagonally colored Young diagrams are replaced by objects called Young walls. 1 2 Example (G D4) 4 0 2 3 = 1 2 4 4 0 2 3 3 1 2 4 2 4 0 2 3 2 1

wi number of blocks (squares or triangles) with label i. 2⇡i The( substitution) = works with the root of unity ⇣ e 1 h ,whereh is the Coxeter number. + = There is no theory of Young diagrams/walls for type E. The generating functions Theorem (Nakajima, 90ies) Hilb C2 G is a smooth quiver variety associated with the corresponding ane Dynkin quiver. H Hilb C2 G carries an action([ of the]) corresponding ane Lie algebra.∗ ( ([ ]) Corollary (using the Weyl–Kac character formula)

m1 mn 1 2 m C m n q q q m m1,...,mn Z 1 n Z 2 q ,...,q C G 0 n 1 qm n 1 ⋅ ⋅ ∑ =( )∈ m 1 ⋅⋅⋅⋅⋅ ( ) [ ]( ) = ∞ + n di = where q i 0 qi , di dim ⇢i ,and∏C is the( − corresponding) ﬁnite type Cartan matrix. = ∏ = = Corollary (Previous Corollary + GyNSz) 2⇡i For G of type A or D (conjecturally of type E), let ⇣ e 1 h . + m1 m2 mn 1 2 m C m n ⇣ q = m m1,...,mn Z Z 2 q C G 1+ q+m⋅⋅⋅+n 1 ⋅ ⋅ ∑ =( )∈ m 1 ( ) ( ) = ∞ + ∏ = ( − ) Simple singularities - McKay correspondence (again)

Finite subgroups of Sl2 C An n 1 , Dn n 4 , E6, E7, E8. To the quotient C2 G we can associate its resolution graph: ( ) ∶ ( ≥ ) ( ≥ )

The (irred.) representations of the ﬁnite group are described by another graph (McKay quiver, ⇢def ⇢i j aij ⇢j ):

⊗ = Doubled framed McKay quiver

Q Q0, Q1 : starting from the McKay quiver add one framing vertex at 0: Q , 0, 1,...,n , = ( ) 0 double the arrows: Q1. = {∞ } Example ⇒ G Z7 A6

= = 2 1 3

∞ 4 6 5 Moduli space of quiver representations

Q0 Dimension vector: v vi i Q0 N with v 1. The space of stability conditions: ∶= ( ) ∈ ∈ ∞ = Q0 ⇥v ✓ Z Q ✓ v 0

The space of representations:∶= { ∶ → ( ) = }

vt a vh a Rep Q, v a Q1 Hom C , C ( ) ( ) ∈ Symmetry group: G( v ) ∶= ⊕0 i n GL vi(, C . ) Let G G v diagonal . C ≤ ≤ Its action on Rep Q(, v) =induces∏∗ a moment( ) map: ∶= ( )( ) ( )µ Rep Q, v g ∗ ∶ ( ) → Moduli space of quiver representations

Any ✓ ⇥v induces a character:

∈ ✓ g G C g ∗ det g ✓i ( ) ∶ → i i Q0 − ( ) A representation M is ✓-stable (respectively∈ semistable) if ✓ dimM 0 and for every proper, nonzero subrepresentation N M,wehave✓ dimN 0(respectively✓ dimN 0 . ( ) = Deﬁnition⊂ (Nakajima( quiver) > variety) ( ) ≥ ) For any ✓ ⇥v ,let

1 ∈ M✓ v µ 0 ✓G − be the GIT quotient of the( locus) ∶= of ✓(-semistable) points of µ 1 0 by the action of G. − ( ) Quiver varieties - examples

Example (Kronheimer–Nakajima)

Choose v1 vreg dim ⇢i . Then for generic stability condition ✓, the GIT quotient M✓ v1 is independent of ✓, and is isomorphic to the minimal= resolution= { Y of} the surface singularity X C2 G. ( ) Example (folklore) = Let v1 n vreg n dim ⇢i for some natural number n. Choose the stability condition ✓ 0. Then the GIT quotient M0 vn is ane (general= ⋅ fact),= { ⋅ and is} isomorphic to the n-th symmetric n product S X .Inparticular,= M0 vn X . ( )

( ) ( ) ≅ Generic and special stability parameters

We continue to work with this setup: ﬁx vn n vreg n dim ⇢i , and study the space M✓ vn as the stability parameter ✓ varies. = ⋅ = { ⋅ } By general principles of variation( ) of GIT (Thaddeus, Dolgachev-Hu), one expects a wall-and-chamber structure, with stability parameters in open chambers giving nice GIT quotients

M✓ vn , while the quotient M✓0 vn becomes more singular for parameters ✓0 lying in walls. ( ) ( ) The general setup will also induce morphisms

M✓ vn M✓0 vn

relating di↵erent quiver varieties.( ) → ( )

Example (continued)

With v1 dim ⇢i as above, moving from a generic stability condition ✓ to ✓ 0 gives a morphism M✓ v1 M0 v1 which can be identiﬁed= { with} the minimal resolution Y X C2 G. = ( ) → ( ) → = Adistinguishedchamberinstabilityspace

Theorem (Varagnolo–Vasserot, Kuznetsov) Fix n 1. There exists a distinguished open chamber C ⇥ inside stability space, so that for ✓ C , + ≥ + ⊂ n ⇢reg 2 M✓ vn Hilb∈ C ⋅ where on the right we have( the) ≅ G-equivariant( ) Hilbert scheme of 2 C corresponding to n ⇢reg Rep G ,with⇢reg the regular representation. ⋅ ∈ ( ) The morphism to the zero of the stability space can be identiﬁed with n ⇢ 2 n 2 Hilb reg C S C ⋅ which is a minimal resolution( of singularities.) → ( ) Example (continued again, thm of Kapranov–Vasserot) ⇢ For n 1, Hilb reg C2 Y , the minimal resolution of C2 .

= ( ) ≅ The wall-and-chamber structure of stability space

Theorem (Bellamy–Craw)

1. ⇥ has a wall-and-chamber structure described combinatorially by the ane root system. 2. The simplicial cone

F ,⇢1,...,⇢n ⇥ ∨ is a fundamental domain∶= for the Weyl group⊂ action on ⇥. 3. There is a closed cone C¯ that can be identiﬁed with the nef n ⇢ 2 cone (closed ample cone)+ of the variety Hilb reg C . 4. Every open chamber of F corresponds to the⋅ ample cone of a n ⇢ ( ) birational model of Hilb reg C2 . ⋅ ( ) Space of stability conditions

Example

The height 1 transversal slice of F ,⇢1,⇢2 for A2 and ⇢ 3⇢reg : ∨ = =

⇢1 ⇢2 3 ⇢ C Hilb reg C2 G ⋅ + ([ ]) Adistinguishedcornerofthestabilityspace Theorem (Craw–Gammelgaard–Gy–Szendr˝oi, 2019)

For a distinguished ray ✓0 @C¯ , we have an isomorphism + n 2 M✓0∈ vn Hilb C

between a quiver variety and( the) ≅ Hilbert( scheme) of points of the surface singularity. Example

⇢1 ⇢2 3 ⇢ C Hilb reg C2 G ⋅ 3 2 + Hilb ([C G ]) ✓0 ( ) Adistinguishedcornerofstabilityspace(continued) Theorem (continued) The resulting chain of morphisms

M✓ vn M✓0 vn M0 vn

can be identiﬁed with( the) chain→ ( ) → ( )

n ⇢ 2 n 2 n 2 Hilb reg C Hilb C G S C G ⋅ which includes the Hilbert–Chow( ) → morphism( ) → of the( variety ) C2 G.

Corollary n The Hilbert scheme Hilb X of the surface singularity X C2 G is an irreducible, normal quasiprojective variety with a unique symplectic (Calabi–Yau) resolution.( ) = This is about as nice as one could hope for! Conjecturally this property characterises surface rational double points among all varieties of dimension at least 2. Many spaces in one diagram

We get the following diagram of GIT-induced morphisms, including the Hilbert–Chow morphisms of both the singularity X C2 G and its minimal resolution Y . = Example Sn Y

Hilb(n Y)

( ) Sn C2 G ≅ n ⇢ Hilb reg C2 G ( ) ⋅ n Hilb ([C2 G ])

( ) Back to Euler characteristics

We have identiﬁed the resolution of singularities

n ⇢ 2 n Hilb reg C Hilb X ⋅ with a map ( ) → ( )

M✓ vn M✓0 vn

between quiver varieties. ( ) → ( ) This suggests that the conjecture of Gy–Némethi–Szendr˝oiabout the generating function of Euler characteristics of Hilbn X could be approached this way. ( ) Nakajima: the fibres of the map M✓ vn M✓0 vn between quiver varieties are themselves (Lagrangian subvarieties in) quiver varieties associated with finite ADE quivers( ) →. ( ) This looks like it gives an approach to the conjecture. However, computing the Euler characteristics of fibres directly is still hard! Nevertheless, using his earlier results based on supercomputer calculations...

Theorem (Nakajima, 2020) i 2⇡i For G of arbitrary type,withq i qi and ⇠ exp 1 h ,the generating function of the Hilbert scheme of points of the surface + singularity X C2 G is related= to∏ the equivariant= generating( ) function by the formula where h is the Coxeter number of the Lie algebra of the= corresponding type. In other words, the conjecture of Gy–N´emethi–Szendr˝oifrom 2015 holds. Further directions (work in progress) Do other walls in the space of stability parameters correspond to moduli spaces? The answer, at least in type A, seems to be yes. Proposition

1. For every j 0, 1,...,n there is a ray ✓j @C¯ such that + n 2 ∈ { M✓j vn} Quot C G, j∈

where the j are indecomposable( ) ≅ ( reﬂexive R modules) on C2 G, 0 . R 2. The substitution q ⇠,i 0, 1,...,n j , q ⇠ nq into ( ) i j Z C2 G q0,...,qn gives the generating series − = ∈ { } { } = [ ] ( ) n 2 m Z 2 q Quot G, j q . C G,j ∞ C n 0 ( ) = ( R ) 3. Combinatorially: count the= j-colength of j-generated partitions. Thank you for your attention! Questions?