# On the Virtual Fundamental Class

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1 On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14,

2 Overview Donaldson-Thomas theory: counting invariants of sheaves on Calabi-Yau threefolds. Symmetric obstruction theories and the virtual fundamental class. Additivity (over stratifications) of Donaldson-Thomas invariants. Motivic Donaldson-Thomas invariants. Categorification of Donaldson-Thomas invariants. Begin with: review of the local case: critical loci.

3 Local discussion 2/ 22 The singular Gauß-Bonnet theorem M f : M C smooth complex manifold (not compact), holomorphic function, X = Crit f M. Assume X compact. Then X is the intersection of two Lagrangian submanifolds in Ω M (complementary dimensions): X M Γ df M 0 Ω M X is compact: intersection number # virt (X ) = I ΩM (M, Γ df ) = [X ] virt 1 well-defined. [X ] virt A 0 (X ) virtual fundamental class of the intersection scheme X. Theorem (Singular Gauß-Bonnet) µ : X Z constructible function. I ΩM (M, Γ df ) = χ(x, µ)

4 Singular Gauß-Bonnet theorem 3/ 22 Milnor fibre X = Crit f M f : M C holomorphic Theorem: I ΩM (M, Γ df ) = χ(x, µ) F P : Milnor fibre of f at P: intersection of a nearby fibre of f with a small ball around P. ) µ(p) = ( 1) (1 dim M χ(f P ) : Milnor number of f at P X = Crit f. Consider two cases of the theorem: dim X = 0, X smooth.

5 Case dim X = 0 4/ 22 Milnor fibre example: f (x, y) = x 2 + y 2 X = Crit(f ) = {P}. Isolated singularity. I ΩM (M, Γ df ) = 1. Near P, the surface f 1 (0) is a cone over the link of the singularity. The cone is contractible. The Milnor fibre is a manifold with boundary. The boundary is the link. The Milnor fibre supports the ) vanishing cycles.the Milnor number µ(p) = ( 1) (1 dim M χ(f P ) is the number of vanishing cycles. Here, χ(x, µ) = µ(p) = 1, and hence I ΩM (M, Γ df ) = χ(x, µ).

6 Case dim X = 0: Milnor s theorem 5/ 22 Milnor fibre example: f (x, y) = x 2 + y 3 X = Crit(f ) = {(x, y) 2x = 0, 3y 2 = 0} = Spec C[y]/y 2. Isolated singularity of multiplicity 2. I ΩM (M, Γ df ) = 2. Link: (2, 3) torus knot (trefoil). The singularity is a cone over the knot. The link bounds the Milnor fibre. Homotopy type (Milnor fibre) = bouquet of 2 circles. χ(f P ) = 1 2 = 1. The Milnor number is µ(p) = ( 1) 2( 1 ( 1) ) = 2. There are 2 vanishing cycles. In this example, I ΩM (M, Γ df ) = 2 = χ(x, µ). Theorem (Milnor, 1969) For the case dim X = 0 (isolated singularities) I ΩM (M, Γ df ) = Milnor number = χ(x, µ).

7 Case X smooth 6/ 22 The excess bundle X = Crit f M f : M C holomorphic Theorem: I ΩM (M, Γ df ) = χ(x, µ) Suppose X is smooth. So N X /M = I X /I 2 X Epimorphism T M df I X O M. df Restrict to X : T M X I /I 2 d Ω M X. H(f ) is a vector bundle on X. (Recall: I /I 2 d Ω M X Ω X 0.) The Hessian matrix H(f ) is symmetric, so taking duals we get the same diagram, so that I /I 2 = N X /M. So the excess bundle (or obstruction bundle) is X M M 0 Ω M Γ df Intrinsic to the intersection X. Ω M X N X /M = Ω M X I /I 2 = Ω X. Always so for Lagrangian intersections.

8 Case X smooth 7/ 22 The excess bundle X = Crit f M f : M C holomorphic Theorem: I ΩM (M, Γ df ) = χ(x, µ) In the case of clean intersection, the virtual fundamental class is the top Chern class of the excess bundle, so: Proposition [X ] virt = c top Ω X [X ] Hence, I ΩM (M, Γ df ) = c top Ω X [X ] = ( 1) dim X [X ] c top T X = ( 1) dim X χ(x ), by Gauß-Bonnet = χ(x, µ X ), with µ X = ( 1) dim X. and for smooth X it turns out the µ X ( 1) dim X. For X smooth, the theorem is equivalent to Gauß-Bonnet.

9 General case 8/ 22 Additive nature of # virt (X ). f : M C, X = Crit f. Theorem (Singular Gauß-Bonnet) # virt (X ) = χ(x, µ) is a result of microlocal geometry, in the 1970s. Main ingredient in proof: microlocal index theorem of Kashiwara, MacPherson. Also: determination of the characteristic variety of the perverse sheaf of vanishing cycles. Major significance: intersection number is motivic, i.e., intersection number makes sense for non-compact schemes: # virt (X ) = χ(x, µ), intersection number is additive over stratifications: χ(x, µ X ) = χ(x \Z, µ X ) + χ(z, µ X ), if Z X is closed. This is unusual for intersection numbers, only true for Lagrangian intersections.

10 Generalization: 9/ 22 Motivic critical loci Group of motivic weights K(Var): Grothendieck group of C-varieties modulo scissor relations: [X ] = [X \ Z] + [Z], whenever Z X is a closed immersion. There exists a lift (motivic virtual count of critical loci): critical loci Φ # virt K(Var) Z χ where Φ(M, f ) = q dim M 2 [φ f ], q = [C] motivic weight of the affine line, [φ f ] motivic vanishing cycles of Denef-Loeser (2000): motivic version of Milnor fibres. (From their work on motivic integration.)

11 Generalization: 10/ 22 Example: Hilb n (C 3 ) ( Hilb n (C 3 ): scheme of three commuting matrices: critical locus of Mn n (C) 3 C n) stab /GL n C, (A, B, C, v) tr([a, B]C). Theorem (B.-Bryan-Szendrői) Φ ( Hilb n (C 3 ) ) t n = n=0 Specialize: q 1 2 1, get # virt( Hilb n (C 3 ) ) t n = n=0 m m=1 k=1 1 1 q k+1 m 2 t m ( 1 ) m. 1 ( t) m m=1 This is (up to signs) the generating function for 3-dimensional partitions #{3D partitions of n} t n = n=0 ( 1 ) m 1 t m m=1

12 Generalization: 11/ 22 Categorified critical locus f : M C, X = Crit f. Let Φ f = Φ f [dim M 1] D c (C X ) be the perverse sheaf of (shifted) vanishing cycles for f (Deligne, 1967). Φ f globalizes the reduced cohomology of the Milnor fibre: H i (Φ f P ) = H i (F P ). Theorem i ( 1)i dim H i (X, Φ f ) = χ(x, µ) So (X, Φ f ) categorifies the virtual count. De Rham model: twisted de Rham complex. Pass to ground field C(( )). Theorem (Sabbah, 2010) ( Ω M (( )), df + d ) [dim M] D c (C(( )) X ) is a perverse sheaf, and i ( 1)i dim C(( )) H i( Ω M (( )), df + d) = χ(x, µ).

13 Donaldson-Thomas invariants 12/ 22 Calabi-Yau threefolds Definition A Calabi-Yau threefold is a complex projective manifold Y of dimension 3, endowed with a nowhere vanishing holomorphic volume form ω Y Γ(Y, Ω 3 Y ). Example. Y = Z(x x 5 4 ) P4 the Fermat quintic. Example. More generally, g(x 0,..., x 4 ) a generic polynomial of degree 5 in 5 variables. Y = Z(g) P 4 the quintic threefold. Example. Algebraic torus C 3 /Z 6 (sometimes excluded, because it is not simply connected). CY3: the compact part of 10-dimensional space-time according to superstring theory.

14 Donaldson-Thomas invariants 13/ 22 Moduli spaces of sheaves Y : Calabi-Yau threefold. Fix numerical invariants, and a stability condition. X : associated moduli space of stable sheaves (derived category objects) on Y. Example: Fix integer n > 0. X = Hilb n (Y ), Hilbert scheme of n points on Y. E X E is the ideal sheaf of a (degenerate) set of n points in Y. Example: Fix integers n Z, d > 0. X = I n,d (Y ), [MNOP] moduli space of (degenerate) curves of genus 1 n, degree d in Y. E X E ideal sheaf of a 1-dimensional subscheme Z Y. Example: Fix r > 0, and c i H 2i (Y, Z). X : moduli space of stable sheaves (degenerate vector bundles) of rank r, with Chern classes c i on Y.

15 Donaldson-Thomas invariants 14/ 22 Donaldson-Thomas theory X : can be a finite set of points. Example. Y : quintic 3-fold in P 4. X = I 1,1 (Y ) moduli space of lines on Y. X = I 1,2 (Y ) moduli space of conics in Y. X : 2875 discrete points. X : discrete points. Slogan. If the world were without obstructions, all instances of X would be finite sets of points. Goal (of Donaldson-Thomas theory) Count the (virtual) number of points of X. Bad news. X almost never zero-dimensional, almost always very singular. Good news. X is quite often compact: always for examples Hilb n (Y ) and I n,d (Y ), sometimes in the last example (depending on the c i ). Thomas: constructs a virtual fundamental class [X ] virt A 0 (X ), and defines # virt (X ) = [X ] virt 1 Z, if X compact. Kuranishi: X is locally isomorphic to Crit f, for suitable f (restrict Chern-Simons to local Kuranishi slices).

16 Obstruction theories 15/ 22 Derived schemes: virtual fundamental class More fundamental geometrical object, the derived moduli scheme X X. Induces morphism T X T X X in D(O X ) of tangent complexes. This morphism is an obstruction theory for X. All derived schemes come with an amplitude of smoothness: T X X D [0,n] (X ) amplitude n. (e.g. classical smooth schemes are derived schemes of amplitude 0) Derived schemes X of amplitude 1 have a virtual fundamental class [X ] virt A rk TX X (X ). [X ] virt = 0! V [C], V: the vector bundle stack associated to the obstruction theory T X X, if T X X = [V 0 V 1 ], V = [V 1 /V 0 ], C: the intrinsic normal cone of X, [C] its fundamental cycle A 0 (E), C = [C X /M /T M X ], if X M, C V (cone stack in vector bundle stack) comes from T X T X X, [X ] virt = 0! V [C] the Gysin pullback, via 0 V : X V, of [C].

17 Symmetric obstruction theories 16/ 22 Shifted symplectic structures X : moduli space π : X Y X E on X Y universal sheaf. T X X = ( τ [1,2] Rπ R H om(e, E ) ) [1] D [0,1] (X ). If P = [E], H 0 (T X P ) = Ext 1 O Y (E, E) = T X P, deformation space, H 1 (T X P ) = Ext 2 O Y (E, E), obstruction space. Serre duality: Deformation space dual of obstruction space H 0 (T X P ) = H 1 (T X P ). X = Crit f, T X X = [T M X H(f ) Ω M X ]. In both cases, T X X is a symmetric obstruction theory, i.e., isomorphism θ : T X X (TX X ) [ 1], such that θ [ 1] = θ. As a pairing: θ : Λ 2 T X X O X [ 1]. This is the classical shadow on the classical locus X X of a shifted symplectic structure on X. Shifted Darboux theorem. Every 1 shifted symplectic structure is locally a derived critical locus.

18 Symmetric obstruction theories 17/ 22 Global version and generalization of singular Gauß-Bonnet Y : is a complex projective Calabi-Yau threefold. X : a moduli space of sheaves on Y, or any scheme endowed with a symmetric obstruction theory. Theorem (B.) Suppose that X is compact. Then the Donaldson-Thomas virtual count is [X ] virt 1 = χ(x, ν X ). ν X : X Z constructible function ν X (P) Z invariant of the singularity of X at P X. Contribution of P X to the virtual count ν X (P) = µ(p) if there exists a holomorphic function f : M C, such that X = Crit f, near P. Construction. ν X is the local Euler obstruction of the image of [C] in X. Proof. Globally embed X M. When performing deformation to the normal cone (locally) inside Ω M, you get Lagrangian cone. Then use K-M.

19 Applications 18/ 22 Additivity of DT invariants Now Donaldson-Thomas invariants exist for X not compact, and are additive over stratifications. Example: global version of n=0 ( # virt Hilb n (C 3 ) ) t n = m=1 ( ) 1 m 1 ( t) m Theorem (B.-Fantechi, Levine-Pandharipande, Li, 2008) Y : Calabi-Yau threefold. ( # virt Hilb n Y ) ( t n = n=0 m=1 ( 1 1 ( t) m ) m ) χ(y ) Simplest non-trivial computation of Donaldson-Thomas invariants using additive nature of the invariants. More in Toda s talk.

20 Applications 19/ 22 Motivic Donaldson-Thomas invariants X moduli space of sheaves on Calabi-Yau threefold Y. To define motivic Donaldson-Thomas invariants, use X is locally Crit f, motivic vanishing cycles orientation data moduli spaces Φ # virt K(Var) Z χ Theorem (B.-Bryan-Szendrői, 2013) ( m Φ(Hilb n Y ) t n 1 ) [Y ] = 1 q k 2 m 2 t m n=0 m=1 k=1 This formula uses the power structure on K(Var). Elaborate theory of motivic invariants by Kontsevich-Soibelman.

21 Applications 20/ 22 Categorification by gluing perverse sheaves Kiem-Li, Joyce et al, (2013) have constructed a perverse sheaf Φ on X, such that # virt (X ) = χ(x, ν X ) = ( 1) i dim H i (X, Φ), by gluing the locally defined perverse sheaves of vanishing cycles for locally existing Chern-Simons potentials.

22 Applications 21/ 22 Categorification via quantization To globalize the de Rham categorification to moduli spaces X, expect to need derived geometry, not just its classical shadows, such as T X X. (More: see Toën s talk.) Consider the local case X = Crit f, f : M C, M smooth. The derived critical locus: A, with A i = Λ i T M, the graded algebra of polyvector fields. Contraction with df defines a derivation Q : A i A i+1, such that Q Q = 1 2 [Q, Q] = 0. The differential graded scheme X = (M, A, Q) is one model of the derived scheme X. X has a 1-shifted symplectic structure on it, of which [T M X H(f ) Ω M X ] is the classical shadow. A has the Lie Schouten bracket {, } of degree +1 on it. This is the Poisson bracket on the algebra of functions of the shifted symplectic scheme X. (Q is a derivation with respect to this bracket.)

23 Applications 22/ 22 Categorification via quantization X = (M, A, Q) dg scheme, A = ΛT M, Q = df {, } Suppose given a volume form on M, (or just a flat connection on the canonical line bundle on M.) This defines a divergence operator : T M O M, wich extends to : A A [1], such that 2 = 0. generates the bracket {, } (xy) ( 1) x x (y) (x)y = {x, y} and commutes with Q. (Batalin-Vilkovisky operator). Then (A (( )), Q + ) categorifies # virt (Crit f ). (Using a volume form on M, giving rise to the divergence, we can identify ΛT M = Ω M [dim M], and then (A (( )), Q + ) = (Ω M (( )), df + d)[dim M] the twisted de Rham complex from above.) For Lagrangian intersections in complex symplectic manifolds, Kashiwara-Schapira (2007) globalized this construction, thus categorifying Lagrangian intersection numbers.

24 Thanks!

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