Cosection localization and the Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$

Let $\mathcal{E}$ be a locally free sheaf of rank $r$ on a smooth projective surface $S$. The Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ of length $l$ coherent sheaf quotients of $\mathcal{E}$ is a natural higher rank generalization of the Hilbert scheme of $l$ points of $S$. We study the virtual intersection theory of this scheme. If $C\subset S$ is a smooth canonical curve, we use cosection localization to show that the virtual fundamental class of $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ is $(-1)^{l}$ times the fundamental class of the smooth subscheme $\mathrm{Quot}^{l}_{C}(\mathcal{E}\vert_{C})\subset\mathrm{Quot}^{l}_{S}(\mathcal{E})$. We then prove a structure theorem for virtual tautological integrals over $\mathrm{Quot}^{l}_{S}(\mathcal{E})$. From this we deduce, among other things, the equality of virtual Euler characteristics $\chi^{\mathrm{vir}}(\mathrm{Quot}^{l}_{S}(\mathcal{E}))=\chi^{\mathrm{vir}}(\mathrm{Quot}^{l}_{S}(\mathcal{O}^{\oplus r}))$.


Introduction
Let S be a smooth projective complex surface. The Hilbert scheme of l points S [l] is a canonical smooth compactification of the configuration space of l unordered points of S, which occurs naturally in several branches of mathematics. A point of S [l] corresponds to a zero-dimensional closed subscheme Z ⊂ S of length l; since a subscheme can be specified by either the quotient O → O Z or by its kernel (i.e. the ideal sheaf of Z ⊂ S), there are two natural generalizations of S [l] : (1) the Quot schemes Quot l S (E) of length l quotients of a locally free sheaf E of higher rank r; (2) the moduli spaces of stable sheaves of higher rank.
Common to both higher rank generalizations is that they are typically singular, and both admit a virtual fundamental class. By integrating universal cohomology classes over it, one obtains invariants of these moduli spaces. For (2), these are the (algebraic) Donaldson invariants of S, which are well studied, see for instance [6,7,10,13,14,22,23,26,29]; the study of the invariants corresponding to (1) is the subject matter of this paper. The schemes Quot l S (E) satisfy three basic properties: Moreover, a locally free sheaf F on S induces an associated tautological sheaf F [l] on Quot l S (E), which is compatible with (I)-(III); we then obtain invariants of Quot l S (E) by taking integrals of Chern classes of tautological sheaves over the virtual fundamental class [Quot l S (E)] vir ∈ A lr (Quot l S (E)).
In the special case E = O ⊕r , the fixed locus of the action of the automorphism group of E is given by products of Hilbert schemes of points of S. It is this observation which is the point of departure of the recent work of Oprea and Pandharipande [30], which studies these integrals using (III) and the virtual torus localization technique of [15]. It is natural to study these invariants for arbitrary locally free sheaves E, by systematically exploiting all three properties (I)-(III). Our first result says that using property (I), the virtual fundamental class can be localized along the zero locus of a 2-form.  Localizing along the zeros of a 2-form is a well-known technique in the study of moduli spaces associated to surfaces [20,27,33,34]. In the case E = O ⊕r Theorem 1.1 was proven in [30] using torus localization; in practice, this result allows one to compute tautological integrals over [Quot l S (E)] vir in terms of tautological integrals over the smooth scheme Quot l C (E| C ). It thus plays a key role in all the computations of [30] and those contained in the follow-up papers [17,24,1]. Our proof has three ingredients: for a 2-form ω ∈ H 0 (Ω 2 S ) with zero scheme C, we construct a cosection σ(ω) of the obstruction sheaf of Quot l S (E) with zero scheme Quot l C (E| C ). By the work of Kiem and Li on cosection localization [20], we obtain a σ(ω)-localized virtual fundamental class by using (a) the dimension of Quot l C (E| C ) equals the virtual dimension of Quot l S (E), (b) the smooth locus of Quot l S (E) intersects the smooth locus of Quot l C (E| C ). Our proof in particular answers a question raised by Oprea and Pandharipande [30]; it applies more generally to any scheme with a perfect obstruction theory and a cosection of its obstruction sheaf satisfying properties analogous to (a) and (b). Although cosection localization has found many applications in enumerative geometry [4,5,21,25], primarily in combination with Graber-Pandharipande's virtual torus localization [15], in the absence of a torus action, the cosection localized virtual fundamental class is poorly understood. In our situation we can compute it completely explicitly, which is fairly rare.
The fundamental observation which guides our approach is that property (I) yields relations in the intersection theory of the schemes Quot l S (E). Apart from Theorem 1.1, another instance of this is the following theorem, which the author proved in [32]. It provides a way of changing the sheaf E, and as such it plays a key role in extending the results of [30] to arbitrary locally free sheaves.
This is a virtual, higher rank analogue of a result of Ellingsrud, Göttsche, and Lehn [8], which has proven to be very useful in a number of contexts. The idea of the proof is as follows: by (II) we can assume that E is globally generated; Theorem 1.2 allows us reduce to the case E = O ⊕r , in which case we can use (III), torus localization and the result of [8]. Finally, we use Theorem 1.1 to eliminate second Chern classes, and (II) to obtain the dependence through the tensor products. Combined with the work of Oprea and Pandharipande, our theorems lead to a fairly complete understanding of the virtual intersection theory of Quot l S (E). As an application of Theorem 1.3, we prove the equality of virtual Euler characteristics using (II). The computation of χ vir (Quot l S (O ⊕r )) is one of the main results of [30]. We also deduce the symmetry property of the virtual Segre series from Theorem 1.3 and the result of [1] in the E = O ⊕r case. As shown in [3], (1) and (2) can also be deduced from Joyce's wall-crossing formula [18]. However, our approach is more geometric and natural, and does not rely on the deep results of [18]. We also show that for any invertible sheaf L on S

Localized Gysin map
Consider a scheme X, and a cosection σ : F → O of a locally free sheaf F on X. Let Z(σ) denote the zero scheme of σ, and F → X the vector bundle associated to F , and write F (σ) for the union F | Z(σ) ∪ Ker(σ), endowed with the reduced subscheme structure. [20]). There exists a σ-localized Gysin map which is compatible with the usual Gysin map 0 ! F of F , in the sense that the diagram commutes.
More precisely, Kiem and Li construct 0 ! σ as follows. If C ⊂ F (σ) is a closed integral subscheme, then either C is contained in F | Z(σ) or not; in the former case one simply puts In the latter case one considers the blow upX = Bl Z(σ) X, forms the Cartesian diagram and definesC to be the proper transform of C. Let J be the ideal sheaf of Z(σ) in X, and write D = ρ −1 (Z(σ)) for the exceptional divisor of ρ, with inclusion i : D →X. Moreover, let K be the vector bundle associated to the kernel of the morphism of sheaves The proper transformC is a closed integral subscheme of K, and 0 is commutative.

Localized virtual fundamental class
Let X be a projective scheme with cotangent complex L X . Recall [2] that a perfect obstruction theory on X is a morphism φ : E → L X in the derived category of X, where E is a perfect 2-term complex concentrated in degrees −1 and 0, such that h 0 (φ) is an isomorphism and h −1 (φ) is surjective. We write Ob = h 1 (E ∨ ) for the obstruction sheaf, and v = rk(E) for the virtual dimension. Behrend and Fantechi show that any perfect obstruction theory on X yields a virtual fundamental class on X, which can be constructed as follows [15].
is a global resolution of E, then by considering the mapping cone of (a representative of) the truncation one obtains a closed subcone C ⊂ F , where F denotes the vector bundle on X associated to (F −1 ) ∨ . The virtual fundamental class of X is defined by which is independent of the choice of resolution of E.
Consider now a cosection σ : Ob → O of the obstruction sheaf, and denote by ι : Z(σ) → X the inclusion of the zero scheme of σ. By abuse of notation, we also write σ for the induced cosection (F −1 ) ∨ → Ob → O. Kiem and Li ([20], Corollary 4.6) prove that cone C ⊂ F is in fact set theoretically contained in F (σ), and therefore defines a class [C] ∈ A * (F (σ)). Kiem-Li's σ-localized virtual fundamental class of X is then given by The diagram (3) shows that Remark 2.2. Let U ⊂ X be an open subscheme, with the induced perfect obstruction theory. Then the restriction map Indeed, the cone C ⊂ F over X restricts to the corresponding cone over U, and so this follows from Remark 2.1.

Explicit computations
We now describe a situation in which the cosection localized virtual fundamental class can be computed explicitly. In the following, we write X 0 for the smooth locus of a scheme X.
Proposition 2.1. Let X be a projective scheme with a perfect obstruction theory of virtual dimension v, and let σ : Ob → O be a cosection of the obstruction sheaf. Assume that the zero scheme Z(σ) is irreducible of dimension v, and that Z(σ) 0 ∩ X 0 is nonempty. Then where c is the codimension of Z(σ) in X.
for some integer n. Let σ 0 be the restriction of σ to the smooth locus X 0 of X. By assumption Z(σ 0 ) = Z(σ) ∩ X 0 is nonempty, and applying Remark 2.2 to (5) gives In fact, since Z(σ) 0 ∩ X 0 is nonempty, we can also replace Z(σ) by Z(σ) 0 , and thus assume that both Z(σ) and X are smooth. In this situation we show, by an explicit computation, that the coefficient in (5) is where c is the codimension of Z(σ) in X. In fact more generally, dropping the assumption that the dimension of Z(σ) is the virtual dimension of X, we show that where c is the codimension of Z(σ) in X. Since X is smooth, the obstruction sheaf Ob is locally free of rank l = dim(X) − v, and the kernel of σ| Z(σ) is locally free of rank dim Z(σ) − v. In the notation of 2.2, the cone C ⊂ F is the the vector bundle associated to the kernel of the surjection (F −1 ) ∨ → Ob. We now consider the construction of the σlocalized Gysin map, which we have described in detail in 2.1. In the notation of 2.1,C is in our situation a subbundle of K, so by a standard result in intersection theory [11], and in particular The inclusion K →F induces a morphism of exact sequences whose associated exact sequence of kernels and cokernels is of the form since C is the the vector bundle associated to the kernel of ( By the equation and the projection formula, we obtain

Virtual intersection theory 3.1 General properties
Let E be a coherent sheaf on a smooth projective surface S. The Quot scheme Quot l S (E) is the moduli space of length l coherent sheaf quotients of E. A point q of Quot l S (E) thus corresponds to a quotient E → Q q on S, with dim Q q = 0 and h 0 (Q q ) = l.
It satisfies the following basic properties [16]: (II) an invertible sheaf L on S induces an isomorphism (III) the automorphism group of E acts on Quot l S (E).
We denote by π : S × Quot l S (E) → Quot l S (E) the projection to Quot l S (E), by E → Q the universal quotient 1 on S × Quot l S (E), and by S its kernel. We will frequently take the Fourier-Mukai transform with kernel Q: a locally free sheaf F on S induces a tautological sheaf It is locally free with fibre H 0 (F ⊗ Q q ) over a point q corresponding to the quotient E → Q q on S. An elementary but important observation is [32]: In all that follows, we will take E to be locally free of rank r. In this case, it is well-known that the canonical deformation-obstruction theory RHom π (S, Q) ∨ → L Quot l S (E) 1 Here and elsewhere we will sometimes suppress notationally the pullback along projections. is perfect of virtual dimension lr [30], and therefore gives rise to a virtual fundamental class [Quot l S (E)] vir ∈ A lr (Quot l S (E)).
We will denote the two-term complex RHom π (S, Q) by T vir Quot l S (E) , and use the identification of the obstruction sheaf Ob = Ext 1 π (S, Q) ∼ − → Ext 2 π (Q, Q), obtained by applying Ext π (−, Q) to the universal exact sequence.

Localization along canonical curves
By definition, the universal quotient sheaf Q is flat with respect to the smooth projective morphism π, in particular perfect. Therefore, we obtain in particular a trace map Tr : RHom(Q, Q) → O, and denote by the induced map on top degree cohomology sheaves (we refer to [19] for the standard properties of these trace maps). We then have a canonical map taking a 2-form ω ∈ H 0 (Ω 2 S ) to the cosection σ(ω) : Ob → O given by Moreover, if C denotes the zero scheme of ω, (I) gives a closed immersion We can now prove Theorem 1.1.
Theorem 3.1. Let ω be a 2-form on S whose zero scheme is a smooth irreducible curve C. Then (i) the zero scheme of the cosection σ(ω) can be identified with Quot l C (E| C ); (ii) the σ(ω)-localized virtual fundamental class of Quot l S (E) is given by In particular, Proof. (i) Let q be a point of Quot l S (E). By the base change property of the trace map Tr 2 π , the fibre of σ(ω) over q is given by S ω ∪ Tr 2 q , where is the global trace map corresponding to q. The multiplicativity property of the trace map asserts that the diagram is commutative, where ∪ denotes the Yoneda pairing. It implies in particular that under Serre duality Ext 2 (Q q , Q q ) ∨ ≃ Hom(Q q , Q q ⊗ Ω 2 S ) the fibre of σ(ω) over q corresponds to the map 1 ⊗ ω : Q q → Q q ⊗ Ω 2 S . Hence σ(ω) vanishes at q if and only if the support of Q q is contained in C, which in turn is equivalent to q being contained in Quot l C (E| C ) ⊂ Quot l S (E). (ii) It suffices to show that the assumptions of Proposition 2.1 are satisfied. Let q be a point of Quot l C (E| C ). Since C is a curve, the subsheaf S q ⊂ E| C is locally free of rank r, the tangent space Hom(S q , Q q ) ≃ H 0 (S ∨ q ⊗ Q q ) has dimension lr, and the obstruction space Hence Quot l C (E| C ) is smooth of dimension lr, which is the virtual dimension of Quot l S (E). To see that Quot l C (E| C ) is irreducible, one can use induction on l and the flag Quot scheme: we have Quot 1 C (E| C ) ≃ P(E| C ), and for the induction step one observes that the canonical map Quot l,l+1 is the projectivization of the universal subsheaf, which is locally free. To show that the intersection of Quot l C (E| C ) with the smooth locus Quot l S (E) 0 is nonempty, recall [32] that a point q of Quot l S (E) is smooth if and only if End(Q q ) has dimension l. Thus any point q of Quot l C (E| C ) represented by a quotient of the form where L is an invertible sheaf on C and Z ⊂ C a subscheme of length l, lies in the intersection.
Example 3.1. Theorem 3.1 applies to a large class of surfaces S. If the canonical sheaf Ω 2 S is very ample -this is satisfied by any smooth complete intersection S ⊂ P n of type (d 1 , . . . , d n−2 ) with n−2 i=1 d i n + 2 (n 3) -then Bertini's theorem implies that the zero scheme of a generic 2-form is smooth and irreducible. If S has trivial canonical sheaf, then by taking ω to be nonzero, the cosection σ(ω) is surjective, and thus [Quot l S (E)] vir = 0; in contrast, however, the exceptional divisor of a blow up of S is cut out by a 2-form.
Consider now a locally free quotient E → E ′′ with kernel E ′ . By (I) we have a closed immersion ι : Quot l S (E ′′ ) → Quot l S (E). As observed by the author in [32], the inclusion E ′ → E induces a regular section of E with zero scheme Quot l S (E ′′ ), and we have an exact triangle in the derived category of Quot l S (E ′′ ). This led to the following result: Proof. Indeed, we have a commutative diagram of morphisms of Quot schemes

of morphisms of sheaves on S and property (I). The image of [Quot
Indeed, this follows from the same argument used to prove Theorem 3.2, but here the section cutting out Quot l C (E ′′ | C ) ⊂ Quot l C (E| C ) is automatically regular (as these Quot schemes are smooth). By Lemma 3.1, the tautological sheaf E| on Quot l S (E), and the map

Universality
Lemma 3.2. Let C ⊂ S be a smooth curve. Then we have an exact triangle of the form in the derived category of Quot l C (E| C ), where Q C denotes the universal quotient sheaf on C × Quot l C (E| C ).
Proof. By definition of the embedding Quot l C (E| C ) → Quot l S (E), we have a morphism of exact sequences of sheaves where the top exact sequence is obtained by restricting the universal exact sequence on S × Quot l S (E), and the bottom exact sequence is the pushforward of the universal exact sequence on C × Quot l C (E| C ). As the middle vertical map is given by restriction, taking the long exact sequence of derived pullbacks with respect to the embedding gives an exact sequence of the form where Q C (−C) is identified with the first derived pullback of the pushforward of Q C to S × Quot l C (E| C ). Applying the functor RHom π (−, Q C ) gives the required exact triangle, as Hom π (S C , Q C ) is the tangent sheaf of Quot l C (E| C ) and RHom π (S, Q C ) ≃ Lι * RHom π (S, Q) by the projection formula and base change.
m and T vir Quot l S (E) are compatible with (II), we may assume that E is globally generated. Consider an exact sequence of the form By Theorem 3.2, Lemma 3.1 and (7), we have We can thus apply the universality theorem for tautological integrals over Hilbert scheme of points [8], and using c(K) = c(E) −1 we obtain that the integral C , RHom π (Q C , Q C (C)), and Quot l C (E| C ). The argument above, mutatis mutandis, also shows that Quot l C (E| C ) P C is given by a universal polynomial U C in the genus g of C and the intersection numbers it suffices to show that U C is a polynomial in g and c 1 (E| C ⊗F i | C ). For simplicity of notation, we take m = 1 and U C to be homogenous of degree d, By (II) and Lemma 3.1, the polynomial U C = U C (F | C , E| C ) satisfies for any invertible sheaf L on C. Taking the coefficient of where f is the rank of F . In particular Substituting this into (11), we obtain It is clear from the proof that c 1 (E)c 1 (S) can only occur if the first Chern classes of the F i can be expressed in terms of the first Chern class of E. Remark 3.2. As pointed out to us by R. Thomas, one can also approach Theorem 3.3 as follows. Assume for simplicity that E has rank r = 2; as in the proof above, we may assume that E is globally generated. By Bertini's theorem, the zero scheme Z = Z(s) of a generic section s of E consists of n = S e(E) reduced points, and the Koszul complex of s gives a nontrivial extension By a standard construction, this yields a deformation of E into the torsion-free sheaf O ⊕ J Z ⊗ det(E), inducing a deformation of Quot l S (E) into Quot l S (O ⊕ J Z ⊗ det(E)). One could then use (III) and torus localization, as well as the results of [8], as the Quot scheme Quot l S (J Z ) can be identified with the locus of all [Z ′ ] in S [n+l] such that Z ′ contains Z.

Virtual Euler characteristic
The topological Euler characteristic of Quot l S (E) is given by Göttsche's formula Indeed, using χ(Quot l S (E)) = χ(Quot l S (O ⊕r )) -which follows for instance from [31]) -and (III), one can reduce to the r = 1 case originally considered by Göttsche [12]. By analogy with the Chern-Gauss-Bonnet theorem, one can define the virtual Euler characteristic of Quot l S (E) as following Fantechi and Göttsche [9]. For these invariants, the generating series is given by a rational function: where x 1 , . . . , x r are the roots of x r − q(x − 1) r = 0.
Our results allow us to generalize this formula to Quot l S (E) for arbitrary E. Proof. Indeed, if S is the disjoint union of surfaces S 1 and S 2 , then where E 1 and E 2 are the restrictions of E to S 1 and S 2 , respectively. It is clear this splitting is compatible with the perfect obstruction theory, and hence This multiplicativity property, Theorem 3.3, and a standard cobordism argument [8] show that

Segre integrals
For any locally free sheaf F consider the generating series of Segre integrals. As observed by Oprea and Pandharipande [30] (in a special case, later generalised in [1]), this Segre series has a remarkable symmetry property, which can be stated as follows. Remark 4.2. Our proof of Proposition 4.2 only partially explains the symmetry property S F E = S E F , through the dependence on E ⊗ F ≃ F ⊗ E provided by Theorem 3.3. It would be interesting to obtain a deeper understanding, and we believe that in this regard, the implications of Theorem 3.2 are yet to be fully explored. It is also a simple matter to deduce from Theorem 3.2 the rationality of descendent series over [Quot l S (E)] vir from the corresponding result in the E = O ⊕r case; the latter is the main theorem of the paper [17]. This, Proposition 4.1, and Proposition 4.2 can also be deduced from a wall-crossing formula of Joyce [18], see [3].  , which is well-known (see e.g. Exercise 9.23 in [28]). For r 2, consider E = L ∨ ⊕ E ′ with E ′ = (L ∨ ) ⊕(r−2) ⊕ O, and the standard exact sequence

Top intersections of Euler classes
We then have c 1 (E ⊗ L) = c 1 (E ′ ⊗ L) = c 1 (L) and for any L on S, which implies A r = A r−1 .