1-point Gromov-Witten invariants of the moduli spaces of sheaves over the projective plane

The Gieseker-Uhlenbeck morphism maps the Gieseker moduli space of stable rank-2 sheaves on a smooth projective surface to the Uhlenbeck compactification, and is a generalization of the Hilbert-Chow morphism for Hilbert schemes of points. When the surface is the complex projective plane, we determine all the 1-point genus-0 Gromov-Witten invariants extremal with respect to the Gieseker-Uhlenbeck morphism. The main idea is to understand the virtual fundamental class of the moduli space of stable maps by studying the obstruction sheaf and using a meromorphic 2-form on the Gieseker moduli space.


Introduction
Recently there have been intensive interests in studying the quantum cohomology and Gromov-Witten theory of Hilbert schemes of points on algebraic surfaces.Two main reasons are the connections with the Donaldson-Thomas theory of 3-folds and with Ruan's Cohomological Crepant Resolution Conjecture.Roughly speaking, the Crepant Resolution Conjecture asserts that the quantum cohomology of an orbifold Z coincides with the quantum cohomology of a crepant resolution Y of Z after analytic continuation and specialization of quantum parameters.For an algebraic surface X, let X [n] be the Hilbert scheme of n-points on X and Sym n (X) be the n-th symmetric product of X.It is well-known that X [n] is smooth of dimension 2n and the Hilbert-Chow morphism Φ : X [n] → Sym n (X) is a crepant resolution of the global orbifold Sym n (X).
A natural generalization of the Hilbert-Chow morphism Φ is the Gieseker-Uhlenbeck morphism Ψ from the moduli space of Gieseker semistable rank-2 torsionfree sheaves on X to the Uhlenbeck compactification space.This morphism was constructed in [LJ1,Mor], and was shown to be crepant [LJ2, when the Gieseker moduli space is smooth.For the projective plane X = P 2 , the moduli space M(n) of Gieseker semistable sheaves V on X with c 1 (V ) = −1 and c 2 (V ) = n is a smooth irreducible projective variety of dimension (4n − 4) when n ≥ 1.In [Q-Z], it is proved that there is exactly one primitive integral class f ∈ H 2 (M(n); Z) contracted by the Gieseker-Uhlenbeck morphism Ψ : M(n) → U(n).
When n = 1, the moduli space M(n) is a point.When n = 2, the fourth Betti number b 4 of the moduli space M(n) is equal to 3 which is different from the case n ≥ 3. The result for n = 2 will appear elsewhere via a different method (see Remark 3.2).
An interesting observation is that the 1-point genus-0 Gromov-Witten invariants α 0,df are independent of the second Chern class n.
There are two main ideas in our proof of Theorem 1.1.The first one is to determine the restriction of the obstruction sheaf of the Gromov-Witten theory for M(n) to certain open subset of the moduli space M 0,1 (M(n), df) of stable maps.This enables us to determine the 1-point invariants α 0,df for the first four cohomology classes α = PD(Ξ 1 ), . . ., PD(Ξ 4 ).
The second one is to study the support of the virtual fundamental class [M 0,1 (M(n), df)] vir ∈ A 4n−6 M 0,1 (M(n), df) using the techniques developed in [K-L, L-L].By introducing a suitable meromorphic 2-form Θ on the Gieseker moduli space M(n), we show that where ev 1 : M 0,1 (M(n), df) → M(n) is the evaluation map, and is the subset of M(n) consisting of all the non-locally free sheaves V such that V | C 0 contains torsion (respectively, V | C 0 is torsion-free and unstable).This allows us to show that α 0,df = 0 for α = PD(Ξ 5 ), PD(Ξ 6 ).This paper is organized as follows.In §2, the Gromov-Witten theory is reviewed.In §3, we recall some properties of the Gieseker moduli space M(n) and the Gieseker-Uhlenbeck morphism Ψ.We study the boundary divisor of M(n) consisting of non-locally free sheaves in M(n).In §4, the basis {Ξ 1 , . . ., Ξ 6 } for H 4 (M(n); C) is constructed.In §5, we analyze the obstruction sheaf of the Gromov-Witten theory for M(n).In §6, (1.1) is proved.In §7, we verify Theorem 1.1.

Stable maps and Gromov-Witten invariants
Let Y be a smooth projective variety.A k-pointed stable map to Y consists of a complete nodal curve D with k distinct ordered smooth points p 1 , . . ., p k and a morphism µ : D → Y such that the data (µ, D, p 1 , . . ., p k ) has only finitely many automorphisms.In this case, the stable map is denoted by [µ : (D; p 1 , . . ., p k ) → Y ].For a fixed homology class β ∈ H 2 (Y, Z), let M g,k (Y, β) be the coarse moduli space parameterizing all the stable maps [µ : (D; p 1 , . . ., p k ) → Y ] such that µ * [D] = β and the arithmetic genus of D is g.Then, we have the evaluation map: The Gromov-Witten invariants are defined by using the virtual fundamental class Then, we have the k-point Gromov-Witten invariant defined by: Next, we summarize certain properties concerning the virtual fundamental class.To begin with, we recall that the excess dimension is the difference between the dimension of M g,k (Y, β) and the expected dimension d in (2.2).For 0 ≤ i < k, use to stand for the forgetful map obtained by forgetting the last (k − i) marked points and contracting all the unstable components.It is known that f k,i is flat when β = 0 and 0 ≤ i < k.The following can be found in [LT1,Beh,Get,.
Proposition 2.1.Let β ∈ H 2 (Y, Z) and β = 0. Let e be the excess dimension of M g,k (Y, β), and M ⊂ M g,k (Y, β) be a closed subscheme.Then, 3. The moduli space of stable rank-2 sheaves on P 2 3.1.Some basic facts of the moduli space.
Throughout the rest of this paper, let X = P 2 be the projective plane, and let ℓ be a line in X.For an integer n, let M(n) be the moduli space parametrizing all Gieseker-semistable rank-2 sheaves V over X with c 1 (V ) = −ℓ, c 2 (V ) = n.Note that every such sheaf V is actually slope-stable and hence is Gieseker-stable.It is well-known that, when n ≥ 1, M(n) is nonempty, smooth, irreducible and rational with the expected dimension (4n − 4); in addition, a universal sheaf over M(n) × X exists.By the Theorem 1 in [Mar], the cohomology groups H i M(n); Z are torsion-free for all i and vanish for odd i.So are the homology groups H i M(n); Z .Let b i (M(n)) be the i-th Betti number of M(n), and put By the Theorem 0.1 of [Yos], p(M(n); q) equals the coefficient of t n of the series (3.1) In the rest of the subsection, we review Strømme's work in [Str] and give a basis of H 4 (M(n), Z) in terms of the classes from [Str].A basis of H 4 (M(n), Z) with geometric flavors will be constructed in Section 4.
Fix n ≥ 2. Let E be a universal sheaf over M(n) × X, and let π 1 and π 2 be the two natural projections on M(n) × X.For 0 ≤ k ≤ 2, define by the Proposition 1.5 in [Str].It follows that the three sheaves A 0 , A 1 , A 2 over M(n) are locally free of rank (n − 1), n, (n − 1) respectively.
Definition 3.1.Let 0 ≤ k ≤ 2 and r k be the rank of A k .Define Note that these classes ǫ, δ, τ k are independent of the choices of the universal sheaf E over M(n) × X.Let K M(n) be the canonical class of M(n), M(n) be the open subset of M(n) parametrizing stable bundles, and (3.4) By the Theorem in [Str], Pic(M(n)) is freely generated by ǫ and δ, and (3.6)By (3.1), the rank of H 4 (M(n); Z) is 3 when n = 2, and is 6 when n ≥ 3. Therefore, if n ≥ 3, then a linear basis of H 4 (M(n); Z) is given by the six classes in (3.6).
Remark 3.2.When n = 2, the rank of H 4 (M(n); Z) is different from that of the case n ≥ 3. Therefore the construction of a basis of H 4 (M(n); Z) in §4 for n ≥ 3 needs to be modified.However, there is a method to describe the moduli space M(2) using the moduli spaces of stable sheaves on the Hirzebruch surface F 1 and chamber structures, which enable us to compute all the Gromov-Witten invariants of M(2) (instead of a special kind considered in this paper for n ≥ 3).Since the method is different, the result for n = 2 will appear elsewhere.

The boundary and the Uhlenbeck compactification.
The quasi-projective variety M(n) has a Uhlenbeck compactification according to [Uhl,LJ1,Mor].Moreover, there exists a birational morphism, called the Gieseker-Uhlenbeck morphism, It follows that the boundary divisor B in (3.4) is contracted by Ψ to the codimension-2 subset It is an open dense subset of the boundary divisor B.
To construct a universal sheaf over B * × X, let E 0 n−1 be a universal sheaf over be the natural projection.Let ∆ X be the diagonal of X × X.Consider the obvious isomorphism α : M(n−1)×∆ X → M(n−1)×X.Then, we have the isomorphisms: where π : × X which denotes the projection to the product of the first and third factors, and α : The following lemma will be used in later sections.
Lemma 3.3.Let πi be the i-th projection on M(n − 1) × X, and let (3.12) Proof.Since B * is open and dense in B, we see from (3.5) and Definition 3.1 that Next, let 0 ≤ k ≤ 2, and let π i be the i-th projection on B * × X.Then the restriction of π 1 to the subset (π 2 O X (−kℓ) and then applying the functor π 1 * , we get the exact sequence Therefore, rewriting the 3rd term in the above exact sequence, we obtain (3.15) Finally, we conclude from (3.13) and (3.14) that We shall construct two curves in the Gieseker moduli space M(n) which freely generate the homology group H 2 (M(n), Z).One such curve is a fiber f of the morphism π from (3.9).The following is the Lemma 3 Next, we shall construct the other curve.Let n ≥ 3, and let ξ consist of n distinct points in general position in Tensoring by π * 2 O X (−kℓ) and applying π 1 * lead to the exact sequence: where 0 ≤ k ≤ 2.An easy computation gives rise to the following: where k = 1, 2. It follows immediately from Definition 3.1 that Lemma 3.5.Let n ≥ 3 and l be a line in the projective space E n .
(i) The homology group H 2 (M(n); Z) is freely generated by f and l; (ii) The class af In this section, we assume n ≥ 3. Then the integral homology group H 4 (M(n); Z) is free of rank 6.In the following, we construct a basis {Ξ 1 , . . ., Ξ 6 } for H 4 (M(n); C).This construction makes use of a result due to Hirschowitz and Hulek.
We review the results in [H-H] where complete rational curves were found in M(n).Let n ≥ 2 and Γ = P 1 .Fix lines ℓ 1 , . . ., ℓ n ⊂ X = P 2 in general position.For 1 ≤ i ≤ n, let φ i : Γ → ℓ i be an isomorphism, and define Y i ⊂ Γ × X to be the graph of φ i .For generic choices of φ 1 , . . ., φ n , it was proved in such that E n | {p}×X ∈ M(n) for all p ∈ Γ, and E n induces a non-constant morphism Let π1 and π2 be the natural projections on Γ × X.Let By the Lemma 3.5 in [H-H], the degrees of the bundles A n,k are 3), the condition in Corollarie (6.9.9) of [Gro] is satisfied.Hence the base-change theorem of the first direct image holds for every projective morphism to M(n) and for the sheaf E ⊗π * 2 O X (−kℓ) with 0 ≤ k ≤ 2. For a finite morphism ϕ from Y onto a subvariety of M(n), the intersection numbers of ϕ(Y ) with ǫ 2 , ǫ•δ, δ 2 , τ 0 , τ 1 , τ 2 can be computed on Y , via the projection formula, by pulling back the Chern classes of the first direct images of the sheaf E ⊗π * 2 O X (−kℓ) with 0 ≤ k ≤ 2.

The homology classes
Let n ≥ 3, and assume Γ ⊂ M(n − 1) as pointed out in Remark 4.1 (ii).Let x → 0 for some p ∈ Γ and x ∈ X.We still use π to denote the natural projection By (3.11), a universal sheaf E over P × X sits in the exact sequence where π is the composition of π × Id X : P × X → Γ × X × X and the projection Γ × X × X → Γ × X to the first and third factors.Let π i (respectively, πi ) be the natural projections on P×X (respectively, Γ×X).By (3.14) and (3.15), where A n−1,k is defined in (4.2), and (4.6) In addition, we conclude from (4.1) that . By Definition 3.1, (4.5), (4.6) and (4.7), we obtain where by abusing notations, π denotes the natural projection where Γ, P and π are from the previous subsection.By Definition 3.1, (4.3), (4.5) and (4.6), where x ∈ Γ × ℓ is a fixed point and by abusing notations, π : W → Γ × ℓ ∼ = P 1 × P 1 stands for the natural projection.In addition, by (4.7), we conclude that Since n ≥ 3, the moduli space M(n − 2) is nonempty.Fix a vector bundle V 2 ∈ M(n − 2) and two distinct points x 1 , x 2 ∈ X.Let Ξ 5 ⊂ M(n) parametrize all the sheaves V ∈ M(n) sitting in exact sequences: Then we have the isomorphisms Ξ 5 ∼ = P(V | x 1 ) × P(V | x 2 ) ∼ = P 1 × P 1 .Moreover, a universal sheaf E ′ over Ξ 5 × X sits in the exact sequence: → 0 where π i and πi denote the i-th projection on Ξ 5 × X and P 1 × P 1 × X respectively.Tensoring the above exact sequence by π * 2 O X (−kℓ)) and applying π 1 * yield Regarding Ξ 5 ∈ H 4 (M(n); C), we obtain the intersection numbers on M(n): Fix a vector bundle V 2 ∈ M(n − 2) and a point x ⊂ X. Fix a trivialization of V 2 in an open neighborhood O x of x.Let X [2] be the Hilbert scheme parametrizing the length-2 closed subschemes of X, and let For ξ ∈ M 2 (x), let ι ξ : O X → O ξ be the natural quotient morphism.
Let Ξ 6 = P 1 × M 2 (x) be the subset of M(n) parametrizing all the sheaves V ∈ M(n) sitting in extensions of the form Next, we construction a universal sheaf over Ξ 6 × X = P 1 × M 2 (x) × X.Let π i 1 ,...,im denote the projection of P 1 × M 2 (x) × X to the product of the i 1 -th, . . ., i m -th factors.Over Ξ 6 = P 1 × M 2 (x), there is a tautological surjection O ⊕2 Ξ 6 → O Ξ 6 (1, 0) → 0. This pulls back to a surjection over P 1 × M 2 (x) × X: Let Z 2 (x) be the universal codimension-2 subscheme in M 2 (x) × X, and let O M 2 (x)×X → O Z 2 (x) → 0 be the natural surjection.Pulling back to P 1 × M 2 (x) × X yields a surjection: Tensoring (4.15) and (4.16), we obtain a surjection where π1 and π2 are the two natural projections on Ξ 6 = P 1 × M 2 (x), and O [2] X is the tautological rank-2 bundle over the Hilbert scheme X [2] whose fiber at a point ξ ∈ X [2] is the space . By Definition 3.1, we get the six intersection numbers on M(n): Now we can summarize the above in the following proposition.

The restriction of the obstruction sheaf on certain open subset
From the previous section, we see that, if we let In this section, we use the geometric construction of B * in Subsect.3.2 to effectively compute the virtual cycle restricted to ev −1 1 (B * ).The result will be used to compute the Gromov-Witten invariants α 0,df when α is dual to the classes Ξ (5.1) Note that the fiber Proof.(i) Take a stable map u = [µ : D → M(n)] in O 0 , and consider Since B * is smooth of codimension-1 in M(n), we obtain the exact sequence Applying ( ev 1 ) * and ( f1,0 ) * to the exact sequence (5.2), we get where we have used R 2 ( f1,0 ) * ( ev 1 ) * T B * = 0 since f1,0 is of relative dimension 1.
If [µ : D → M(n)] is a stable map in O 0 , then µ(D) is a fiber of the projection π in (3.9).Hence the normal bundle of µ(D) in B * is trivial.Therefore we have . The kernel of the tautological surjection π * E 0 n−1 → O B * (1) → 0 is a line bundle.By comparing the first Chern classes, we get (5.4) Applying the functor ( f1,0 ) * to (5.4), we get the exact sequence → 0 where we have used the projection formula, ( f1,0 . By Lemma 5.1 and Lemma 3.3, we obtain the desired exact sequence for V. where by abusing notations, we still use f 1,0 and ev 1 to denote the forgetful map and the evaluation map from M 0,1 (P 1 , d[P 1 ]) to M 0,0 (P 1 , d[P 1 ]) and P 1 respectively.6.The virtual fundamental class M 0,1 (M(n), df) vir As we saw in the construction of the classes Ξ i , Ξ 5 and Ξ 6 don't lie in B * .The method to compute the virtual cycle restricted to ev −1 1 (Ξ i ) in the previous section won't work for i = 5, 6.In this section, we shall employ the localization method of Kiem-Li to find a sufficiently small closed subset of M(n) containing the image of the virtual cycle under the evaluation map ev 1 .The result will be used to show the vanishing of the Gromov-Witten invariants α 0,df when α is dual to Ξ 5 , Ξ 6 .
Let X = P 2 , and let C 0 ⊂ X be a smooth cubic curve.Recall that the Zariski tangent space of M(n show that the meromorphic 2-form Θ on the moduli space M(n) induces a meromorphic homomorphism: where Λ ⊂ M 0,1 (M(n), df) is the degeneracy loci consisting of points at which either the map η is undefined or not surjective.
stable.We will draw a contradiction by showing that η is both defined and surjective at [µ : ( Applying the functor Hom(V, •), we obtain a long exact sequence: Hence the meromorphic 2-form Θ is defined at V ∈ µ(D).This proves that Θ is holomorphic along µ(D).So η is defined at [µ : (D; p) → M(n)].
The above argument also shows that Θ| µ(D) is an isomorphism.Since µ is not a constant map, the image of µ * : T Dreg → T M(n) does not lie in the null space of Θ : T M(n) T M(n) * , where D reg denotes the smooth part of D. By the Lemma 6.2.Let C 0 ⊂ X = P 2 be a smooth cubic curve.Let n ≥ 1, and let C 0 and H 0 C 0 , O X (1)| C 0 have the same dimension, f 0 is an isomorphism and H 0 C 0 , Ω X ⊗ O X (1) | C 0 = 0. Hence we obtain a contradiction.
Chern classes of the bundles A0 , A 1 , A 2 .It follows that H 4 (M(n); Z)is the Z-linear span of the six integral classes: and H 2 M(n); Z) is freely generated by ǫ and δ.By (3.19), l • ǫ = 1 and l • δ = 0.By Definition 3.1 and (3.14), f • ǫ = 0 and f • δ = 1.bl is effective, then a, b ≥ 0 since the divisor cǫ + dδ is ample if and only if c, d > 0. (iii) By (ii), C = df + bl ∈ H 2 (M(n); Z) where d and b are nonnegative integers not both zero.Let L be a very ample divisor on U Since H 2 (M(n); Z) is torsion-free, H 2 (M(n); Z) is freely generated by f and l. (ii) Since f and l are effective, af + bl is effective if a, b ≥ 0. Conversely, if af +