A NOTE ON MODULI OF STABLE SHEAVES ON RULED SURFACES

KŌTA YOSHIOKA

0. Introduction

Let π : X → C be a ruled surface defined over ℂ and H an ample divisor on H. Let f be a fiber of π and σ a section of π. For a pair (L, c_{2}) of line bundle L on X and a non-negative integer c_{2}, M_{H}(L, c_{2}) is the moduli space of stable sheaves E of rank 2 with (det(E), c_{2}(E)) = (L, n). We note that M_{σ+nf}(L, c_{2}), n ≫ 0 does not depend on n. We denote this space by M(L, c_{2}). We assume that (K_{X}, H) < 0. If (c_{1}(L), f) is odd, then by using [G-S], Göttsche [G] and the author [Y2, appendix] computed the Hodge polynomial of M_{H}(L, c_{2}). If (c_{1}(L), f) is even, then the author [Y2] computed Poincaré polynomial of M_{H}(L, c_{2}). In this note, we shall compute the Hodge polynomial of M_{H}(L, c_{2}). By the same argument as in [G], [Y2, appendix], it is sufficient to compute the Hodge polynomial of M(L, c_{2}). Our method is the same as that in [G]. That is, we shall use virtual Hodge polynomials. By virtue of the computations of Cheah [Ch], we can use almost all computations in [Y2] except the use of Siegel formula [H-N]. By using the idea of Drezet-Narasimhan [D-N], we can avoid to use Siegel formula.

For a vector bundle E of rank r on C, they considered filtrations O_{C}(−n) ⊂ E ^{1}, where n is a sufficiently large integer. Then there is an exact sequence

0 → O_{C}(−n)(D) → E → F → 0,  (0.1)

where D is an effective divisor on C and F is a vector bundle of rank r −1. If E/O_{C}(−n) is locally free, then D = 0. Since they are only interested in the computation of the Picard groups, it is sufficient to consider the case D = 0. But if we treat all D, we can get more precise informations. In particular, we can recover Siegel formula in [H-N]. So we shall follow Drezet-Narasimhan’s method. Unfortunately, it seems that filtrations on non-locally free sheaves are not easy to treat. Hence we restrict ourselves to rank 2 case and consider extensions

0 → O_{X}(D − nf) → E → I_{Z}(L − nf − D) → 0,  (0.2)

where E is locally free, D is an effective divisor coming from C and Z is a locally complete intersection subscheme of X. For such extensions, it is difficult to check the stability. Hence we must treat bigger bounded subset S of vector bundles. So we need to extend the definition of virtual Hodge polynomial to S. For a line bundle L coming from C, let T (L, c_{2}) be the set of torsion free sheaves E of rank 2 such that E_{|π^{−1}(P)} ≅ O_{π^{−1}(P)}^{⊕2} for general P ∈ C and (c_{1}(E), c_{2}(E)) = (L, c_{2}). Let T (L, c_{2})^{ss} be the subset of T (L, c_{2}) consisting of μ-semi-stable sheaves with respect to σ + nf, n ≫ 0. Then the following theorem holds.

Theorem 0.1. We assume that deg L := (σ, L) is 0 or 1. Then,

\begin{aligned}∑_{n} e(T (L, n)^{ss})t^{n} = & \frac{(x^{2}y − 1)^{g}(xy^{2} − 1)^{g}}{((xy)^{2} − 1)(xy − 1)^{2}} ∏_{i=0,1} ∏_{b≥1} ∏_{p,q} (\frac{1}{1 − x^{p}y^{q}(xy)^{2(b−i)}t^{b}} )^{e^{p,q}(X)} \\ & − \frac{(x − 1)^{g}(y − 1)^{g}(xy)^{deg L+g−1}}{((xy)^{2} − 1)(xy − 1)^{2}} ∏_{b≥1} ∏_{p,q} (\frac{1}{1 − x^{p}y^{q}(xy)^{2b−1}t^{b}} )^{2e^{p,q}(X)} , \end{aligned}

where g is the genus of C and e^{p,q}(X) = (−1)^{p+q}h^{p,q}(X).

For the definition of virtual Hodge polynomial, see Section 1.1. By this theorem, we can compute the Hodge polynomial of M_{H}(L, c_{2}), if M_{H}(L, c_{2}) is compact. For example, we get the following.

Corollary 0.2. If deg L = 1, then

\begin{aligned}∑_{n} e(M(L, n))t^{n} = & \frac{(x^{2}y − 1)^{g}(xy^{2} − 1)^{g}}{((xy)^{2} − 1)(xy − 1)} ∏_{i=0,1} ∏_{b≥1} ∏_{p,q} (\frac{1}{1 − x^{p}y^{q}(xy)^{2(b−i)}t^{b}} )^{e^{p,q}(X)} \\ & − \frac{(x − 1)^{g}(y − 1)^{g}(xy)^{g}}{((xy)^{2} − 1)(xy − 1)} ∏_{b≥1} ∏_{p,q} (\frac{1}{1 − x^{p}y^{q}(xy)^{2b−1}t^{b}} )^{2e^{p,q}(X)} . \end{aligned}  (0.3)

1991 Mathematics Subject Classification. 14D20.

1In fact, they used filtration O_{C}(−n)^{⊕(r−1)} ⊂ E, but the idea is similar.

1. Preliminaries

Notation.

Throughtout this note, we use the following notations. Let X be a smooth surface. For a scheme S, we denote the projection S × X → S by p_{S}. For a coherent sheaf E on S × X and a morphism f : S′ → S, we denote the pull-back of E to S′ × X by f^{#}E.

Let ℚ[[1/x, 1/y]] be the formal power series ring of two variable 1/x, 1/y and R the localization of ℚ[[1/x, 1/y]] by 1/(xy). For f, g ∈ R and an ideal I of ℚ[[1/x, 1/y]], f ≡ g mod I, if f − g belongs to I.

1.1. Virtual Hodge polynomial. For a variety Y over ℂ, cohomology with compact support H_{c}^{∗}(Y, ℚ) has a natural mixed Hodge structure. Let e^{p,q}(Y) := ∑_{k}(−1)^{k}h^{p,q}(H_{c}^{k}(Y)) be the virtual Hodge number and e(Y) := ∑_{p,q} e^{p,q}(Y)x^{p}y^{q} the virtual Hodge polynimial of Y. For more details, see [Ch, 0.1]. By the properties of virtual Hodge polynomials, we can extend the definition of e(Y) to constructible sets.

Let S be a bounded set of coherent sheaves on X with Hilbert polynomial h(x). Under some conditions, we shall define the virtual Hodge polynomial of S as an element of R. So it is not a polynomial in general. Let H be an ample divisor on X. Since S is bounded, there is an integer m such that for every element E,

1. E(mH) is generated by global sections,

2. H^{i}(X, E(mH)) = 0 for i > 0.

We set N := h(m). We shall consider the quot scheme Q := Quot_{O_{X}(−mH)^{⊕N} /X/ℂ}^{h(x)}. Let O_{Q×X}(−mH)^{⊕N} → Q be the universal quotient. We assume that Q_{S} := {q ∈ Q|Q_{q} ∈ S} is a constructible set. We shall show that this condition does not depend on the choice of Q. Let T be a scheme and E a coherent sheaf on T × X such that E is flat over T. Let T^{0} be the open subscheme of T consisting of point t ∈ T_{S} such that

1. E_{t}(mH) is generated by global sections,

2. H^{i}(X, E_{t}(mH)) = 0 for i > 0. By base change theorem, p_{T ∗}(E(mH)) is a locally free sheaf on T^{0}. Let τ : P → T be the associated principal GL(N) bundle over T^{0}. Then there is a surjective homomorphism O_{P×X}(−mH)^{⊕N} → τ^{#}E. Hence we get a morphism η : P → Q. We set T_{S} := {t ∈ T |E_{t} ∈ S}. Then we see that T_{S} = τ(η^{−1}(Q_{S})). Thus T_{S} is constructible, if Q_{S} is constructible. In particular, constructibility does not depend on the choice of Q.

Definition 1.1. 1. S is constructible, if Q_{S} is constructible.

2. For a constructible set S, we define the virtual Hodge polynomial by

e(S) := \frac{e(Q_{S})}{e(GL(N))} ∈ R.

It is easy to see that this definition does not depend on the choice of Q.

1.2. Quot scheme. We shall prove a variant of [Y1, Thm. 0.4].

Lemma 1.1. Let E be a vector bundle of rank r on S × X. Then

e(Quot_{E/S×X/S}^{n}) = e(Quot_{O_{X}^{⊕r}/X/ℂ}^{n})e(S).  (1.1)

Proof. We shall prove that e(Quot_{E/S×X/S}^{n}) does not depend on E. Let Δ be the diagonal of X×X and I_{X} the ideal of Δ. We note that E ⊗O_{X×X}/I_{Δ}^{n+1} is flat over S ×X. Hence V := p_{S×X∗}(E ⊗O_{X×X}/I_{Δ}^{n+1}) is a locally free sheaf on S×X and p_{S×X}^{∗}V → E ⊗O_{X×X}/I_{Δ}^{n+1} is surjective. Let S×X = ∪_{i}U_{i} be a stratification of S×X such that U_{i} are locally closed subset of S × X, V_{|U_{i}} are free and O_{U_{i}×X×X}/I_{Δ}^{n+1} ≅ O_{U_{i}}[x, y]/(x, y)^{n+1}. Indeed I_{Δ}/I_{Δ}^{2} ≅ Ω_{X} is a locally free sheaf on X, and hence we can choose such a pair (x, y). We set Y := Spec(ℂ[x, y]/(x, y)^{n+1}) and Q^{k} := Quot_{O_{Y}^{⊕r}/Y/ℂ}^{k}. Let O_{Q^{k}×Y}^{⊕r} → Q^{k} be the universal quotient on

Q^{k} × Y. Then we get a homomorphism E → (O_{U_{i}×X×X}/I_{Δ}^{n+1})^{⊕r} → O_{U_{i}} ⊗_{ℂ} Q^{k} on U_{i} × Q^{k} × Y. Let Q_{i,n}^{k} be the complement of the big diagonal of n-th fiber product (U_{i} × Q^{k}) ×_{S} (U_{i} × Q^{k}) ×_{S} ⋯ ×_{S} (U_{i} × Q^{k}). For a partition n = k_{i}^{1}n_{i}^{1} + ⋯ + k_{i}^{s}n_{i}^{s}, 1 ≤ k_{i}^{1} < k_{i}^{2} < ⋯ < k_{i}^{s} of n, there is a morphism

Q_{i,n_{i}^{1}}^{k_{i}^{1}} ×_{S} Q_{i,n_{i}^{2}}^{k_{i}^{2}} ×_{S} ⋯ ×_{S} Q_{i,n_{i}^{s}}^{k_{i}^{s}} → Quot_{E/S×X/S}^{n}

which is 𝔖_{(n_{i}^{1},...,n_{i}^{s})} := 𝔖_{n_{i}^{1}} × ⋯ × 𝔖_{n_{i}^{s}}-invariant. Hence we get that

e(Quot_{E/S×X/S}^{n}) = ∑_{i} ∑_{\begin{matrix}n=k_{i}^{1}n_{i}^{1}+···+k_{i}^{s}n_{i}^{s} \\ 1≤k_{i}^{1}<k_{i}^{2}<···<k_{i}^{s} \end{matrix}} e(Q_{i,n_{i}^{1}}^{k_{i}^{1}} ×_{S} Q_{i,n_{i}^{2}}^{k_{i}^{2}} ×_{S} ⋯ ×_{S} Q_{i,n_{i}^{s}}^{k_{i}^{s}}/𝔖_{(n_{i}^{1},...,n_{i}^{s})}).

Since we can choose the same stratification for E and O_{S×X}^{⊕r}, we get that

e(Quot_{E/S×X/S}^{n}) = e(Quot_{O_{S×X}^{⊕r}/S×X/S}^{n}).

Since Quot_{O_{S×X}^{⊕r}/S×X/S}^{n} ≅ Quot_{O_{X}^{⊕r}/X/ℂ}^{n} ×S, we get our lemma.

Lemma 1.2. [N-Y] For a smooth surface X,

∑_{n} e(Quot_{O_{X}^{⊕r}/X/ℂ}^{n})t^{n} = ∏_{i=1}^{r} ∏_{b≥1} ∏_{p,q} (\frac{1}{1 − x^{p}y^{q}(xy)^{rb−i}t^{b}} )^{e^{p,q}(X)} .  (1.2)

By virtue of computations in Cheah [Ch], this lemma follows from the local case.

Lemma 1.3. For X = Spec(O_{P}), P ∈ X,

∑_{n} e(Quot_{O_{P}^{⊕r}/X/ℂ}^{n})t^{n} = ∏_{i=1}^{r} ∏_{b≥1} ∏_{p,q} (\frac{1}{1 − (xy)^{rb−i}t^{b}} ) .  (1.3)

Moreover Quot_{O_{P}^{⊕r}/X/ℂ}^{n} has a celluler decomposition. Hence h^{p,q}(H_{c}^{k}(Quot_{O_{P}^{⊕r}/X/ℂ}^{n})) = 0 unless k = p + q.

The method of the proof can be seen in [N]. Also we can derive the first claim from a modification of the proof of [Y1, Thm. 0.4]. More precisely, it is not difficult to modify the proof of [Y1, Prop. 5.3] (cf. the proof of Proposition 2.1).

1.3. Locally free extensions. Let π : X → C be a ruled surface defined over C and f a fiber of π. We shall prove a variant of [Y1, Prop. 0.2]. For a constructible subset Y of C, we set

H^{n} : = {Z|Z ∈ Hilb_{X}^{n}, Z is a locally complete intersection subscheme},
H_{n,a,Y} : = {Z ∈ H^{n}|π(Z) ∈ Y, deg π_{∗}(I_{Z}) = −a}.

For a point P of C, we also set

H_{n,a,P} (d) := {Z ∈ H_{n,a,P} |I_{Z|π^{−1}(P)} ≅ O_{π^{−1}(P)}(−d) ⊕ T, # Supp T < ∞}.

Let Z ⊂ H^{n} × X be the universal subscheme and D a divisor on X which comes from C. Then W := p_{H^{n}∗}(∧^{2}(I_{Z}/I_{Z}^{2})^{∨}(D)) is a locally free sheaf on H^{n}. We also denote the associated vector bundle by q : 𝕍(W^{∨}) → H^{n}. Then there is a canonical homomorphism O_{𝕍(W^{∨})} →^{s} W. Let 𝕍(W^{∨})^{0} be the open subscheme

of 𝕍(W^{∨}) such that p_{H^{n}}^{∗}O_{𝕍(W^{∨})} →^{p_{H^{n}}^{∗} s} p_{H^{n}}^{∗}W → ∧^{2}(I_{Z}/I_{Z}^{2})^{∨}(D) is surjective. We set N_{n,a,Y} := q^{−1}(H_{n,a,Y})∩

𝕍(W^{∨})^{0} and N_{n,a,P} (d) := q^{−1}(H_{n,a,P} (d)) ∩ 𝕍(W^{∨})^{0}. We shall prove the following.

Lemma 1.4.

∑_{n,a} e(N_{n,a,Y})u^{a}t^{n} = ∏_{b≥1} ∏_{p,q} (\frac{1 − x^{p}y^{q}(xy)^{2b−2}ut^{b}}{1 − x^{p}y^{q}(xy)^{2b}ut^{b}} )^{e^{p,q}(Y)} .  (1.4)

By the same argument of Cheah [Ch]^{2}, it is sufficient to consider the case that Y is a point P. In this case, similar computations in [Y1, sect. 1, 2] show our results. So we shall give an outline of our computations for Y = {P }.

We may assume that C = ℙ^{1}, D = −2f, −f and π^{−1}(P) = f. By base change theorem, we get that

1. Hom_{p_{H^{n}}} (I_{Z}, O_{H^{n}×X}(D)) = Ext_{p_{H^{n}}}^{2} (I_{Z}, O_{H^{n}×X}(D)) = 0.

2. Ext_{p_{H^{n}}}^{1} (I_{Z}, O_{H^{n}×X}(D)) is a locally free sheaf.

Then ℙ_{D} := ℙ(Ext_{p_{H^{n}}}^{1} (I_{Z}, O_{H^{n}×X}(D))^{∨}) parametrizes all non-trivial extensions of I_{Z}, Z ∈ H^{n} by O_{X}(D) up to multiplication by constants. Since Rp_{H^{n}∗}^{2}(Hom(I_{Z}, O_{H^{n}×X}(D))) ≅ Rp_{H^{n}∗}^{2}(O_{H^{n}×X}(D)) = 0, the restriction homomorphism

ϕ : Ext_{p_{H^{n}}}^{1} (I_{Z}, O_{H^{n}×X}(D)) → p_{H^{n}∗}(Ext^{1}(I_{Z}, O_{H^{n}×X}(D)))  (1.5)

is surjective. Let ℙ_{D}^{0} be the open subscheme parametrizing locally free extension sheaves. Since

Ext^{1}(I_{Z}, O_{H^{n}×X}(D)) ≅ ∧^{2}(I_{Z}/I_{Z}^{2})^{∨}(D),  (1.6)

we get that ℙ_{D}^{0} = ϕ^{−1}(𝕍(W^{∨})^{0})/ℂ^{×}. Hence we see that

\begin{matrix}e(ℙ_{D}^{0} ×_{H^{n}} H_{n,a,P} (d)) = & (xy)^{−χ(X,O_{X}(D))}e(𝕍(W^{∨})^{0} ×_{H^{n}} H_{n,a,P} (d))/(xy − 1) \\ = & (xy)^{−χ(X,O_{X}(D))}e(N_{n,a,P} (d))/(xy − 1). \end{matrix}  (1.7)

On the otherhand, by considering elementary transformations of locally free extension sheaves along O_{f}(k), k ∈ ℤ, we obtain relations on e(ℙ_{−2f}^{0} ×_{H^{n}} H_{n,a,P} (d)) and e(ℙ_{−f}^{0} ×_{H^{n′}} H_{n′,a′,P} (d′)). By (1.7), we get the following relation.

e(N_{n,a,P} (d)) = ∑_{0≤d′<d}(xy)^{2d−2}((xy)^{2} − 1)e(N_{n−d,a−1,P} (d′)) + (xy)^{2d}e(N_{n−d,a−1,P} (d)),  (1.8)

2see appendix

where we set e(N_{0,0,P} (0)) = 1. By using this relation successively, we get Lemma 1.4 for Y = {P }.

2. Computation of virtual Hodge polynomials

2.1. Notations. Let C be a smooth projective curve of genus g and π : X → C a ruled surface. Let σ be a section of π and f a fiber of π. For simplicity, we denote O_{X}(nf) by O_{X}(n). For a coherent sheaf E such that det E comes from C, we set deg E := (σ, c_{1}(E)). We shall fix a line bundle L of deg L = 0, 1 which comes from C.

Definition 2.1. 1. T (L, c_{2}) is the set of torsion free sheaves E of rank 2 such that E_{|π^{−1}(P)} ≅ O_{π^{−1}(P)}^{⊕2} for general points P of C and (det(E), c_{2}(E)) = (L, c_{2}), and V (L, c_{2}) is the subset of T (L, c_{2}) consisting of locally free sheaves.

2. For an integer d ≥ 1, T (L, c_{2}, d) is the subset of T (L, c_{2}) consisting of E which is defined by an exact sequence

0 → I_{1} → E → I_{2} → 0,  (2.1)

where I_{1} and I_{2} are torsion free sheaves of rank 1 and deg(c_{1}(I_{1})) = d, and V (L, c_{2}, d) := T (L, c_{2}, d) ∩ V (L, c_{2}).

3. We set T (L, c_{2})^{ss} := T (L, c_{2}) \ ∪_{d≥1}T (L, c_{2}, d) and V (L, c_{2})^{ss} := T (L, c_{2})^{ss} ∩ V (L, c_{2}).

We set

V_{c_{2},a} := {E ∈ V (L, c_{2})| deg(π_{∗}(E)) = deg(E) − a}

For a positive integer k, we set

V_{c_{2},a}(k) := {E ∈ V_{c_{2},a}|E ∉ ∪_{j≥k}V (L, c_{2}, j)}.

For a fixed k, we shall choose a sufficiently large integer n ≥ k such that H^{1}(C, π_{∗}(E(n))) = 0 for all E ∈ V_{c_{2},a}(k). We set

F V_{c_{2},a} := {O_{X}(−n) ⊂ E|E ∈ V_{c_{2},a}}.

Then the filtration O_{X}(−n) ⊂ E defines an exact sequence

0 → O_{X}(D − nf) → E → I_{Z}(−D + L + nf) → 0

where D is an effective divisor on X which comes from C. Let F V_{c_{2},a}^{l} be the set of equivalence classes of extensions

0 → O_{X}(D − nf) → E → I_{Z}(−D + L + nf) → 0  (2.2)

such that E is locally free, D is effectice, deg(D) < n + k, deg(π_{∗}(I_{Z})) = −a and deg(Z) = c_{2}, where equivalence relation is defined by the natural action of Aut(O_{X}(D − nf)) × Aut(I_{Z}(−D + L + nf)). Then there is a natural bijective correspondence F V_{c_{2},a} → ∪_{l}F V_{c_{2},a}^{l}. By this identification, we can regard F V_{c_{2},a}^{l} as a subset of F V_{c_{2},a}. There is an integer k′ such that every filtration O_{X}(−n) ⊂ E satisfies that E ∈ V_{c_{2},a}(k′).

2.2. Virtual Hodge polynomial of F V_{c_{2},a}. Let H be an ample divisor on X. Since F V_{c_{2},a} is a bounded set, there is an integer m such that for every extension of (2.2) and the Harder-Narasimhan filtration (2.1) of E ∈ T (L, c_{2}, d), d < k′,

1. O_{X}(D − nf + mH) and I_{Z}(−D + L + nf + mH) are generated by global sections,

2. H^{i}(X, O_{X}(D − nf + mH)) = H^{i}(X, I_{Z}(−D + L + nf + mH)) = 0, i > 0,

3. I_{1}(mH) and I_{2}(mH) are generated by global sections,

4. H^{i}(X, I_{1}(mH)) = H^{i}(X, I_{2}(mH)) = 0, i > 0. We set N_{1}^{l} := χ(X, O_{X}(D − nf + mH)), N_{2}^{l} := χ(X, I_{Z}(−D + L + nf + mH)) and N := N_{1}^{l} + N_{2}^{l}, where l = deg D. We note that N does not depend on the choice of l. We set V_{i} = O_{X}(−mH)^{⊕N_{i}^{l}}, i = 1, 2 and V = O_{X}(−mH)^{⊕N}. We shall consider an open subscheme Q of Quot_{V/X/ℂ} consisting of quotients V → E, E ∈ T (L, c_{2}) \ ∪_{j≥k′}T (L, c_{2}, j).

Let O_{Q} ⊠ V → Q be the universal quotient on Q × X. We set

Q_{c_{2},a} := {q ∈ Q|Q_{q} ∈ V_{c_{2},a}},
Q_{c_{2},a}(k) := {q ∈ Q|Q_{q} ∈ V_{c_{2},a}(k)}.

We shall define virtual Hodge polynimial e(F V_{c_{2},a}). For this purpose, we shall construct a parameter space of F V_{c_{2},a}. Let E_{1} → E_{0} → O_{X}(−n) be a locally free resolution of O_{X}(−n) such that H^{j}(X, E_{0}^{∨} ⊗ Q_{q}) = H^{j}(X, E_{1}^{∨} ⊗ Q_{q}) = 0 for q ∈ Q and j > 0. By base change theorem, p_{Q∗}(E_{i}^{∨} ⊗ Q), i = 0, 1 are locally free sheaves. Let I be the ideal of ℙ(p_{Q∗}(E_{0}^{∨} ⊗ Q)^{∨}) generated by the image of the homomorphism ξ : p_{Q∗}(E_{1}^{∨} ⊗ Q)^{∨} → p_{Q∗}(E_{0}^{∨} ⊗ Q)^{∨}. Let F Q be the subscheme of ℙ(p_{Q∗}(E_{0}^{∨} ⊗ Q)^{∨}) defined by I. Then there is a universal family of filtrations

0 ⊂ O_{F Q×X}(−λ) ⊂ O_{F Q} ⊗_{Q} Q  (2.3)

on F Q × X, where O_{F Q×X}(λ) is the pull-back of the tautological line bundle on ℙ(p_{Q∗}(E_{0}^{∨} ⊗ Q)^{∨}). We set F Q_{c_{2},a} := F Q×_{Q}Q_{c_{2},a} and F Q_{c_{2},a}(k) := F Q×_{Q}Q_{c_{2},a}(k). Let F Q_{c_{2},a}^{l} be the subset of F Q_{c_{2},a} parametrizing filtrations which belong to F V_{c_{2},a}^{l}.

We shall define the virtual Hodge polynomials of F V_{c_{2},a}, F V_{c_{2},a}(k) and F V_{c_{2},a}^{l} by

\begin{aligned}e(F V_{c_{2},a}) & := \frac{e(F Q_{c_{2},a})}{e(GL(V))} ∈ R, \\ e(F V_{c_{2},a}(k)) & := \frac{e(F Q_{c_{2},a}(k))}{e(GL(V))} ∈ R, \\ e(F V_{c_{2},a}^{l}) & := \frac{e(F Q_{c_{2},a}^{l})}{e(GL(V))} ∈ R. \end{aligned}  (2.4)

2.3. Computation of e(F V_{c_{2},a}). We shall compute e(F V_{c_{2},a}) by using the stratification F V_{c_{2},a} = ∪_{l}F V_{c_{2},a}^{l}. We get the following proposition.

Proposition 2.1.

e(F V_{c_{2},a}^{l}) = \frac{e(S^{l}C)e(N_{c_{2},a,C})}{(xy)^{2l−deg L−2n+1−g}(xy − 1)^{2}} .  (2.5)

Proof. The following argument is quite similar to that in [A-B], or [Y2, appendix]. Let D ⊂ S^{l}C × C be the universal family of effective divisors. For simplicity, we denote the pull-back of D to S^{l}C × X by the same D. We set

\begin{aligned}E_{1} : & = O_{S^{l}C×H^{c_{2}} ×X}(D − nf + mH), \\ E_{2} : & = I_{Z}(−D + L + nf + mH). \end{aligned}  (2.6)

By the choice of m, p_{S^{l}C×H^{c_{2}} ∗}(E_{i}), i = 1, 2 are locally free sheaves of rank N_{i}^{l} on S^{l}C × H^{c_{2}}. Let P_{i} → S^{l}C ×H^{c_{2}} be the associated principal fiber bundles over S^{l}C ×H^{c_{2}}. We set P := (P_{1}/ℂ^{×})×_{S^{l}C×H^{c_{2}}} (P_{2}/ℂ^{×}) and we denote the projection P → S^{l}C × H^{c_{2}} by α. α : P → S^{l}C × H^{c_{2}} is a P GL(N_{1}^{l}) × P GL(N_{2}^{l})-bundle over S^{l}C × H^{c_{2}}. Then there are line bundles P_{1}, P_{2} on P, and we get surjective homomorphisms:

\begin{matrix}ψ_{1} : O_{P} ⊠ V_{1} → α^{#}E_{1} ⊗ p_{P}^{∗}P_{1} → 0, \\ ψ_{2} : O_{P} ⊠ V_{2} → α^{#}E_{2} ⊗ p_{P}^{∗}P_{2} → 0. \end{matrix}  (2.7)

By base change theorem, F := Hom_{p_{P}} (ker ψ_{2}, α^{#}E_{1} ⊗ p_{P}^{∗}P_{1}) is a locally free sheaf. Let β : 𝕍(F^{∨}) → P be the projection. We set E′_{i} := β^{#}(α^{#}E_{i} ⊗ p_{P}^{∗}P_{i}), i = 1, 2. Then there is an exact and commutative diagram.

\begin{aligned}0 & ⟶ & O_{𝕍(F^{∨})} ⊠ V_{1} & ⟶ & O_{𝕍(F^{∨})} ⊠ V & ⟶ & O_{𝕍(F^{∨})} ⊠ V_{2} & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & E′_{1} & ⟶ & E & ⟶ & E′_{2} & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0 \end{aligned}  (2.8)

We note that

φ : F → Ext_{p_{P}}^{1} (E′_{2}, E′_{1}) → p_{P∗}(Ext_{O_{P×X}}^{1} (E′_{2}, E′_{1})) =: G

is surjective and

dim ker φ = dim F − c_{2}.  (2.9)

We set W̃ := p_{P}(∧^{2}(I_{Z}/I_{Z}^{2})^{∨}(2D − L − 2nf)). In the same way as in section 1.3, we define 𝕍(W̃^{∨})^{0}. Then it is easy to see that

e(𝕍(W̃^{∨})^{0} ×_{H^{c_{2}}} H_{c_{2},a,C}) = e(N_{c_{2},a,C})e(S^{l}C)e(P GL(V_{1}))e(P GL(V_{2})).  (2.10)

We set Y := φ^{−1}(𝕍(W̃^{∨})^{0}). Let P be the parabolic subgroup of GL(V) fixing the subsheaf V_{1}. Then there is an injective morphism Y ×_{P} GL(V) → F Q whose image is the subset F Q_{c_{2},a}^{l}. Hence we get that

\begin{aligned}e(F V_{c_{2},a}^{l}) & = \frac{e(F Q_{c_{2},a}^{l})}{e(GL(V))} = \frac{e(Y)}{e(P)} \\ & = \frac{e(S^{l}C)e(N_{c_{2},a,C})}{(xy)^{2l−deg L−2n+1−g}(xy − 1)^{2}} , \end{aligned}  (2.11)

where we used (2.9) and the equality

dim F − dim Hom(V_{2}, V_{1}) = −χ((E′_{2})_{y}, (E′_{1})_{y})

= −(2l − deg L − 2n − c_{2} + 1 − g), y ∈ Y. (2.12)

Thus we get that

e(F V_{c_{2},a}) = ∑_{0≤l<n+k} \frac{e(S^{l}C)}{(xy)^{2l−deg L−2n+1−g}} \frac{e(N_{c_{2},a,C})}{(xy − 1)^{2}}.  (2.13)

2.4. Proof of Theorem 0.1. We shall first estimate e(F V_{c_{2},a}) − e(F V_{c_{2},a}(k)). We can easily show the following.

Lemma 2.2. For E ∈ V_{c_{2},a} ∩ V (L, c_{2}, d), h^{0}(X, E(n)) ≤ max{n + d + 1, 2n + deg L − a + 2}.

By using deformation theory of Drezet-Le Potier [D-L] or the argument of Atiyah-Bott [A-B], we see that

codim_{Q}{q ∈ Q|Q_{q} ∈ T (L, c_{2}, d)} = 2d + c_{2} − deg L + g − 1  (2.14)

for d < k′. By Lemma 2.2, χ(X, π_{∗}E(n)) − h^{0}(X, E(n)) ≥ min{n − d + deg L + 1 − 2g, −2g}. Since n ≥ k, we see that codim(F Q_{c_{2},a} \ F Q_{c_{2},a}(k)) ≥ 2k + A_{1}, where A_{1} is a constant which does not depend on k and n. Since dim F Q_{c_{2},a} − dim GL(V) − 2n does not depend on k and n, we get that

e(F V_{c_{2},a}) ≡ e(F V_{c_{2},a}(k)) mod (\frac{(xy)^{2n}}{(xy)^{2k+A_{2}}} ) ,  (2.15)

where A_{2} is a constant which does not depend on k and n. Since F Q_{c_{2},a}(k) → Q_{c_{2},a}(k) is a projective bundle with fiber ℙ(H^{0}(X, E(n))^{∨}), E ∈ V_{c_{2},a}(k), we get that

e(F V_{c_{2},a}(k)) = \frac{(xy)^{deg L+2n−a+2(1−g)} − 1}{xy − 1}e(V_{c_{2},a}(k)).  (2.16)

By (2.15) and (2.16),

e(V_{c_{2},a}(k)) ≡ ∑_{0≤l<n+k} \frac{e(S^{l}C)}{(xy)^{2l}}(xy)^{3g−3}(xy)^{a} \frac{e(N_{c_{2},a,C})}{(xy − 1)} mod (\frac{1}{(xy)^{2k+A_{3}}} ) ,  (2.17)

where A_{3} is a constant which does not depend on k and n. By using the argument of Atiyah-Bott [A-B] (see [Y2, appendix]), we get that

e(V_{c_{2},a}(1)) = e(V_{c_{2},a}(k)) − ∑_{j=1}^{k−1} e(V (L, c_{2}, j) ∩ V_{c_{2},a})

= e(V_{c_{2},a}(k)) − ∑_{j=1}^{k−1} \frac{e(Pic^{0}(C))}{(xy)^{2j−deg L+(1−g)}} \frac{e(N_{c_{2},a,C})}{(xy − 1)^{2}}.

Lemma 2.3.

∑_{0≤l<n+k} \frac{e(S^{l}C)}{(xy)^{2l}} ≡ \frac{(1 − x/(xy)^{2})^{g}(1 − y/(xy)^{2})^{g}}{(1 − 1/(xy)^{2})(1 − 1/(xy))} mod (\frac{1}{(xy)^{n+k}} ) .  (2.18)

Proof. By Macdonald’s formula [Ch], we get that

∑_{l≥0} e(S^{l}C)t^{l} = \frac{(1 − xt)^{g}(1 − yt)^{g}}{(1 − t)(1 − xyt)} .  (2.19)

Since deg_{x}(e(S^{l}C)) ≤ l and deg_{y}(e(S^{l}C)) ≤ l, ∑_{l≥0} e(S^{l}C)/(xy)^{2l} converges to

\frac{(1 − x/(xy)^{2})^{g}(1 − y/(xy)^{2})^{g}}{(1 − 1/(xy)^{2})(1 − 1/(xy))}

and

∑_{0≤l<n+k} \frac{e(S^{l}C)}{(xy)^{2l}} ≡ ∑_{l≥0} \frac{e(S^{l}C)}{(xy)^{2l}} mod (\frac{1}{(xy)^{n+k}} ) .

By Lemma 2.3, we obtain that

e(V_{c_{2},a}(1)) ≡ \frac{(1 − x/(xy)^{2})^{g}(1 − y/(xy)^{2})^{g}}{(1 − 1/(xy)^{2})(1 − 1/(xy))} (xy)^{3g−3}(xy)^{a} \frac{e(N_{c_{2},a,C})}{xy − 1}

− \frac{e(Pic^{0}(C))}{(1 − 1/(xy)^{2})(xy − 1)^{2}} (xy)^{deg L+g−3}e(N_{c_{2},a,C}) mod (\frac{1}{(xy)^{2k+A_{4}}} ) , (2.20)

where A_{4} is a constant which does not depend on k and n. Since we can choose any sufficiently large integer k, we get that

\begin{aligned}e(V_{c_{2},a}(1)) = & \frac{(1 − x/(xy)^{2})^{g}(1 − y/(xy)^{2})^{g}}{(1 − 1/(xy)^{2})(1 − 1/(xy))} (xy)^{3g−3}(xy)^{a} \frac{e(N_{c_{2},a,C})}{xy − 1} \\ & − \frac{e(Pic^{0}(C))}{(1 − 1/(xy)^{2})(xy − 1)^{2}} (xy)^{deg L+g−3}e(N_{c_{2},a,C}). \end{aligned}  (2.21)

By using Lemma 1.4, we see that

\begin{aligned}∑_{n} e(V (L, n)^{ss})t^{n} = & \frac{(x^{2}y − 1)^{g}(xy^{2} − 1)^{g}}{((xy)^{2} − 1)(xy − 1)^{2}} (∑_{n,a} e(N_{c_{2},a,C})(xy)^{a}t^{n}) \\ & − \frac{(x − 1)^{g}(y − 1)^{g}}{((xy)^{2} − 1)(xy − 1)^{2}} (xy)^{deg L+g−1} (∑_{n,a} e(N_{c_{2},a,C})t^{n}) \\ = & \frac{(x^{2}y − 1)^{g}(xy^{2} − 1)^{g}}{((xy)^{2} − 1)(xy − 1)^{2}} ∏_{b≥1} (\frac{1 − x^{p}y^{q}(xy)^{2b−1}t^{b}}{1 − x^{p}y^{q}(xy)^{2b+1}t^{b}} )^{e^{p,q}(C)} \\ & − \frac{(x − 1)^{g}(y − 1)^{g}}{((xy)^{2} − 1)(xy − 1)^{2}} (xy)^{deg L+g−1} ∏_{b≥1} (\frac{1 − x^{p}y^{q}(xy)^{2b−2}t^{b}}{1 − x^{p}y^{q}(xy)^{2b}t^{b}} )^{e^{p,q}(C)} . \end{aligned} (2.22)

By using Lemma 1.1, 1.2, we see that

∑_{n} e(T (L, n)^{ss})t^{n} = (∑_{n} e(V (L, n)^{ss})t^{n}) ∏_{i=1}^{2} ∏_{b≥1} ∏_{p,q} (\frac{1}{1 − x^{p}y^{q}(xy)^{2b−i}t^{b}} )^{e^{p,q}(X)} .  (2.23)

Since e(X) = e(C)e(ℙ^{1}), (2.22) and (2.23) implies Theorem 0.1.

For the proof of Corollary 0.2, we note that there is a universal family on M(L, n) × X. In the notation of this section, we set Q(L, n)^{ss} := {q ∈ Q|Q_{q} ∈ T (L, n)^{ss}}. Then the quotient map Q(L, n)^{ss} → M(L, n) is a (Zariski locally trivial) P GL(V)-bundle over M(L, n). Hence

e(Q(L, n)^{ss}) = e(M(L, n))e(P GL(V)).

By the definition of e(T (L, n)^{ss}) and Theorem 0.1, we get Corollary 0.2.

Remark 2.1. Kirwan [K, sect. 14] defined a natural Hodge structure of the equivariant cohomology group H_{GL(V)}^{∗}(Q(L, n)^{ss}, ℂ). By [K, sect. 14], h^{p,q}(H_{GL(V)}^{k}(Q(L, n)^{ss}, ℂ)) = 0 unless k = p + q. Moreover [K, sect. 14] implies that

∑_{p,q}(−1)^{p+q}h^{p,q}(H_{GL(V)}^{p+q}(Q(L, n)^{ss}, ℂ))x^{−p}y^{−q} = \frac{e(T (L, n)^{ss})}{(xy)^{dim T (L,n)^{ss}}} ,  (2.24)

where dim T (L, n)^{ss} = dim M(L, n) − 1 (cf. [Y2, Prop. 4.3]).

3. appendix

In this appendix, we shall explain what will be useful to modify Cheah’s argument [Ch].

Definition 3.1. 1. A constructible set Y has a grading if there is a stratification Y = ∪_{n≥0}Y_{n} and Y_{n}

are constructible.

2. For constructible sets, Y, Z with gradings Y = ∪_{n≥0}Y_{n}, Z = ∪_{n≥0}Z_{n}, we define the grading of Y × Z by (Y × Z)_{n} := ∪_{i+j=n}Y_{i} × Z_{j}.

Let Y = ∪_{n≥0}Y_{n} be a graded constructible set. Then m-th product Y^{m} has a natural grading and the natural action of 𝔖_{m} preserves the grading. Hence m-th symmetric product S^{m}(Y) also has a natural grading.

For a graded constructible set Y = ∪_{n≥0}Y_{n}, we define the virtual Hodge polynomial by e(Y) := ∑_{p,q,r} e^{p,q,r}(Y)x^{p}y^{q}z^{r}, where e^{p,q,r}(Y) := e^{p,q}(Y_{r}). Then [Ch, Prop. 1.2] is replaced by

∑_{m} e(S^{m}(Y))t^{m} = ∏_{p,q,r} (\frac{1}{1 − x^{p}y^{q}z^{r}t})^{e^{p,q,r}(Y)} .  (3.1)

Proof. By the definition of the grading, we get

(S^{m}Y)_{n} = ⋃_{\begin{matrix}∑_{r} rm_{r}=n \\ ∑_{r} m_{r}=m \end{matrix}} ∏_{r} S^{m_{r}}Y_{r}.  (3.2)

By using [Ch, Prop. 1.2], we see that

\begin{aligned}∑_{m,n} e((S^{m}Y)_{n})z^{n}t^{m} & = ∑_{m,n} ( ∑_{\begin{matrix}∑_{r} rm_{r}=n \\ ∑_{r} m_{r}=m \end{matrix}} ∏_{r} e(S^{m_{r}}Y_{r})) z^{n}t^{m} \\ & = ∏_{r} (∑_{m} e(S^{m}Y_{r})(z^{r}t)^{m}) \\ & = ∏_{r} ∏_{p,q} (\frac{1}{1 − x^{p}y^{q}z^{r}t})^{e^{p,q}(Y_{r})} . \end{aligned}  (3.3)

Thus (3.1) holds.

In the notation of section 1.3, N_{n,Y} := ∪_{a≥0}N_{n,a,Y} becomes a graded constructible set. Replacing [Ch, Prop. 1.2] by (3.1), we can argue the same way as in [Ch].

Acknowledgement. The author would like to thank Max-Planck-Institut für Mathematik for support and hospitality.

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Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111, Bonn, Germany & Department of Mathematics, Faculty of Science, Kobe University, 657, Japan

   E-mail address: yoshioka@mpim-bonn.mpg.de, yoshioka@math.kobe-u.ac.jp