Framed Sheaves over Treefolds

and Symmetric Obstruction Theories

Dragos Oprea

Received: September 20, 2012

Communicated by Edward Frenkel

Abstract. We note that open moduli spaces of sheaves over local Calabi-Yau surface geometries framed along the divisor at infinity admit symmetric perfect obstruction theories. We calculate the corresponding Donaldson-Thomas weighted Euler characteristics (as well as the topological Euler characteristics). Furthermore, for blowup geometries, we discuss the contribution of exceptional curves.

2010 Mathematics Subject Classification: 14N35, 14D20

Keywords and Phrases: Moduli spaces of sheaves, Donaldson-Thomas invariants

1 Introduction

Moduli spaces of sheaves over threefolds admit virtual fundamental classes in a lot of examples, yielding Donaldson-Thomas invariants [T]. The rank 1 case is particularly interesting, bearing connections with virtual curve counts [MNOP].

In this note, we study open moduli spaces of higher rank sheaves over local Calabi-Yau surface geometries, framed along the divisor at infinity. We prove that the moduli spaces admit symmetric perfect obstruction theories, and in this context, we compute the ensuing Donaldson-Thomas Euler characteristics. In addition, we find the topological Euler characteristics of the compactified moduli spaces of framed modules. We also discuss a “blowup” formula. Finally, we point out other geometries which can be studied by the same methods.

This way, we extend previously known results in two directions ^{1}.

1The search for such generalizations motivated our interest this topic.

Documenta Mathematica 18 (2013) 323–342

(i) First, there is quite a bit of literature on moduli spaces of framed sheaves over surfaces. An exhaustive survey is not our intention here, but we refer the reader to [BPT], [N] for calculations which we partially carry out in the higher dimensional setting, and also for a more comprehensive bibliography.

(ii) Second, as suggested above, we partially generalize to higher rank results about the Hilbert scheme of points over threefolds. For these, the Donaldson-Thomas Euler characteristics, in the form needed here, were calculated in [BF].

2 Framed sheaves over local Calabi-Yau surface geometries

We now detail the discussion. Let S be a smooth complex projective surface, and let X^{∘} denote the total space of the canonical bundle K_{S} → S. We are concerned with moduli spaces of sheaves over the open Calabi-Yau threefold X^{∘}. The noncompact geometry does not allow for a good moduli space of semistable sheaves. Instead, we will consider the compact threefold

π : X = ℙ(K_{S} + O_{S}) → S.

This comes equipped with two divisors

S_{∞} = ℙ(K_{S} + 0) and S_{0} = ℙ(0 + O_{S})

corresponding to the summands K_{S} and O_{S}. Clearly,

X \ S_{∞} = X^{∘}.

We form the moduli space 𝔐_{n} of semistable framed modules (E, ϕ) of rank r with

c_{1}(E) = c_{2}(E) = 0, χ(F) = N := rχ(O_{S}) − n,

with a non-zero framing over S_{∞}:

ϕ : E → O_{S_{∞}} ⊗ ℂ^{r}.

The moduli space 𝔐_{n} was constructed in [HL]. Semistability was defined with respect to a polynomial δ of degree ≤ 2 with positive leading coefficient, as well as an ample divisor H on X. We will pick

H = π^{⋆}H_{0} + ϵc_{1}(O_{ℙ}(1)),

for an ample divisor H_{0} on S and a sufficiently small rational ϵ > 0. By definition, (E, ϕ) is semistable provided that

(i) for all proper subsheaves F of E, the Hilbert polynomials satisfy

P_{F} − δ ≤ \frac{rkF}{rkE} (P_{E} − δ);

Framed Sheaves and Symmetric Obstruction Theories 325

(ii) if F is contained in the kernel of ϕ, then

P_{F} ≤ \frac{rkF}{rkE} (P_{E} − δ).

Semistable framed modules admit Harder-Narasimhan filtrations, yielding the notion of S-equivalence. There is a projective moduli space 𝔐_{n} of Sequivalence classes of framed modules, cf. [HL].

We will consider the open subset

𝔐_{n}^{∘} ↪ 𝔐_{n}

corresponding to what are called framed sheaves in [L], [N]. These are stable framed modules (E, ϕ) such that

(iii) E is torsion free, locally free near S_{∞}, and ϕ is an isomorphism along S_{∞}.

Over curves and certain surfaces and for special framings, the stability conditions (i) and (ii) are automatic for the framed sheaves of (iii), cf. [BM], but for threefolds stability is not yet known to follow on general grounds.

Example We describe the moduli space in perhaps the simplest example, that of the Hilbert polynomial

P_{E} = P_{O_{X}} − ℓ.

Intuitively, in this case we should get the Hilbert scheme of points. This is not entirely obvious because framed modules are not required to be torsion free and because of the stability condition. The exact description will be determined by comparing δ to the polynomial

Δ = χ(mH|_{S_{∞}}) = \frac{m^{2}H_{0}^{2}}{2} + l.o.t.

To avoid strictly semistables, we assume that Δ − δ is not a constant a with 0 ≤ a ≤ ℓ.

The sheaves E in the moduli space have rank 1 and can be written in the form

0 → T → E → E^{∘} → 0

where T is torsion and E^{∘} is torsion free. In fact,

E^{∘} = I_{Z} ⊗ L

for some line bundle L → X, and some subscheme Z of dimension at most 1. Now, by stability, or using Lemma 1.2 of [HL], the kernel K of the restricted framing

ϕ_{∞} : T → O_{S_{∞}}

must satisfy

p_{K} ≤ 0 ⟹ K = 0.

Therefore ϕ_{∞} gives an inclusion of the torsion module T into the framing O_{S_{∞}}, showing that

T = 0 or T = i_{⋆}(I_{W} ⊗ M),

for some line bundle M over i : S_{∞} → X, and a subscheme W ⊂ S_{∞} of dimension zero.

In the first case, since c_{1}(E) = c_{2}(E) = 0, we must have c_{1}(L) = 0 and Z is zero dimensional. We claim that L = O_{X}. Indeed, L restricts trivially to the fibers of X → S, hence it must be a pullback

L = π^{⋆}N

of a degree zero line bundle N on S. The framing condition implies that there must exist a nonzero morphism N → O_{S}, hence N must be trivial. Therefore, up to isomorphisms, the only framed modules are

(E, ι) : I_{Z} → O_{S_{∞}}

for zero dimensional subschemes Z of length ℓ. We analyze semistability. The kernel of ι takes the form I_{U}(−S_{∞}) for some zero dimensional scheme U. Thus, we must have

χ(I_{U}(mH − S_{∞})) ≤ χ(mH) − ℓ − δ ⟺ ℓ − ℓ(U) ≤ Δ − δ.

If Δ − δ has negative leading term, the inequality cannot be satisfied. If Δ − δ has positive leading term, then the inequality is automatic and stability follows. Hence, the moduli space is either empty, or isomorphic to X^{[ℓ]}. ^{2}

We claim the second case cannot occur under our assumptions. If it did, then

0 → i_{⋆}(I_{W} ⊗ M) → E → L ⊗ I_{Z} → 0.

Calculating the Chern class c_{1}(E) = 0, we find

L = O(−S_{∞}) ⊗ π^{⋆}N,

for some degree 0 line bundle N overs S. Therefore

0 → i_{⋆}(I_{W} ⊗ M) → E → I_{Z}(−S_{∞}) ⊗ π^{⋆}N → 0, ϕ : E → O_{S_{∞}}.

2We also remark here that if Δ − δ = a for some a ∈ {0, …, ℓ}, then there are strictly semistable framed modules. Indeed, pairs of subschemes Z^{∘} of X^{∘} and Z_{∞} of S_{∞} with ℓ(Z_{∞}) = a yield the strictly semistable framed modules (I_{Z^{∘}}(−S_{∞}), 0) ⊕ (i_{⋆}I_{Z_{∞}/S_{∞}}, ι).

Documenta Mathematica 18 (2013) 323–342

We already argued above that the restriction ϕ_{∞} of ϕ to the torsion module i_{⋆}(I_{W} ⊗ M) must be injective. Since ϕ_{∞} ≠ 0, there must exist a non-zero morphism M → O_{S}, hence M^{∨} must be effective. Since c_{2}(E) = 0, we have

ι_{⋆}c_{1}(M) = [Z].

Therefore, M is trivial and Z is of dimension zero. The Hilbert polynomial gives

ℓ(Z) + ℓ(W) = ℓ.

Furthermore, up to scalars, ϕ_{∞} must be the natural inclusion. Semistability implies that

P_{i_{⋆}I_{W}} − δ ≤ 0 ⟹ Δ − δ ≤ ℓ(W).

Since ϕ_{∞} is the natural inclusion, the image of ϕ in O_{S_{∞}} must be the ideal sheaf I_{U} of a scheme U ⊂ W. Let K be the kernel of ϕ. We have

P_{K} = P_{E} − P_{i_{⋆}I_{U}} = P_{E} − (Δ − ℓ(U)).

Semistability implies

P_{K} ≤ P_{E} − δ ⟹ Δ − δ ≥ ℓ(U).

Therefore, we conclude that

Δ − δ = a

for some constant a such that

ℓ(W) ≥ a ≥ ℓ(U).

In particular 0 ≤ a ≤ ℓ which contradicts our assumption.

To summarize, when Δ − δ is not equal to a constant between 0 and ℓ, we obtain the following description of the moduli space:

(a) if (the leading term of) Δ − δ < 0, then we get ∅;

(b) if (the leading term of) Δ − δ > 0, then the moduli space is the Hilbert scheme X^{[ℓ]}.

3 Obstruction theory

We note now that the obstruction theory of framed sheaves is symmetric. To this end, we assume that δ is good, i.e. it satisfies the following conditions:

• if deg δ = 0, then δ > (r − 1)n;

• if deg δ = 2, the quadratic term of δ is sufficiently small compared to that of Δ.

In particular, any δ of degree 1 is good for all n.

Theorem 1. When δ is good, the moduli space 𝔐_{n}^{∘} admits a symmetric perfect obstruction theory at the stable points (E, ϕ).

Proof. The deformation theory of stable framed sheaves was worked out in [HL], [S]. Write

(E, Φ) → 𝔐_{n}^{∘} × X

for the universal family, which exists by [HL], and let p and q be the natural projections. The complex

F = Rp_{⋆}RHom(E(−S_{∞}), E ⊗ q^{⋆}K_{X})[2]

is an obstruction theory over 𝔐_{n}^{∘}. The obstruction theory is symmetric in the sense that there is a symmetric isomorphism

F → F^{∨}[1].

This is a consequence of Grothendieck duality and of the the crucial observation that

K_{X} = O(−2S_{∞}).

The calculation of the canonical bundle standardly follows from the Euler sequence of the projective bundle X.

The obstruction theory is perfect with amplitude contained in [−1, 0]. Indeed, the amplitude is clearly contained in [−2, 1]. By symmetry, it suffices to explain that the degree −2 term is zero. In turn, this is implied by the vanishing

Hom(E, E(−S_{∞})) = 0

which holds for all sheaves E in 𝔐_{n}^{∘}. Indeed, assuming there is a non-zero morphism

E → E(−S_{∞}),

we let K and I denote its kernel and image, and write r_{K} and r_{I} for their ranks. We have r_{I} > 0. By stability

P_{K} − δ ≤ \frac{r_{K}}{r} (P_{E} − δ), P_{I(S_{∞})} − δ ≤ \frac{r_{I}}{r} (P_{E} − δ).

Considering the quadratic terms of these inequalities, we obtain

c_{1}(K) · H^{2} ≤ δ_{0} (1 − \frac{r_{K}}{r} ) , c_{1}(I) · H^{2} + r_{I}S_{∞} · H^{2} ≤ δ_{0} (1 − \frac{r_{I}}{r} ) ,

where δ_{0} is half the leading term of δ. We also have

c_{1}(K) + c_{1}(I) = 0.

Adding, we obtain

r_{I}S_{∞} · H^{2} ≤ δ_{0}

which is impossible when the leading term δ_{0} < S_{∞} · H^{2} is sufficiently small. This completes the proof.

Example We determine the obstruction theory for the previous example. We consider case (b) corresponding to Δ − δ > 0, Δ − δ does not equal a constant a with 0 ≤ a ≤ ℓ. The tangent space at the ideal sheaf I_{Z} was found in [HL] to be

T_{Z}𝔐 = Ext^{1}(I_{Z}, [I_{Z} → O_{S_{∞}}]).

This can be calculated from the exact triangle

[I_{Z} → O_{S_{∞}}] → [O_{X} → O_{S_{∞}}] ≅ O(−S_{∞}) → O_{Z}.

We have

Ext^{0}(I_{Z}, O_{X}(−S_{∞})) = Ext^{3}(O_{X}, I_{Z}(−S_{∞}))^{∨} = H^{3}(I_{Z}(−S_{∞}))^{∨}
= H^{3}(O_{X}(−S_{∞}))^{∨} = 0,

and similarly for Ext^{1}. From the exact triangle, we obtain

0 → Ext^{0}(I_{Z}, O_{Z}) ≅ T_{Z}X^{[ℓ]} → T_{Z}𝔐 → 0.

In particular, this agrees with the identification

𝔐 ≅ X^{[ℓ]}.

Thus, by symmetry, the obstruction theory of 𝔐 coincides with the usual obstruction theory for the Hilbert scheme only along the open part (X^{∘})^{[ℓ]}.

4 Calculations

Symmetric perfect obstruction theories have associated Behrend functions [B]. In particular, the open moduli space

𝔐_{n}^{∘} ↪ 𝔐_{n}

is endowed with a constructible function

ν : 𝔐_{n}^{∘} → ℤ.

We will calculate the Donaldson-Thomas weighted Euler characteristic

χ̃(𝔐_{n}^{∘}) = ∑_{k} kχ(ν^{−1}(k)).

Since the obstruction theory is not perfect symmetric over the boundary, these weighted Euler characteristics do not calculate intersection theoretic Donaldson-Thomas invariants of 𝔐_{n}.

4.1 Virtual localization

Our computation is via equivariant localization. The following result was proved in [BF] for torus actions with isolated fixed points, and in [LQ] in arbitrary generality. Let 𝔐 be a moduli space admitting a C^{⋆}-action compatible with the symmetric perfect obstruction theory. Then the fixed point set 𝔐^{C^{⋆}} also inherits a symmetric perfect obstruction theory. Furthermore, the Behrend functions of 𝔐 and 𝔐^{C^{⋆}} at torus fixed points p are related by

ν_{𝔐}(p) = (−1)^{ϵ_{p}}ν_{𝔐^{C^{⋆}}} (p),

where ϵ_{p} is given by the difference in the dimension of the Zariski tangent spaces

ϵ_{p} = dim T_{p}𝔐 − dim T_{p}𝔐^{C^{⋆}}.

This observation is used in [LQ] as follows. The torus acts on the subscheme

{p ∈ 𝔐 \ 𝔐^{C^{⋆}} : ν_{𝔐}(p) = k}

with no fixed points, hence its Euler characteristic must be zero, cf. [LY]. Therefore,

χ({p ∈ 𝔐 : ν_{𝔐}(p) = k}) = χ({p ∈ 𝔐^{C^{⋆}} : ν_{𝔐}(p) = k})

yielding

χ̃(𝔐) = ∑_{k} kχ({p ∈ 𝔐^{C^{⋆}} : ν_{𝔐^{C^{⋆}}} (p) = k(−1)^{ϵ_{p}}}).

We will apply these remarks to the action of C^{⋆} on 𝔐_{n} induced by the scaling action in the fibers of the projective bundle X → S and the scaling action on the framing coming from a generic embedding

C^{⋆} ↪ GL_{r}.

We will find the torus fixed points in 𝔐_{n}.

Lemma 2. Assume δ is good. The C^{⋆}-fixed framed modules in 𝔐_{n} take the form

E = ⨁_{i=1}^{r} I_{Z_{i}}

where Z_{i} are zero dimensional subschemes of X invariant under the action of the torus, of total length n. The framing ϕ is the natural composition E ↪ O_{X}^{r} → O_{S_{∞}}^{r}.

Proof. We first prove that all invariant framed modules are torsion free. Indeed, the torsion module T of E is C^{⋆}-fixed. By stability, the framing ϕ gives a C^{⋆}- invariant injection

ϕ_{∞} : T ↪ O_{S_{∞}} ⊗ ℂ^{r}.

Therefore, the torsion module splits

T = ⊕_{j=1}^{ℓ}i_{⋆}(I_{W_{j}} ⊗ M_{j}),

for zero dimensional subschemes W_{j} of S_{∞} and line bundles M_{j} over S_{∞}. Again by stability applied to the torsion submodule T we find

p_{T} − δ ≤ 0 ⟹ ℓ · H^{2} · S_{∞} ≤ δ_{0}

where δ_{0} is half the quadratic coefficient of δ. By assumption, we may take δ_{0} < H^{2} · S_{∞}, implying that ℓ = 0 and showing that the torsion module vanishes.

Now, since E is torsion free and C^{⋆}-invariant, the argument of [BPT] shows that

E = ⊕_{i=1}^{r}I_{Z_{i}} ⊗ L_{i}

where L_{i} are line bundles over X and Z_{i} are subschemes of dimension at most 1. The subschemes Z_{i} must be torus invariant. Since c_{1}(E) = c_{2}(E) = 0, we find

∑ c_{1}(L_{i}) = 0  (1)

and furthermore

∑_{i=1}^{r} c_{1}(L_{i})^{2} = 2 ∑_{i=1}^{r}[Z_{i}].  (2)

Since the framed module (E, ϕ) is semistable, for all submodules F of E of positive rank we must have

\frac{c_{1}(F) · H^{2} − δ_{0}}{rk F} ≤ \frac{c_{1}(E) · H^{2} − δ_{0}}{rk E} .

Taking F = I_{Z_{i}} ⊗ L_{i} we find

c_{1}(L_{i}) · H^{2} ≤ δ_{0} (1 − \frac{1}{r}) .

Now, since δ_{0} is sufficiently small compared to the denominator of the rational divisor H, we conclude

c_{1}(L_{i}) · H^{2} ≤ 0.

In fact, c_{1}(L_{i}) · H^{2} = 0 for all i, because of (1). We argue that L_{i} are trivial and Z_{i} are zero dimensional.

Write

c_{1}(L_{i}) = π^{⋆}D_{i} + d_{i}ζ,

where D_{i} are divisors on the surface S and ζ = c_{1}(O_{ℙ}(1)). We calculate

∑_{i} c_{1}(L_{i})^{2} = π^{⋆} (∑_{i}D_{i}^{2}) + 2 ∑_{i} d_{i} (π^{⋆}D_{i} · ζ) + (∑_{i} d_{i}^{2})ζ^{2} = 2 ∑_{i}[Z_{i}], (3)

Set

M = 2 ∑_{i} d_{i}D_{i} − (∑_{i} d_{i}^{2}) K_{S}.

Using

ζ^{2} + K_{S} · ζ = 0,

we conclude from (3) that

π^{⋆} (∑_{i} D_{i}^{2}) + π^{⋆}M · ζ = 2 ∑_{i}[Z_{i}].  (4)

Pushing (4) forward under π we find

M = 2 ∑_{i} π_{⋆}[Z_{i}].

As a consequence, M is effective. The requirement that the slopes of L_{i} are trivial translates into the condition

(π^{⋆}D_{i} + d_{i}ζ)(π^{⋆}H_{0} + ϵζ)^{2} = 0

which rewrites as

(D_{i} − d_{i}K_{S}) · Σ = −d_{i}

where

Σ = \frac{ϵ(2H_{0} − ϵK_{S})}{H_{0}^{2}}

is an ample rational curve class on S for small ϵ. Write

F_{i} = D_{i} − d_{i}K_{S}

so that

F_{i} · Σ = −d_{i}.

Since

M = 2 ∑_{i} d_{i}F_{i} + (∑_{i} d_{i}^{2})K_{S}

is effective, its intersection with Σ must be positive. This gives

−2 ∑_{i} d_{i}^{2} + (∑_{i} d_{i}^{2})K_{S} · Σ ≥ 0.

For small ϵ we have K_{S} · Σ < 2. We conclude from here that d_{i} = 0 for all i. Therefore M = 0, and by (4) we must have

π^{⋆} (∑_{i} D_{i}^{2}) = 2 ∑_{i}[Z_{i}]

is effective. Note that the left hand side is supported on fibers. Therefore,

∑_{i} D_{i}^{2} ≥ 0.  (5)

We moreover proved

F_{i} · Σ = 0 ⟹ D_{i} · Σ = 0.

Since Σ is ample, by Hodge index theorem we have

D_{i}^{2} ≤ 0,

with equality only if D_{i} is numerically equivalent to 0. In fact equality must occur because of (5). This yields c_{1}(L_{i}) = 0. In turn,

L_{i} = π^{⋆}N_{i}

for some line bundles N_{i} → S of first Chern class 0. Furthermore, from (3) we find [Z_{i}] = 0 hence Z_{i} must be zero dimensional.

Thus

E = ⊕_{i=1}^{r}I_{Z_{i}} ⊗ π^{⋆}N_{i},

where

∑ ℓ(Z_{i}) = n.

Clearly, Z_{i} must be torus invariant and ϕ = ⊕ϕ_{i} where

ϕ_{i} : I_{Z_{i}} ⊗ π^{⋆}N_{i} → O_{S_{∞}}.

We next claim that ϕ_{i} ≠ 0 for all i. Indeed, if ϕ_{i} = 0 for some i, then I_{Z_{i}} ⊗π^{⋆}N_{i} is in the kernel of ϕ, yielding by stability

χ(I_{Z_{i}}(mH) ⊗ π^{⋆}N_{i}) ≤ \frac{1}{r} (∑_{j} χ(I_{Z_{j}}(mH) ⊗ π^{⋆}N_{j}) − δ) .

This gives

n ≥ ℓ(Z_{i}) ≥ \frac{n}{r} + \frac{δ}{r} .

This is a contradiction since δ > (r − 1)n. Therefore ϕ_{i} ≠ 0, showing that there exists a non-zero morphism N_{i} → O_{S}. Therefore N_{i} must be trivial, completing the proof.

Over the open moduli space 𝔐_{n}^{∘}, the same result holds without any restrictions on δ:

Lemma 2A. The C^{⋆}-fixed framed sheaves E in 𝔐_{n}^{∘} must split

E = ⨁_{i=1}^{r} I_{Z_{i}}

where Z_{i} are zero dimensional subschemes of X^{∘} invariant under the action of the torus, of total length n.

Proof. By assumption E is torsion free, hence

E = ⊕_{i=1}^{r}I_{Z_{i}} ⊗ L_{i}.

Since the framing is an isomorphism, we conclude Z_{i} is contained in X^{∘} and L_{i} is trivial on S_{∞}. Hence,

L_{i} = O(d_{i}S_{0})

for some integers d_{i}. We claim that d_{i} = 0 for all i. This in turn implies that Z_{i} are zero dimensional by using c_{2}(E) = 0.

Assume first that the quadratic term of δ is sufficiently small. This case is already covered by Lemma 2, but a simpler argument is possible over 𝔐_{n}^{∘}; we record it here for future reference. Indeed, the stability condition applied to L_{i} ⊗ I_{Z_{i}} gives

d_{i}S_{0} · H^{2} = c_{1}(L_{i}) · H^{2} ≤ δ_{0} (1 − \frac{1}{r}) ⟹ d_{i} ≤ 0.

Since c_{1}(E) = 0, we have ∑_{i} d_{i} = 0. Hence d_{i} = 0 for all i, as claimed.

We now give the general argument. Using that c_{1}(E) = c_{2}(E) = 0, we find

(∑_{i} d_{i}^{2})S_{0}^{2} = 2 ∑_{i}[Z_{i}].  (6)

Assume not all d_{i} are equal to 0. Since the Z_{i}’s are torus invariant and disjoint from S_{∞}, their cohomology classes are supported on the surface S_{0}. Using that

S_{0}^{2} = (K_{S}^{2})f + ζ · π^{⋆}K_{S},

from equation (6) we find K_{S}^{2} = 0. From here, pushing forward under π, we conclude

(∑_{i} d_{i}^{2})K_{S} = 2 ∑_{i} π_{⋆}[Z_{i}]

which is effective. Hence

K_{S} · H_{0} ≥ 0.

Now, I_{Z_{i}}(d_{i}S_{0} − S_{∞}) is contained in the kernel of ϕ. Hence by stability

(d_{i}S_{0} − S_{∞}) · H^{2} ≤ −\frac{δ_{0}}{r} .

Pick an index i such that d_{i} ≥ 1. The above inequality implies

(S_{0} − S_{∞}) · H^{2} ≤ −\frac{δ_{0}}{r} ⟹ π^{⋆}K_{S} · H^{2} ≤ −\frac{δ_{0}}{r} .

However,

π^{⋆}K_{S} · H^{2} = ϵ · (2H_{0} − ϵK_{S})K_{S} = 2ϵ · K_{S} · H_{0} ≥ 0.

Therefore, K_{S} · H_{0} = δ_{0} = 0. Since the quadratic term of δ is 0, the previous paragraph applies, showing that in fact all d_{i} = 0.

Lemma 3. If δ is good, all torus fixed framed sheaves E in 𝔐_{n} described above are stable.

Proof. Let F be a subsheaf of E = ⊕I_{Z_{i}} of rank r′. Since F is a subsheaf of O_{X}^{r}, by Gieseker semistability we have

P_{F} ≤ r′χ(mH) < \frac{r′}{r} P_{E} + \frac{r − r′}{r} δ,

at least when r′ ≠ r, using that δ > (r − 1)n. When r′ = r, induction on r yields the claim. For the inductive step, consider the non-zero map F → I_{Z_{r}}, and write F′ for the kernel. Then, apply the induction hypothesis to F′ which is contained in ⊕_{i=1}^{r−1}I_{Z_{i}}.

Next, assume F is in the kernel of ϕ. The kernel of ϕ is contained in O_{X}(−S_{∞})^{r} (and it is isomorphic to ⊕_{j}I_{Z_{j}}(−S_{∞}) for E in 𝔐_{n}^{∘}). By Gieseker-semistability, we have

P_{F} ≤ r′χ(mH − S_{∞}) < \frac{r′}{r} (rχ(mH) − n − δ) = \frac{r′}{r} (P_{E} − δ),

using that δ is good.

Lemma 4. For all torus fixed sheaves E in 𝔐_{n}^{∘}, we have

dim T_{E}𝔐_{n} ≡ rn mod 2.

Proof. Since E = ⊕I_{Z_{i}} is stable, the tangent space is calculated in [HL]:

T_{E}𝔐_{n} = Ext^{1}(E, E(−S_{∞})) = ∑_{i,j} Ext^{1}(I_{Z_{i}}, I_{Z_{j}}(−S_{∞})).

We consider first the contributions of terms corresponding to pairs of indices (i, j) and (j, i) for i ≠ j:

Ext^{1}(I_{Z_{i}}, I_{Z_{j}}(−S_{∞})) + Ext^{1}(I_{Z_{j}}, I_{Z_{i}}(−S_{∞}))

= Ext^{1}(I_{Z_{i}}, I_{Z_{j}}(−S_{∞})) + Ext^{2}(I_{Z_{i}}, I_{Z_{j}}(−S_{∞}))

by Serre duality. Now, considering the above expression modulo 2 we obtain

χ(I_{Z_{i}}, I_{Z_{j}}(−S_{∞})) + Ext^{0}(I_{Z_{i}}, I_{Z_{j}}(−S_{∞})) + Ext^{3}(I_{Z_{i}}, I_{Z_{j}}(−S_{∞})).

Next, it is easily seen that Ext^{0} vanishes, and same for Ext^{3} by duality. Thus, we are left with

χ(I_{Z_{i}}, I_{Z_{j}}(−S_{∞})) = χ(O_{X}, O_{X}(−S_{∞})) − χ(O_{X}, O_{Z_{j}}) − χ(O_{Z_{i}}, O_{X}(−S_{∞}))
= χ(O_{X}(−S_{∞})) − ℓ(Z_{j}) + ℓ(Z_{i})
= ℓ(Z_{i}) + ℓ(Z_{j}) mod 2.

We consider now the terms with i = j:

Ext^{1}(I_{Z_{i}}, I_{Z_{i}}(−S_{∞})).

This term was already worked out in the deformation theory of Example 1. We obtained

Ext^{1}(I_{Z_{i}}, I_{Z_{i}}(−S_{∞})) = Ext^{0}(I_{Z_{i}}, O_{Z_{i}}) ≡ ℓ(Z_{i}) mod 2,

where for the last congruence we used [BF] or [MNOP]. The lemma follows by collecting the above facts.

We can now put together the calculation of Lemma 4 and the remarks about Behrend functions in Subsection 4.1 to calculate the Donaldson-Thomas Euler characteristic of 𝔐_{n}^{∘}. We write 𝔛_{ℓ} for the subset of the Hilbert scheme of points in X^{∘} which parametrizes torus fixed Z’s of length ℓ. For each partition ℓ⃗ into r parts (ℓ_{1}, …, ℓ_{r}) with

ℓ_{1} + … + ℓ_{r} = n

we write

𝔛ℓ⃗ = 𝔛^{ℓ_{1}} × … × 𝔛^{ℓ_{r}}.

Then, 𝔛ℓ⃗ are the C^{⋆}-fixed loci of 𝔐_{n}^{∘}. With the convention that

Z⃗ = (Z_{1}, …, Z_{r})

represents an r-tuple of schemes in 𝔛ℓ⃗, we calculate

χ̃ (𝔐_{n}^{∘}) =
= ∑_{ℓ⃗} ∑_{k} k χ({Z⃗ ∈ 𝔛ℓ⃗ : ν_{𝔛ℓ⃗}(Z⃗) = k(−1)^{rn−dim TZ⃗𝔛ℓ⃗}})

= (−1)^{(r−1)n} ∑_{ℓ⃗} ∑_{k⃗} ∏_{i=1}^{r} k_{i} χ({Z_{i} ∈ 𝔛^{ℓ_{i}} : ν_{𝔛^{ℓ_{i}}} (Z_{i}) = k_{i}(−1)^{ℓ_{i}−dim T_{Z_{i}} 𝔛^{ℓ_{i}}} })

= (−1)^{(r−1)n} ∑_{ℓ⃗} ∏_{i=1}^{r} (∑_{k} kχ({Z : ν_{𝔛^{ℓ_{i}}} (Z) = k(−1)^{ℓ_{i}−dim T_{Z}𝔛^{ℓ_{i}}} })

By applying these results when r = 1, and using the identification of the rank 1 moduli space with the Hilbert scheme worked out in Example 1, we obtain

χ̃((X^{∘})^{[ℓ]}) = ∑_{k} kχ({Z : ν_{𝔛^{ℓ}}(Z) = k(−1)^{ℓ−dim T_{Z}𝔛_{ℓ}}}).

This yields

χ̃(𝔐_{n}^{∘}) = (−1)^{(r−1)n} ∑_{ℓ⃗} χ̃((X^{∘})^{[ℓ_{1}]}) · . . . · χ̃((X^{∘})^{[ℓ_{r}]}).

We form the generating series

∑_{n} q^{n}χ̃(𝔐_{n}^{∘}) = (∑_{ℓ}((−1)^{r−1}q)^{ℓ}χ̃((X^{∘})^{[ℓ]}))^{r} .

Now, from [BF] we lift the calculation

∑_{ℓ} q^{ℓ}χ̃((X^{∘})^{[ℓ]}) = M(−q)^{e(X^{∘})} = M(−q)^{e(S)},

where M(q) is the MacMahon function

M(q) = ∏_{k=1}^{∞}(1 − q^{k})^{−k}.

To summarize, for δ good, we proved

Theorem 5. The following equality holds

∑_{n≥0} q^{n}χ̃(𝔐_{n}^{∘}) = M((−1)^{r}q)^{re(S)}.  (7)

We are unable to define (and calculalte) the virtual motive [𝔐_{n}^{∘}]^{vir}, as it is done in rank 1 in [BBS] and for surfaces in [N]. This question may deserve further study.

4.2 Topological Euler characteristics

Lemma 1 also allows us to calculate the topological Euler characteristics of the compact spaces 𝔐_{n} via the localization results of [LY]:

e(𝔐_{n}) = e(𝔐_{n}^{C^{⋆}}).

The same calculation as above shows that

∑ q^{n}e(𝔐_{n}) = (∑ q^{n}e(ℑ_{n}))^{r} = M(q)^{re(X)}  (8)

where ℑ_{n} ≅ X^{[n]} denotes the rank 1 moduli space. The series needed here

∑ q^{n}e(ℑ_{n}) = M(q)^{e(X)}

is computed in [C]. The answer we found is valid whenever δ is good.

4.3 Blow-up surfaces

A slightly more complicated example arises by considering blow-up geometries. Indeed, assume that the surface S contains a (−1)-curve C. Then, C ↪ S_{0} is super-rigid in X:

N_{C/X} = N_{C/S} ⊕ N_{S_{0}/X}|_{C} = O_{C}(−1) ⊕ O_{C}(−1).

We consider the moduli space 𝔐_{n,k} of rank r modules over X framed by a trivial rank r bundle at infinity, with numerics

c_{1} = 0, c_{2} = k[C], χ = rχ(O_{X}) − n.

In order not to worry about stability, we assume that δ is good of degree 2. By the argument in the first two paragraphs of Lemma 2A, the torus fixed sheaves in 𝔐_{n,k}^{∘} take the form

E = ⊕_{i}I_{Z_{i}}

where Z_{i} may have at most 1 dimensional components contained in X^{∘}. Furthermore,

∑_{i}[Z_{i}] = k[C], ∑_{i} χ(O_{Z_{i}}) = n.

In fact, [Z_{i}] = k_{i}[C] for non-negative integers k_{i} adding up to k. Indeed, after projecting to the blowdown surface

X → S → S̄

the classes of the effective curves Z_{i} add up to 0, hence they must be trivial. This shows that the components of Z_{i} are supported on the fibers of X → S or are contained in the Hirzebruch surface

𝔽 = ℙ(K_{S}|_{C} ⊕ O_{C}) → C.

In fact, by torus invariance, all components of Z_{i} must be supported over fibers or over the zero section C ↪ S_{0}. Since

∑[Z_{i}] = k[C]

contains no fiber classes, or alternatively since the framing must be an isomorphism along S_{∞}, we conclude that Z_{i} has no support over fibers, hence [Z_{i}] = k_{i}[C] as claimed.

We carry out the computation of the Donaldson-Thomas Euler characteristics. In the new setting, for all ℂ^{⋆}-fixed sheaves E in 𝔐_{n,k}^{∘} we have

dim T_{E}𝔐_{n,k} ≡ rn − k mod 2.

The proof follows that of Lemma 4. The only change is the calculation

dim Ext^{1}(I_{Z_{i}}, I_{Z_{i}}(−S_{∞})) ≡ χ(O_{Z_{i}}) − k_{i} mod 2.

To this end, consider the exact sequence

0 → I_{Z_{i}}(−S_{∞}) → I_{Z_{i}} → O_{S_{∞}} → 0.

Since the map Ext^{0}(I_{Z_{i}}, I_{Z_{i}}) → Ext^{0}(I_{Z_{i}}, O_{S_{∞}}) is an isomorphism, we obtain the exact sequence

0 → dim Ext^{1}(I_{Z_{i}}, I_{Z_{i}}(−S_{∞})) → Ext^{1}(I_{Z_{i}}, I_{Z_{i}}) → Ext^{1}(I_{Z_{i}}, O_{S_{∞}}).

To find the last group, we use the local to global spectral sequence

H^{p}(Ext^{q}(I_{Z_{i}}, O_{S_{∞}})) ⟹ Ext^{p+q}(I_{Z_{i}}, O_{S_{∞}}).

The terms with q ≥ 1 vanish since Z_{i} avoids S_{∞}, while the q = 0 terms equal H^{p}(O_{S}). Therefore,

Ext^{1}(I_{Z_{i}}, O_{S_{∞}}) = H^{1}(O_{S}).

From the exact sequence, we conclude

Ext^{1}(I_{Z_{i}}, I_{Z_{i}}(−S_{∞})) = Ext^{1}(I_{Z_{i}}, I_{Z_{i}})_{0}.

The dimension of the last vector space was found in [BB] using Theorem 2 of [MNOP]. The answer is

Ext^{1}(I_{Z_{i}}, I_{Z_{i}}(−S_{∞})) ≡ χ(O_{Z_{i}}) − k_{i} mod 2

as claimed above.

We form the generating series

∑_{n,k} χ̃ (𝔐_{n,k}^{∘}) q^{n}v^{k} = (∑_{n,k}((−1)^{r−1}q)^{n}v^{k} χ̃(ℑ_{n,k}^{∘}))^{r}

where ℑ_{n,k}^{∘} denotes the rank 1 framed moduli space. This is isomorphic to the Hilbert scheme. The rank 1 Donaldson-Thomas invariants of super-rigid curves were calculated in [BB]:

∑_{n,k} q^{n}v^{k}χ̃(ℑ_{n,k}^{∘}) = M(−q)^{e(X^{∘})} · ∏_{m=1}^{∞}(1 − (−q)^{m}v)^{m}.

Therefore, we obtain

∑_{n,k} χ̃ (𝔐_{n,k}^{∘}) q^{n}v^{k} = M((−1)^{r}q)^{re(S)} · ∏_{m=1}^{∞}(1 − ((−1)^{r}q)^{m}v)^{mr}.  (9)

4.4 Other geometries

There are other geometries for which the above methods apply. We discuss some of them here. Most straightforwardly, assuming S_{∞} is a smooth framing divisor with

K_{X} = −2S_{∞},

then our techniques yield

∑_{n} q^{n}χ̃(𝔐_{n}^{∘}) = M((−1)^{r}q)^{re(X^{∘})}.  (10)

In order to make the proof of Lemma 2A work, we need to assume for instance that the restriction

Pic(X) ∩ (H^{2})^{⊥} → Pic(S_{∞})

is injective. If X is Fano of index 2, this requirement is satisfied by the Lefschetz hyperplane theorem applied to the ample class S_{∞}. Examples pertinent to this setting include, among others:

• X is a cubic in ℙ^{4} or a (2, 2) complete intersection in ℙ^{5}, and S_{∞} is a hyperplane section;

• X is a double cover of ℙ^{3} branched along a quartic, and S_{∞} is the pullback of a hyperplane.

Fano threefolds of index higher than 2 also yield symmetric perfect obstruction theories. This can be checked directly using the well-known classification:

• X is a quadric in ℙ^{4} or X = ℙ^{3}, and S_{∞} is a hyperplane section.

In index 3, more examples arise from the curve geometry:

• X = ℙ(O_{C} + E) → C, with E → C any rank 2 bundle of determinant det E = K_{C}, and S_{∞} the divisor at infinity.

Since the same argument works in all cases above, let us only discuss the rank r sheaves over ℙ^{3} framed along the plane at infinity ℙ^{2} ↪ ℙ^{3}. Along 𝔐_{n}^{∘}, the obstruction theory is symmetric since

T_{E}𝔐_{n} = Ext^{1}(E, E(−1)) = Ext^{1}(E, E(−3)) = Ext^{2}(E, E(−1))^{∨} = ob_{E}^{∨}.

The second isomorphism follows from the short exact sequences

0 → E(−k − 1) → E(−k) → O_{ℙ^{2}}^{r}(−k) → 0

for k = 1 and k = 2, and the vanishings

Ext^{0}(E, O_{ℙ^{2}}(−k)) = Ext^{1}(E, O_{ℙ^{2}}(−k)) = 0 for k = 1, 2.

The first vanishing is clear. The second follows from the local to global spectral sequence:

E_{2}^{p,q} = H^{p}(Ext^{q}(E, O_{ℙ^{2}}(−k)) → Ext^{p+q}(E, O_{ℙ^{2}}(−k))

with vanishing E_{2} terms when p + q = 1. This proves the claim about the obstruction theory. Equation (10) still holds by the same methods.

5 Acknowledgements

The author gratefully acknowledges correspondence with Balázs Szendrői, as well as support from the NSF via grant DMS 1001486 and from the Sloan Foundation.

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Dragos Oprea Department of Mathematics University of California, San Diego doprea@math.ucsd.edu