Documenta Math. 323 Framed Sheaves over Treefolds and Symmetric Obstruction Theories

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


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 directions1 .
(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 This comes equipped with two divisors S ∞ = P(K S + 0) and S 0 = P(0 corresponding to the summands K S and O S .Clearly, 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 ∞ : 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 for an ample divisor H 0 on S and a sufficiently small rational ǫ > 0. By definition, (E, φ) is semistable provided that (i) for all proper subsheaves F of E, the Hilbert polynomials satisfy Framed Sheaves and Symmetric Obstruction Theories 325 (ii) if F is contained in the kernel of φ, then 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 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 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 To avoid strictly semistables, we assume that ∆ − 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 where T is torsion and E • is torsion free.In fact, 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 Therefore φ ∞ gives an inclusion of the torsion module T into the framing O S∞ , showing that 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 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) : 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 If ∆ − δ has negative leading term, the inequality cannot be satisfied.If ∆ − 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 Calculating the Chern class c 1 (E) = 0, we find for some degree 0 line bundle N overs S. Therefore 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 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 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 Semistability implies Therefore, we conclude that ∆ − 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 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 < 0, then we get ∅; (b) if (the leading term of) ∆ − δ > 0, then the moduli space is the Hilbert scheme X [ℓ] .

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 = 2, the quadratic term of δ is sufficiently small compared to that of ∆.

Dragos Oprea
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 is an obstruction theory over M • n .The obstruction theory is symmetric in the sense that there is a symmetric isomorphism This is a consequence of Grothendieck duality and of the the crucial observation that 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 which holds for all sheaves E in M • n .Indeed, assuming there is a non-zero morphism 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 Considering the quadratic terms of these inequalities, we obtain where δ 0 is half the leading term of δ.We also have Adding, we obtain which is impossible when the leading term δ 0 < S ∞ • H 2 is sufficiently small.This completes the proof.

Documenta Mathematica 18 (2013) 323-342
Example We determine the obstruction theory for the previous example.We consider case (b) corresponding to ∆ − δ > 0, ∆ − 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 This can be calculated from the exact triangle We have and similarly for Ext 1 .From the exact triangle, we obtain In particular, this agrees with the identification 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] .

Calculations
Symmetric perfect obstruction theories have associated Behrend functions [B].
In particular, the open moduli space We will calculate the Donaldson-Thomas weighted Euler characteristic 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 .

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 where ǫ p is given by the difference in the dimension of the Zariski tangent spaces This observation is used in [LQ] as follows.The torus acts on the subscheme with no fixed points, hence its Euler characteristic must be zero, cf.[LY].Therefore, 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 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 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 Therefore, the torsion module splits 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 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 = ⊕ r i=1 I Zi ⊗ 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 and furthermore Since the framed module (E, φ) is semistable, for all submodules F of E of positive rank we must have Now, since δ 0 is sufficiently small compared to the denominator of the rational divisor H, we conclude In fact, c 1 (L i ) • H 2 = 0 for all i, because of (1).We argue that L i are trivial and where D i are divisors on the surface S and ζ = c 1 (O P (1)).We calculate Using we conclude from (3) that Pushing ( 4) forward under π we find As a consequence, M is effective.The requirement that the slopes of L i are trivial translates into the condition H 2 0 is an ample rational curve class on S for small ǫ.Write is effective, its intersection with Σ must be positive.This gives For small ǫ 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 Framed Sheaves and Symmetric Obstruction Theories 333 is effective.Note that the left hand side is supported on fibers.Therefore, We moreover proved Since Σ is ample, by Hodge index theorem we have 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, 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.
Clearly, Z i must be torus invariant and φ = ⊕φ i where We next claim that φ i = 0 for all i.Indeed, if φ i = 0 for some i, then I Zi ⊗π ⋆ N i is in the kernel of φ, yielding by stability This gives 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: 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 Since the framing is an isomorphism, we conclude Z i is contained in X • and L i is trivial on S ∞ .Hence, 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 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 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 from equation ( 6) we find K 2 S = 0. From here, pushing forward under π, we conclude (

Framed Sheaves and Symmetric Obstruction Theories 335
Pick an index i such that d i ≥ 1.The above inequality implies However, 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 Zi of rank r ′ .Since F is a subsheaf of O r X , by Gieseker semistability we have 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 Zr , and write F ′ for the kernel.Then, apply the induction hypothesis to 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 . By Gieseker-semistability, we have using that δ is good.
Lemma 4. For all torus fixed sheaves E in M • n , we have Proof.Since E = ⊕I Zi is stable, the tangent space is calculated in [HL]: We consider first the contributions of terms corresponding to pairs of indices (i, j) and (j, i) for i = j: by Serre duality.Now, considering the above expression modulo 2 we obtain Next, it is easily seen that Ext 0 vanishes, and same for Ext 3 by duality.Thus, we are left with We consider now the terms with i = j: This term was already worked out in the deformation theory of Example 1.We obtained 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 ℓ into r parts (ℓ 1 , . . ., ℓ r ) with ℓ 1 + . . .
Then, X ℓ are the C ⋆ -fixed loci of M • n .With the convention that represents an r-tuple of schemes in X ℓ , we calculate Framed Sheaves and Symmetric Obstruction Theories 337 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 This yields ).
We form the generating series Now, from [BF] we lift the calculation S) , where M(q) is the MacMahon function To summarize, for δ good, we proved Theorem 5.The following equality holds n≥0 q n χ(M • n ) = M((−1) r q) re (S) .
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.

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]: ).The same calculation as above shows that where I n ∼ = X [n] denotes the rank 1 moduli space.The series needed here is computed in [C].The answer we found is valid whenever δ is good.

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: 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 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 for non-negative integers k i adding up to k.Indeed, after projecting to the blowdown surface X → S → S 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 In fact, by torus invariance, all components of Z i must be supported over fibers or over the zero section C ֒→ S 0 .Since 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 We carry out the computation of the Donaldson-Thomas Euler characteristics.
In the new setting, for all The proof follows that of Lemma 4. The only change is the calculation To this end, consider the exact sequence 0 Since the map Ext 0 (I Zi , I Zi ) → Ext 0 (I Zi , O S∞ ) is an isomorphism, we obtain the exact sequence To find the last group, we use the local global spectral sequence The terms with q ≥ 1 vanish since Z i avoids S ∞ , while the q = 0 terms equal From the exact sequence, we conclude Ext The dimension of the last vector space was found in [BB] using Theorem 2 of [MNOP].The answer is Ext as claimed above.
We form the generating series 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]: Therefore, we obtain n,k χ (M • n,k ) q n v k = M((−1) r q) re (S) • ∞ m=1 (1 − ((−1) r q) m v) mr .(9)

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 χ(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.