Erratum to Donaldson–Thomas invariants and flops (J. reine angew. Math. 716 (2016), 103–145)

1 Explanation

The paper [6] is concerned with proving a comparison formula for Donaldson–Thomas (DT) invariants. Let us recall the main strategy. One starts with a flopping contraction Y→X and a flop Y+→X. The main goal is to prove the formula

The proof goes via the fact that Y and Y+ are derived equivalent. The respective categories of perverse coherent sheaves are mapped to each other under Bridgeland’s equivalence, hence the perverse Hilbert schemes are isomorphic. Up to some additional details, this formally implies that the perverse DT numbers of Y and Y+ are the same [6, Theorem 3.26].

The main focus of the paper actually lies in proving [6, Theorem 3.23], which compares the perverse DT invariants with the ordinary ones. In this sense, all the action happens on just one side of the flop.

Let us go back to our flopping contraction Y→X. Readers solely interested in flops can assume that the singular locus of X is zero-dimensional. In this case, the statements and proofs of [6] can be left untouched and one can safely ignore the present text.

Nonetheless, the actual perverse/ordinary comparison formula is of interest beyond the context of flops (for example in the setting of the crepant resolution conjecture for Donaldson–Thomas invariants [2, 3]).

If we allow the singular locus of X to be a curve then [6, Theorem 3.23] needs to be modified into [5, Theorem 3.23]. The latter statement, however, contains the symbol DT∂ which we now explain.

The proof of [6, Proposition 3.5] contains a mistake, the root of which can be traced back to the following fact. Inside the category of perverse coherent sheaves 𝒜p we find 𝒜≤1p, consisting of those complexes having support of dimension at most one (recall that the support of a complex is the union of the supports of the cohomology sheaves).

The subcategory 𝒜≤1p⊂𝒜p is not closed under subobjects nor quotients.

Let us illustrate this with an example, for which we thank the referee. For simplicity, let us assume p=-1. Suppose f:Y→X is contracting a divisor D≃ℙ1×ℙ1 to a ℙ1, via projection to the second factor. Let C be a horizontal curve, i.e. C≃{l}×ℙ1 for some l∈ℙ1, so that we have f⁢(C)=ℙ1. Take the following short exact sequence:

As f*⁢𝒪D⁢(-C)=0 we see that 𝒪D⁢(-C)∈ℱp. As 𝒪D and 𝒪C are quotients of 𝒪Y they both lie in 𝒯p. Therefore,

is a short exact sequence in 𝒜p where the middle term lies in 𝒜≤1p while the outer terms do not.

To fix the mistake, one needs (among other things) to change the main identity (3.4) of [6]. The element ℋ≤1 needs to be replaced by 𝒦≤1, which corresponds to the subscheme Hilb≤1∂⁡(Y)⊂Hilb≤1⁡(Y) parameterizing those surjections 𝒪Y↠𝒪Z in Coh⁡(Y) whose perverse cokernel lies in ℱ≤1p⁢[1]. This is explained in [5, Remark 3.5]. We remark that the perverse cokernel of any surjection 𝒪Y↠𝒪Z lies in ℱp⁢[1].

When X happens to have singular locus of dimension zero, ℱ≤1p=ℱp and thus Hilb≤1∂⁡(Y)=Hilb≤1⁡(Y). However, in general the two are different. Let us illustrate this by means of an example. Suppose p=-1 and assume again f:Y→X is contracting a divisor D≃ℙ1×ℙ1 to a ℙ1. Let as before C be a horizontal curve and let IC be its ideal sheaf in Y, so that we have a short exact sequence

in Coh⁡(Y). Let now

be the unique exact sequence with respect to the torsion pair (𝒯p,ℱp). It follows that F⁢[1] is the perverse cokernel of 𝒪Y→𝒪C.

We also have a short exact sequence

Since we have 𝒪D⁢(-C)∈ℱp, the surjection IC→𝒪D⁢(-C) factors through a surjection F→𝒪D⁢(-C). Hence the support of F is two-dimensional.

Similarly, we have two perverse Hilbert schemes: Hilbch≤1p and Hilb≤1p. The first parameterizes perverse quotients 𝒪Y↠E with ch0⁡(E)=0=ch1⁡(E), while the second parameterizes quotients 𝒪Y↠E where the dimension of the support of E is at most one. As a consequence, we introduce partial invariants DT∂ by integrating 𝒦≤1 or, equivalently, by taking the weighted Euler characteristic of Hilb≤1∂⁡(Y).

2 Differences

We proceed here by detailing how [6] was modified to obtain [5] (we will omit a few minor changes which have no impact on the mathematical content).

1. For organization purposes, we moved Lemma 3.2 to [5, Lemma 1.4] and added [5, Lemma 1.5]. The latter makes it clear that the subcategory ℱ≤1p⁢[1]⊂𝒜≤1p is closed under quotients and extensions.

2. Section 2 can be left untouched.

3. In [5], we added Remark 3.5 to explain why the element 𝒦≤1 is necessary.

4. In the identity (3.4), ℋ≤1 should be replaced by 𝒦≤1, cf. [5, (3.6)].

5. The definitions of the stacks 𝔐R and 𝔑 need to be changed: one must require in both cases the morphism 𝒪Y→T in the displayed diagram to have perverse cokernel in ℱ≤1p⁢[1].

6. The proof of Proposition 3.5 needs to be modified slightly (by including an application of the snake lemma, cf. [5, Proposition 3.10]).

7. In the identity prior to Section 3.6, one should replace ℋ≤1 by 𝒦≤1.

8. In the statement of Proposition 3.15, ℋ≤1 should be replaced by 𝒦≤1. Similarly, in its proof Hilb should be replaced by Hilb∂, cf. [5, Proposition 3.21].

9. Similarly, in the first identity of Section 3.7, ℋ≤1 should be replaced by 𝒦≤1.

10. Remark 3.21 should be extended to point out that it holds ditto for DT∂⁡(Y).

From here on out, we should introduce the notation DT∂⁡(Y/X) for consistency with DT∂⁡(Y) and DT⁡(Y), cf. [5, Remark 3.28]. Namely, DT⁡(Y/X) comes from integrating Hilbch≤1p(Y/X) while DT∂⁡(Y/X) comes from Hilb≤1p(Y/X). From [6, Remark 3.21] onwards, one should replace DT⁡(Y/X) with DT∂⁡(Y/X). Similarly, 𝒦≤1 should replace ℋ≤1 in all Hall algebra identities.

Finally, [5] contains a new section where we investigate what happens when we restrict to curve classes contracted by f.

3 One-dimensional singular locus

Recall our general setup.

As we have remarked earlier the category 𝒜≤1p, consisting of those E∈𝒜p with dim⁡supp⁡E≤1, is not closed under quotients. This fact is the core reason why we had to replace ℋ with 𝒦 and is what causes the appearance of the “partial” DT numbers.

It is not clear whether there is a compact formula in the Hall algebra relating ℋp with ℋ. Any approach seems to involve dealing with surface classes contracted by f. To complicate matters further, the basic identity (in H∞) 1𝒜p𝒪=1ℱp⁢[1]𝒪*1𝒯p𝒪 does not even hold in the full Hall algebra: given E∈𝒜p, together with its torsion and torsion-free parts F⁢[1] and T, there is an exact sequence

where the last group is equal to H2⁢(Y,F)=H1⁢(X,R1⁢f*⁢F). This group vanishes when F∈ℱ≤1p (this assumption, in conjunction with f*⁢F=0, forces the support of F to be a union of curves contracted to points, hence the support of R1⁢f* is zero-dimensional), but in general it may not be the case.

This being said, let us come to the good news. If we restrict to the subcategory 𝒜excp then we can work around these obstacles.

Before we prove this lemma, we recall the key technical result of Van den Bergh [1, Lemmas 3.1.3, 3.1.5].

What truly makes the category 𝒜excp robust is the following lemma.

Consequently, Section 3 of [6] can be adapted to the category 𝒜excp without needing any “partial” invariants.