Rank $r$ DT theory from rank $0$

Abstract

Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the quintic 3-fold. We express Joyce's generalised DT invariants counting Gieseker semistable sheaves of any rank r ≥ 1 on X in terms of those counting sheaves of rank 0 and pure dimension 2.The basic technique is to reduce the ranks of sheaves by replacing them by the cokernels of their Mochizuki/Joyce-Song pairs and then use wall crossing to handle their stability.It has long been speculated that the higher rank or "nonabelian" DT theory of a Calabi-Yau 3-fold might be governed by its rank one "abelian" theory. 1 This would be a 6dimensional analogue of the correspondence between nonabelian Donaldson theory and abelian Seiberg-Witten theory for smooth 4-manifolds. Combined with the MNOP conjecture [MNOP], now proved for most Calabi-Yau 3-folds [PP], it would express DT theory in any rank entirely in terms of Gromov-Witten invariants. Contents 1. Weak stability conditions 4 2. Semistable objects are sheaves 6 3. Destabilising objects are sheaves 13 4. Enumerative invariants and wall crossing formulae 21 5. Crossing the walls 28 Appendix A. Joyce-Song pairs 31 Appendix B. Rank 2 simplifications 33 Appendix C. Moduli stacks 36 References 39Plan. We work on a smooth complex projective threefold X. In Section 1 we review the weak stability conditions and Bogomolov-Gieseker inequality of [BMT, BMS]. Section 2 proves that for such stability conditions, semistable objects E ∈ D(X) of class v n are always sheaves. Moreover in Section 3 we show that their destabilising objects are also sheaves, with one exception -on the Joyce-Song wall, E can be destabilised by an exact triangleexpressing it in terms of a Joyce-Song pair (2) up to tensoring by some T ∈ Pic 0 (X).From Section 4 we assume X is Calabi-Yau. We review the invariants counting its (weak) semistable sheaves and their wall crossing formulae in Section 4. Section 5 gives the proof of Theorem 1. It hides the role of Joyce-Song stable pairs so we explain their relevance in Appendix A, and stronger results in rank 2 in Appendix B. Neither is necessary for the rest of the paper. Appendix C uses work of Piyaratne-Toda to check that moduli stacks of weak semistable objects are algebraic of finite type.Outlook. Up until Section 4 all of our results apply to any 3-fold satisfying the Bogomolov-Gieseker inequality. From Section 4 we restrict attention to Calabi-Yau 3-folds only to apply the wall crossing formula. Joyce is currently developing a a wall crossing formula that should apply to Fano 3-folds [GJT, Conjecture 4.2]; we would then expect our results to prove a version of Theorem 1 with insertions.On Calabi-Yau 3-folds with h 1 (O X ) > 0 the locally free action of Jac(X) on moduli spaces of rank r > 0 sheaves forces J(v) = 0. So from Section 4 we assume h 1 (O X ) = 0 without loss of generality. But it would be interesting to extend our results to counts of sheaves of fixed determinant when h 1 (O X ) > 0; this should not be too hard because in this case the wall cro...

The unreasonable effectiveness of wall-crossing in algebraic geometry

“…Already the second step, tilt-stability, has geometric consequences when combined with Conjecture 4.7. We present three applications: to a bound for the genus of a curve on a threefold [58], to higher rank Donaldson-Thomas theory on Calabi-Yau threefolds [30,31], and to Clifford-type bounds for vector bundles on curves and quintic threefolds [51].…”

“…Let (X, H) be a complex polarized Calabi-Yau threefold. In a recent sequence of papers [30,31], Feyzbakhsh and Thomas proved the following theorem: if (X, H) satisfies the generalized BG inequality on U, then the higher rank Donaldson-Thomas (DT) theory is completely governed by the rank 1 theory, i.e., Hilbert schemes of curves. There is some flexibility on the assumption on the generalized BG inequality; in particular, their theorem holds for the examples of Calabi-Yau threefolds where Conjecture 4.7 has been proved, e.g.…”

Rank $r$ DT theory from rank $1$

“…, the first by ( 7) and the second by [BMS,Corollary 3.10] or [FT3,Lemma 3.2]. We will show this means there are only finitely many points ch 0 (v i ), ch bH 1 (v i ).H 2 , ch bH 2 (v i ).H ∈ Q 3 corresponding to such decompositions.…”

“…• for any sheaves F of rank r = 1 in [FT2], and • for arbitrary sheaves F of any rank r > 1 in [FT3]. As a result we expressed all rank r ≥ 1 DT invariants J(v) in terms of invariants J of ranks ≤ (r − 1).…”

“…By a further wall crossing already carried out in [Br3,To1], stable pairs invariants can be expressed in terms of the rank 1 DT invariants counting ideal sheaves. And a final wall crossing from [FT3,Section 5] expresses the counts J(v) of Gieseker semistable sheaves of rank 0 and dimension 2 in terms of those which are semistable in the large volume chamber.…”