Curve counting invariants for crepant resolutions (Q2787950)

The paper under review constructs new curve counting invariants for a nonsingular projective Calabi-Yau threefold \(Y\) with \(H^1(\mathcal{O}_Y)=0\), equipped with a birational morphism \(\pi : Y \to X\), where \(X\) is a projective Calabi-Yau threefold that is the coarse space of a smooth orbifold CY3 \( \mathcal{X}\) that satisfies the hard Lefschetz condition. When \(\pi\) is identity these invariants specialize to Pandharipande and Thomas (PT) invariants. The main example is when \(Y\) is the crepant resolution of \(X\). For simplicity, in this review we assume that we are in this situation, and that \(Y\) is the distinguished crepant resolution of \(X\) given by \(Y = \text{Hilb}^{[\mathcal{O}_p]}(X)\), the Hilbert scheme parametrizing substacks in the \(K\)-group class \([\mathcal{O}_p] \in K(\mathcal{X})\), for a generic point (with trivial stabilizer) \(p\in \mathcal{X}\). The hard Lefschetz condition implies that the resolution is semi-small (i.e. the fibres of \(\pi\) are zero- or one-dimensional) and that the singular locus of \(X\) is one-dimensional. In this case, it is known that there is a Fourier-Mukai isomorphism (in the level of K-groups) \(\Psi : K(Y )\to K(\mathcal{X})\), with the inverse is denoted by \(\Phi\).NEWLINENEWLINEThe hard Lefschetz condition implies that \(F_{\mathrm{exc}}K(Y ):=\Phi(F_0 K(\mathcal{X})) \subseteq F_1 K(Y)\) (\(F_i\) is the usual filtration of the \(K\)-group by the \(i\)-dimensional sheaves); its elements can be represented by formal differences of sheaves supported on the exceptional fibres of \(\pi\). The multi-regular part of \(K\)-theory, \(F_{\text{mr}}(\mathcal{X})\), is defined to be the preimage of \(F_1 K(Y)\) under \(\Psi\); its elements can be represented by formal differences of sheaves supported in dimension one where at the generic point of each curve in the support, the associated representation of the stabilizer group of that point is a multiple of the regular representation. One can then define the corresponding Donaldson-Thomas generating series: NEWLINE\[NEWLINEDT_{\mathrm{exc}}(Y ) =\sum_{\alpha \in F_{\mathrm{exc}}K(Y )} DT^\alpha(Y)q^\alpha,NEWLINE\]NEWLINE NEWLINE\[NEWLINEDT_{\text{mr}}(\mathcal{X} ) =\sum_{\alpha \in F_{\text{mr}}K(\mathcal{X} )} DT^\alpha(\mathcal{X})q^\alpha,NEWLINE\]NEWLINE NEWLINE\[NEWLINEDT_{0}(\mathcal{X} ) =\sum_{\alpha \in F_{0}K(\mathcal{X})} DT^\alpha(\mathcal{X})q^\alpha.NEWLINE\]NEWLINENEWLINENEWLINEBryan-Cadman-Young [\textit{J. Bryan} et al., Adv. Math. 229, No. 1, 531--595 (2012; Zbl 1250.14027)] conjectured the following relation among these generating series (crepant resolution conjecture): NEWLINE\[NEWLINEDT_{\text{mr}}(\mathcal{X})/DT_{0}(\mathcal{X} ) = DT(Y)/DT_{\mathrm{exc}}(Y).NEWLINE\]NEWLINENEWLINENEWLINEThe main object of study in the paper under review is called a \(\pi\)-stable pair defined as follows. A pair of a coherent sheaf \(G\) and a section \(s:\mathcal{O}\to G\) is called \(\pi\)-stable, if \(R\pi_* \text{coker}(s)\) is a \(0\)-dimensional sheaf, and \(\text{Hom}(P,G)=0\) for any coherent sheaf \(P\) with \(R\pi_* P\) is a 0-dimensional sheaf. It is proven in this paper that there is a finite-type constructible space, \(\pi\)-\(\text{Hilb}^\alpha\) parametrizing these objects with \([G] = \alpha \in K(Y )\). The invariant \(\pi\)-\(PT^\alpha(Y)\) is defined by taking the weighted Euler characteristic of \(\pi\)-\(\text{Hilb}^\alpha\). Note that if \(\pi\) is the identity then \(\pi\)-\(PT^\alpha(Y)=PT^\alpha(Y)\). Let \(\pi\)-\(PT(Y)\) be the generating function of these invariants. The main result of the paper under review is proving the formula \(\pi\)-\(PT(Y)=DT(Y)/DT_{\mathrm{exc}}(Y)\).NEWLINENEWLINEThis formula can be seen as a step towards proving the crepant resolution conjecture above. To prove the conjecture one needs to prove \(\pi\)-\(PT(Y)=PT(\mathcal{X})\) and then use PT/DT correspondence on \(\mathcal{X}\) claimed by Bayer.