A theory of generalized Donaldson-Thomas invariants (Q2882487)

Let \(X\) be a Calabi--Yau 3-fold, which for us will mean a smooth projective 3-fold \(X\), for simplicity over the field of complex numbers, with trivial canonical bundle and such that the first cohomology group of the structure sheaf vanishes. Given a very ample line bundle \(\mathcal{O}_X(1)\) and a class \(\alpha\) in the numerical Grothendieck group \(K^{\text{num}}(X)\) of \(X\), one can consider the moduli space \(M_{\text{ss}}^{\alpha}(\tau)\) of all Gieseker semistable (with respect to \(\mathcal{O}_X(1)\)) sheaves with class \(\alpha\) and its open subscheme of stable sheaves \(M_{\text{st}}^{\alpha}(\tau)\). Assuming that there are no strictly semistable sheaves, \textit{R. P. Thomas} [J.\ Differ.\ Geom.\ 54, No.\ 2, 367--438 (2000; Zbl 1034.14015)] constructed a perfect symmetric obstruction theory on \(M_{\text{ss}}^{\alpha}(\tau)=M_{\text{st}}^{\alpha(\tau)}\). This can be used to define a virtual fundamental class which is a \(0\)-cycle, and integrating over it gives the Donaldson--Thomas invariant \(DT^{\alpha}(\tau)\), which is an integer, ``counts'' \(\tau\)-stable sheaves of class \(\alpha\) and is invariant under deformations. Later \textit{K. Behrend} [Ann.\ Math.\ (2) 170, No.\ 3, 1307--1338 (2009; Zbl 1191.14050)] proved that any finite type scheme \(Y\) over \(\mathbb{C}\) has a certain constructible function \(\nu_Y\), called Behrend function, and that \(DT^{\alpha}(\tau)\) coincides with the Euler characteristic of \(M_{\text{ss}}^{\alpha}(\tau)\) weighted by \(\nu_{M_{\text{st}}^{\alpha}(\tau)}\).NEWLINENEWLINEThe book under review seeks to answer two questions. Firstly, how does one generalize the above to the case when strictly semistable sheaves do exist, and secondly, how do these invariants change when the line bundle or, more generally, a stability condition \(\tau\) on coherent sheaves is varied? Roughly speaking, the result is that there exist generalized Donaldson--Thomas invariants \(\overline{DT}^{\alpha}(\tau)\) which ``count'' both \(\tau\)-stable and \(\tau\)-semistable sheaves with class \(\alpha\). These, in general rational, numbers are unchanged under deformations of \(X\), transform by a wall-crossing formula when the stability condition is varied and coincide with the usual DT-invariants when no strictly semistable sheaves exist.NEWLINENEWLINEThe approach, in particular, draws ideas from a series of papers by the first author where certain counting invariants were defined for any class \(\alpha\). Roughly, the setting is the following. Consider \(\mathcal{M}\), the Artin stack of all coherent sheaves on \(X\). One can define a certain Lie algebra \(\text{SF}^{\text{ind}}_{\text{al}}(\mathcal{M})\) of stack functions supported on virtual indecomposables and a Lie algebra morphism \(\Psi\) from this algebra to an infinite-dimensional Lie algebra \(L(X)\). The latter is the \(\mathbb{Q}\)-vector space with basis of symbols \(\lambda^{\alpha}\) for \(\alpha\) in \(K^{\text{num}}(X)\) and Lie bracket given by \([\lambda^{\alpha},\lambda^{\beta}]=\overline{\chi}(\alpha,\beta)\lambda^{\alpha+\beta}\), where \(\overline{\chi}\) is the Euler form. Roughly, \(\Psi\) of a function \(f\) is defined by taking the Euler characteristic of the restriction of \(f\) to the stack of sheaves of class \(\alpha\in K^{\text{num}}(X)\) and then summing over all \(\alpha\). Given a stability condition \(\tau\) on coherent sheaves, for example Gieseker stability, there are certain elements \(\overline{\epsilon}^{\alpha}(\tau)\) in the algebra \(\text{SF}^{\text{ind}}_{\text{al}}(\mathcal{M})\) which are built from characteristic functions of the stacks of \(\tau\)-semistable sheaves, and one defines a rational number \(J^{\alpha}(\tau)\) as the coefficient of \(\lambda^{\alpha}\) in \(\Psi(\overline{\epsilon}^{\alpha}(\tau))\). Furthermore, given two stability conditions \(\tau\) and \(\tau'\), one can write \(\overline{\epsilon}^{\alpha}(\tau)\) in terms of an explicit formula involving the \(\overline{\epsilon}^{\beta}(\tau')\) for \(\beta \in K^{\text{num}}(X)\) and the Lie bracket in \(\text{SF}^{\text{ind}}_{\text{al}}(\mathcal{M})\). Since \(\Psi\) is a Lie algebra morphism, the invariants \(J^{\alpha}(\tau)\) and \(J^{\alpha}(\tau')\) also satisfy a transformation law. Furthermore, if there are no strictly semistable sheaves, \(J^{\alpha}(\tau)\) is just the Euler characteristic of the moduli space \(M_{\text{st}}^{\alpha}(\tau)\). The basic idea behind the book is that the Behrend function should be inserted into the just described programme.NEWLINENEWLINEOf course, several things have to be changed for this to work. For instance, instead of \(L(X)\) one considers an algebra \(\tilde{L}(X)\) with basis \(\tilde{\lambda}^{\alpha}\) and bracket given as above but with a sign change by \((-1)^{\overline{\chi}(\alpha,\beta)}\). Next, the map \(\Psi\) is changed to \(\tilde{\Psi}\) by using the weighted Euler characteristic. The proof that this new map from \(\text{SF}^{\text{ind}}_{\text{al}}(\mathcal{M})\) to \(\tilde{L}(X)\) is a Lie algebra morphism uses the fact that \(\Psi\) is one and certain Behrend function identities which need to be established separately. Proving these identities requires a good understanding of the local structure of the moduli stack \(\mathcal{M}\). Roughly speaking, an atlas for this stack may be locally written as the critical locus of a holomorphic function \(f: U\to \mathbb{C}\), where \(U\) is smooth. The advantage of this description lies in the fact that if \(Y\) is the critical locus, then the Behrend function \(\nu_Y(y)\) can be expressed in terms of the Euler characteristic of the Milnor fibre of \(f\) at \(y \in Y\). Having established all of this, the invariant \(\overline{DT}^{\alpha}(\tau)\) is defined as the coefficient of \(\tilde{\lambda}^{\alpha}\) in \(\tilde{\Psi}(\overline{\epsilon}^{\alpha}(\tau)) \in \tilde{L}(X)\). Similarly to the above, the \(\overline{DT}^{\alpha}(\tau)\) also obey a certain transformation law when the stability condition is varied.NEWLINENEWLINETo prove that the generalized DT-invariants are unchanged under deformations, the authors introduce another family of invariants \(PI^{\alpha,n}(\tau)\) (denote the corresponding moduli space by \(M^{\alpha,n}_{\text{stp}}(\tau)\)) which are similar to the above mentioned PT-invariants. These can be checked to be unchanged under deformations. Furthermore, these invariants can be written in terms of the \(\overline{DT}^{\alpha}(\tau)\) and hence the latter are also deformation invariant.NEWLINENEWLINEThe book is organized as follows. Chapter 1 gives a concise introduction into the subject. Chapters 2--4 gather background information about constructible and stack functions, the above mentioned programme by the first author, and Behrend functions and DT theory, respectively. The main results are stated in Chapter 5 and proved in Chapters 8--13. In Chapter 6 several examples are presented. Chapter 7 deals with DT-theory of quivers with superpotentials. Namely, instead of coherent sheaves on \(X\) one can consider the abelian category \(\text{mod}\)-\(\mathbb{C}Q/I\) of representations of a quiver \(Q\) with relations \(I\) defined by a superpotential \(W\). Since the moduli stack of objects in \(\text{mod}\)-\(\mathbb{C}Q/I\) can again be locally written as the critical locus of a holomorphic function and the corresponding bilinear form in the quiver setting satisfies some nice properties, most of the above results can be extended to this situation. Note however, that the analogues of the moduli schemes \(M_{\text{ss}}^{\alpha}(\tau)\) and \(M^{\alpha,n}_{\text{stp}}(\tau)\) in the quiver setting are not proper in general. Therefore, the corresponding quiver analogues of \(\overline{DT}^{\alpha}(\tau)\) and \(PI^{\alpha,n}(\tau)\) will in general not be invariant under deformations of the superpotential \(W\). Interestingly, the latter invariants are known in the literature as non-commutative DT-invariants while the former are new.NEWLINENEWLINEFinally, it should be noted that a paper of \textit{M. Kontsevich} and \textit{Y. Soibelman} [``Stability structures, motivic Donaldson-Thomas invariants and cluster transformations'', Preprint, \url{arxiv:0811.2435}] has some overlaps with this book. See Chapter 1.6 for information concerning this.