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Stable pairs on local \(K3\) surfaces. (English) Zbl 1260.14045
Let \(S\) be a smooth projective complex \(K3\) surface and \(X\) the total space of its canonical bundle, that is, \(X=S\times \mathbb{C}\). A stable pair on \(X\) is a pair \((F,s)\), where \(F\) is a pure one-dimensional coherent sheaf supported on the fibres of the second projection and \(s: \mathcal{O}_X\to F\) is a map with \(0\)-dimensional cokernel. The moduli space \(P_n(X,\beta)\) of pairs satisfying \(\chi(F)=n\) and \([F]=\beta \in H_2(X,\mathbb{Z})\) is a quasi-projective scheme and one considers the associated PT-invariant, which is its Euler characteristic and hence an integer. Furthermore, given an ample divisor \(\omega\) on \(S\), one can consider the moduli stack \(\mathcal{M}_\omega(r,\beta,n)\) of \(\omega\)-Gieseker semistable sheaves on \(X\) with Mukai vector \(v=(r,\beta,n)\in \mathbb{Z}\oplus H^2(S,\mathbb{Z})\oplus\mathbb{Z}\) which are supported on the fibres of the second projection, and its “Euler characteristic” \(J(r,\beta,n)\), which is in fact independent of \(\omega\). Note that this is the usual Euler characteristic of \(\mathcal{M}_\omega(r,\beta,n)\) if \(v\) is a primitive vector and in this case it is in fact the Euler characteristic of a Hilbert scheme of points on \(S\). However, in the general setting Joyce’s theory on counting invariants is needed to define this number, which is then not necessarily an integer. The main result of the paper under review states that the generating function of PT-invariants \(\text{PT}^\chi(X)=\sum_{\beta,n}\chi(P_n(X,\beta))y^\beta z^n\) has a product expansion involving the invariants \(J(r,\beta,r+n)\). Another result is that \(J(v)=J(gv)\) for any Hodge isometry \(g\) of the lattice \(\mathbb{Z}\oplus H^2(S,\mathbb{Z})\oplus \mathbb{Z}\) endowed with the Mukai pairing.
Very roughly, the idea of the proof is as follows. First, compactify \(X\) to \(\overline{X}=S\times \mathbb{P}^1\) and prove that \(\text{PT}^\chi(\overline{X})=\text{PT}^\chi(X)^2\). On \(\overline{X}\) one can use Joyce’s wall-crossing formula to write the series \(\text{PT}^\chi(\overline{X})\) as a product involving certain invariants \(N(r,\beta,n)=2J(r,\beta,r+n)\) and yet another invariants \(L(\beta,n)\). The latter ones count certain objects in \(D^b(\overline{X})\), the bounded derived category of coherent sheaves on \(\overline{X}\). To further investigate these invariants, one studies a certain triangulated subcategory \(\mathcal{D}\) of \(D^b(\overline{X})\) and weak stability conditions \(\sigma_t=(Z_{t\omega},\mathcal{A}_\omega)\), \(t\in \mathbb{R}_{>0}\), on it. This allows to construct an Euler characteristic version of a Donaldson–Thomas type invariant of \(Z_{t\omega}\)-semistable objects in \(\mathcal{A}_\omega\) satisfying some properties and study the generating series \(\text{DT}^\chi_{t\omega}(\overline{X})\) of these invariants. The author investigates the behaviour of this series for big and small \(t\) and the wall-crossing formula shows how these two expressions are related. This is still not quite enough to show the main result, so one has to consider weak stability conditions on an abelian subcategory of \(\mathcal{A}_\omega\) and use the wall-crossing formula again.
One of the motivations for the main result is a conjecture by Katz–Klemm–Vafa (KKV) which arises from physical considerations and mathematically is formulated in terms of generating series of reduced Gromov–Witten invariants. D. Maulik, R. Pandharipande and R. P. Thomas [J. Topol. 3, No. 4, 937–996 (2010; Zbl 1207.14058)] proved this conjecture for primitive classes \(\beta\). An important ingredient of their proof is a formula for reduced PT-invariants proved by T. Kawai and K. Yoshioka [Adv. Theor. Math. Phys. 4, No. 2, 397–485 (2000; Zbl 1013.81043)] and the main result is an attempt to generalize this formula to arbitrary curve classes and hence make a first step towards proving the KKV-conjecture in general.

MSC:
14J28 \(K3\) surfaces and Enriques surfaces
14D20 Algebraic moduli problems, moduli of vector bundles
14J33 Mirror symmetry (algebro-geometric aspects)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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