Higher rank stable pairs on \(K3\) surfaces (Q2392993)

Let \(X\) be an algebraic complex \(K3\) surface, let \(H\) be an ample divisor and recall that a coherent sheaf \(\mathcal{E}\) on \(X\) is Gieseker stable (resp.\ semistable) if for any subsheaf \(\mathcal{F}\subset \mathcal{E}\) the Hilbert polynomials satisfy \(\chi(\mathcal{F}\otimes H^n)<\chi(\mathcal{E}\otimes H^n)\) (resp.\ \(\leq\)) for \(n>0\). Furthermore, we say that a divisor \(D\) is of minimal degree if \(D.H=\text{min}\left\{L.H\;|\;L\in \text{Pic}(X), L.H>0\right\}\). Given a Mukai vector \(v=(r,D,a)\in H^*(X,\mathbb{Z})\) with, for simplicity, \(D\) of minimal degree, we can consider the moduli space \(M(v)\) of (semi-)stable sheaves \(\mathcal{E}\) satisfying \(v(\mathcal{E}):=\text{ch}(\mathcal{E})\sqrt{\text{td}(X)}=v\), which, for generic \(H\), is known to be a smooth projective irreducible symplectic variety deformation equivalent to a Hilbert scheme of points on \(X\). Next, recall that a stable pair \((U,\mathcal{E})\) on \(X\) consists of a stable sheaf \(\mathcal{E}\) and a subspace \(U\subset \text{Hom}(\mathcal{O},\mathcal{E})\), and note that we can write a stable pair as the evalutaion morphism \(U\otimes \mathcal{O}\to \mathcal{E}\). A morphism of stable pairs is defined in the obvious way, and there is also a relative notion of stable pair. It is known that the moduli space of stable pairs with \(v(\mathcal{E})=v\) and \(\dim U=n\) is (coarsely) representable by a projective scheme \(\text{Syst}^n(v)\). The authors view the spaces \(\text{Syst}^n(r,D,k)\) for \(n>1\) or \(r>0\) as moduli spaces of higher stable pairs, which are the higher rank analogues of Pandharipande--Thomas stable pairs originally defined on threefolds in [Invent.\ Math.\ 178, No.\ 2, 407--447 (2009, Zbl 1204.14026)]. The first main theorem of the paper gives an explicit formula for \(\frac{F^r_n(q,y)}{S(q)}\), where \(F^r_n(q,y)\) is the generating function of the Hodge polynomials of the moduli spaces of stable pairs and \(S(q)\) is the generating function of the Hodge polynomials of the Hilbert schemes of \(n\) points on \(X\). The second main result shows that the \(F^r_n(q,y)\) are governed by modular forms. In the first section the authors give a description of the main results and some background information about stable pair invariants on threefolds and about previous work on stable pairs on \(K3\) surfaces. Section 2 recalls some facts about moduli theory of sheaves and stable pairs on \(K3\) surfaces. In particular, there is a stratification of \(M(v)\) which via the forgetful morphism \(p: \text{Syst}^n(v)\to M(v)\) induces a stratification on \(\text{Syst}^n(v)\). Furthermore, \(p\) is an étale-locally trivial fibration whose fibres are Grassmannians. The main results are proved in Section 3. The main ingredients to establish the first statement are the compatibility of the Hodge polynomials with stratifications, the fact that in our situation they are also compatible with the fibration structure, and the deformation equivalence of \(M(v)\) to some Hilbert scheme of points. Finally, in Section 4 the authors present some computations used in Section 3.