Higher rank flag sheaves on Surfaces and Vafa-Witten invariants

Abstract

We study moduli space of holomorphic triples $E_{1}\xrightarrow{\phi} E_{2}$, composed of torsion-free sheaves $E_{i}, i=1,2$ and a holomorphic mophism between them, over a smooth complex projective surface $S$. The triples are equipped with Schmitt stability condition [Alg Rep Th. 6. 1. pp 1-32, 2003]. We observe that when Schmitt stability parameter $q(m)$ becomes sufficiently large, the moduli space of triples benefits from having a perfect relative and absolute deformation-obstruction theory in some cases. We further generalize our construction by gluing triple moduli spaces, and extend the earlier work of Gholampour-Sheshmani-Yau [arXiv:1701.08899] where the obstruction theory of nested Hilbert schemes over the surface was studied. Here we extend the earlier results to the moduli space of chains$$E_{1}\xrightarrow{\phi_{1}} E_{2}\xrightarrow{\phi_{2}} \cdots \xrightarrow{\phi_{n-1}} E_{n},$$ where $\phi_{i}$ are injective morphisms and $rk(E_{i})\geq 1, \forall i$. There is a connection, by wallcrossing in the master space in the sense of Mochizuki, between the theory of such higher rank flags, and the theory of Higgs pairs on the surface, which provides the means to relate the flag invariants to the local DT invariants of threefold given by a line bundle on the surface, $X :=Tot(\mathcal{L} \to S)$. The latter, when $\mathcal{L}=\omega_{S}$, provides the means to compute the contribution of higher rank flag sheaves to partition function of Vafa-Witten theory on $X$.