Sanghyeon Lee, Vafa-Witten invariants on elliptic surfaces and moduli space of quasisections

For a threefold X = C x S, D. Nesterov developed a theory of quasimaps from C to the moduli space of sheaves over S. He compared the moduli space of quasimaps and the moduli space of sheaves over the threefold X and also compared their obstruction theories.

When X is a smooth fibration over C, I will explain how to generalize Nesterov’s theory by using a notion of quasisections, instead of quasimap, and the correspondence between moduli space of sheaves over X and the moduli space of quasisections to the relative moduli space of sheaves. As an application, I will explain how to compute Vafa-Witten invariants of an elliptic surface, which is a smooth fibration over C.

Moreover, for the case when the elliptic surface has nodal fibers, we will discuss how we can generalize the theory of quasisections. This is based on an ongoing project with Yaoxiong Wen.