A generalized Quot scheme and meromorphic vortices

TL;DR

This paper constructs a natural compactification of a moduli space of pairs consisting of holomorphic vector bundles and meromorphic homomorphisms on a Riemann surface, and computes its Poincaré polynomial.

Contribution

It introduces a generalized Quot scheme for meromorphic vortices and provides an explicit computation of the Poincaré polynomial for its compactification.

Findings

Constructed a natural compactification of the moduli space.

Computed the Poincaré polynomial of the compactified space.

Extended the Quot scheme framework to meromorphic vortices.

Abstract

Let X be a compact connected Riemann surface. Fix a positive integer r and two nonnegative integers d_p and d_z. Consider all pairs of the form (F, f), where F is a holomorphic vector bundle on X of rank r and degree d_z-d_p, and f : {\mathcal O}^{\oplus r}_X \rightarrow F is a meromorphic homomorphism which an isomorphism outside a finite subset of X and has pole (respectively, zero) of total degree d_p (respectively, d_z). Two such pairs $(F_1, f_1) and $(F_2, f_2) are called isomorphic if there is a holomorphic isomorphism of F_1 with F_2 over X that takes f_1 to f_2. We construct a natural compactification of the moduli space equivalence classes pairs of the above type. The Poincar\'e polynomial of this compactification is computed.