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# Advances in Theoretical and Mathematical Physics

## Volume 19 (2015)

### Number 4

### A generalized Quot scheme and meromorphic vortices

Pages: 905 – 921

DOI: http://dx.doi.org/10.4310/ATMP.2015.v19.n4.a6

#### Authors

#### 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\[f : \mathcal{O}^{\oplus r}_{X} \longrightarrow \mathcal{F}\]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 $(\mathcal{F}_1, f_1)$ and $(\mathcal{F}_2, f_2)$ are called isomorphic if there is a holomorphic isomorphism of $\mathcal{F}_1)$ with $\mathcal{F}_2$ 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 Poincaré polynomial of this compactification is computed.