On a lower bound for the dimension of non-abelian theta functions of positive genus

Abstract

In this paper we study the sections of the canonical line bundle on the moduli space of parabolic semistable vector bundles with trivial determinant and fixed parabolic structure of type $\underlineλ=(λ_1,..., λ_s)$ (with each weight $λ_i$ in $P_{\ell}(\SL(r))$) on a smooth projective irreducible curve over $\C$ of genus $g \geq 1$. We give a nontrivial lower bound for the dimension of the sections (that are called generalized parabolic SL(r)-theta functions) when $\sum_{1}^{s} λ_i$ is in the root lattice.