Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Non-abelian theta functions of positive genus

Let $C_g$ be a smooth projective irreducible curve (over complex numbers) of genus $g \geq 1$ and let $\{p_1,\dots, p_s\}$ be a set of distinct points on $C_g$. We fix a nonnegative integer $\ell$ and denote by $M_g(\underline{p},\underline{\lambda})$ the moduli space of parabolic semistable vector bundles of rank $r$ on $C_g$ with trivial determinant and fixed parabolic structure of type $\underline{\lambda}=(\lambda_1,\dots, \lambda_s)$ at $\underline{p}=(p_1,\dots, p_s)$, where each weight $\lambda_i$ is in $P_{\ell}(\SL(r))$. On $M_g(\underline{p},\underline{\lambda})$ there is a canonical line bundle $L(\underline{\lambda}, \ell)$, whose sections are called generalized parabolic $\SL(r)$-theta functions of order $\ell$. In this paper we prove the existence of such non-abelian theta functions.