On a Lower Bound for the Dimension of Non–abelian Theta Functions of Positive Genus

Let Cg be a smooth projective irreducible curve over C of genus g ≥ 1 and let {p1, p2, . . . , ps} be a set of distinct points on Cg. We fix a nonnegative integer l and denote by Mg(p, λ) the moduli space of parabolic semistable vector bundles of rank r with trivial determinant and fixed parabolic structure of type λ = (λ1, λ2, . . . , λs) at p = (p1, p2, . . . , ps), where each weight λi is in Pl(SL(r)). On Mg(p, λ) there is a canonical line bundle L(λ, l), whose sections are called generalized parabolic SL(r)-theta functions of order l. The main result of this paper is: if Ps 1 λi is in the root lattice, then dimH0(Mg(p, λ),L(λ, l)) ≥ (♯Pl(SL(r))) , where ♯Pl(SL(r)) = (l + r − 1)!/(l!(r − 1)!). Such nontrivial lower bounds for the number of generalized parabolic theta functions were not previously known.