Maximal Subbundles and Gromov-Witten Invariants

Abstract

Let $C$ be a nonsingular irreducible projective curve of genus $g\ge 2$ defined over the complex numbers. Suppose that $1\le n^{\prime }\le n-1$ and $n^{\prime }d-n{d}^{\prime }=n^{\prime }(n-n^{\prime })(g-1)$. It is known that, for the general vector bundle $E$ of rank $n$ and degree $d$, the maximal degree of a subbundle of $E$ of rank $n^{\prime }$ is $d^{\prime }$ and that there are finitely many such subbundles. We obtain a formula for the number of these maximal subbundles when $(n^{\prime },d^{\prime })=1$. For $g=2$, $n^{\prime }=2$, we evaluate this formula explicitly. The numbers computed here are Gromov-Witten invariants in the sense of a recent paper of Ch. Okonek and A. Teleman (to appear in Commun. Math. Phys.) and our results answer a question raised in that paper. In this revised version some references are added.

11 pages