Counting maximal Lagrangian subbundles over an algebraic curve

Abstract

Let $C$ be a smooth projective curve and $W$ a symplectic bundle over $C$. Let $L{Q}_e(W)$ be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves $E\subset W$ of degree $e$. We give a closed formula for intersection numbers on $L{Q}_e(W)$. As a special case, for $g\ge 2$, we compute the number of Lagrangian subbundles of maximal degree of a general stable symplectic bundle, when this is finite. This is a symplectic analogue of Holla's enumeration of maximal subbundles in [13].

27 pages