Counting maximal Lagrangian subbundles over an algebraic curve

Abstract

Let C be a smooth projective curve and W a symplectic bundle over C. Let LQe(W) be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves E⊂W of degree e. We give a closed formula for intersection numbers on LQe(W). As a special case, for g≥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 [14].