Lagrangian subvarieties of the moduli space of stable vector bundles on a regular algebraic surface withp g >0

Abstract

Mukai [M1] showed that there is a nondegenerate symplectic structures on the moduli space of stable vector bundles on a K3 surface. Later Tyurin [T2] studied (generalized) symplectic structures on the moduli space of stable vector bundles on any smooth regular surface X with P9 > 0. In the work of Tyurin, a symplectic structure means a nonzero regular two form on the moduli spaces, in particular it may degenerate. In this paper, we define a Lagrangian subvariety of the moduli space to be a subvariety on the Zariski tangent space (at any point) of which the given symplectic two form is identically zero. Note that we do not impose any restriction on the dimension of a Lagrangian subvariety. This is because symplectic structures considered here may degenerate. The purpose of this paper is to use Bril l-Noether theory for curves to construct explicit ly a family of Lagrangian subvarieties of the moduli space of stable vector bundles on a regular surface with pg > 0 . Le t . P/~ be a generically smooth and irreducible component of the moduli space of Gieseker-stable bundles of rank r + 1 with respect to a fixed polarization D on a regular algebraic surface X with p~ > 0. By the boundedness of ..Jf/~ (see [Ma]), after possibly twisted by the same negative line bundle . ~ on X, we can assume that for any point [E] C ,//J~, (i) E * is generated by global sections. (ii) hl(E *) = hZ(E *) : hi(E) : O. (iii) h l (de t E *) = hZ(det E *) = 0. For any point [E] c .//Z. Choose a (r + 1) dimensional subspace V C H~ and consider an evaluation map e v : V | P x --~ E* . For a general V, we can make e v degenerate exactly along a smooth curve C C X and coker e v is a line bundle