A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve

Insong Choe, G. H. Hitching
2014 International Journal of Mathematics  
A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V ) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on t(V ), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for
more » ... ctor bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case. Note that this is a priori weaker than the stability condition for V as a vector bundle; compare with Ramanathan [18] . However, Ramanan [17] proved that semistability as an orthogonal bundle is equivalent to semistability of the underlying vector bundle, and moreover that a general stable orthogonal bundle is a stable vector bundle. The same argument (worked through in [5] ) shows that the analogous statement is true for symplectic bundles. We denote by SU X (2n, O X ) the moduli space of semistable vector bundles of rank 2n and trivial determinant, and write MS X (2n) (resp., MO X (2n)) for the sublocus in SU X (2n, O X ) of bundles admitting a symplectic (resp., orthogonal) structure. In the symplectic case it has been proven by Serman [20] that the forgetful map associated to the extension of the structure group Sp 2n C ⊂ SL 2n C, is an embedding, where M X (Sp 2n C) is the moduli space of semistable principal Sp 2n C-bundles over X. So MS X (2n) coincides with the embedded image of M X (Sp 2n C). The orthogonal case is more delicate. By [20] , the forgetful map is generically two-to-one, amounting to forgetting the data of an orientation on a principal SO 2n C-bundle. On the other hand, the map is an embedding. The moduli space M X (O 2n C) of semistable principal O 2n C-bundles over X has several components, which are indexed by the first and second Stiefel-Whitney classes (w 1 , w 2 ) ∈ H 1 (X, Z 2 ) × H 2 (X, Z 2 ). The class w 1 corresponds to the determinant, and there are two components of M X (O 2n C) with w 1 trivial. We write MO X (2n) ± for the embedded images of these components in SU X (2n, O X ).
doi:10.1142/s0129167x14500475 fatcat:6e3buai4sfhbxkamscxllqaify