A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve (Q2874723)

Let \(X\) be a smooth projective algebraic curve over \({\mathbb C}\) of genus \(g\geq2\) and let \(V\) be a vector bundle on \(X\) of rank \(2\). The Segre invariant \(s(V)\) of \(V\) is defined by \(s(V):=\min\{\deg V-2\deg L\}\), the minimum being taken over all line subbundles \(L\) of \(V\). This invariant provides a stratification of the moduli spaces of semistable vector bundles on \(X\), which was investigated by \textit{H. Lange} and \textit{M. S. Narasimhan} [Math. Ann. 266, 55--72 (1983; Zbl 0507.14005)]. A generalisation to higher rank was described by \textit{L. Brambila-Paz} and \textit{H. Lange} [J. Reine Angew. Math. 494, 173--187 (1998; Zbl 0919.14016)] and by \textit{B. Russo} and \textit{M. Teixidor i Bigas} [J. Algebr. Geom. 8, No. 3, 483--496 (1999; Zbl 0942.14013)].NEWLINENEWLINEIn the present paper, the authors consider an analogue of the Segre invariant for symplectic and orthogonal bundles \(V\) of rank \(2n\). The invariant is defined as \(t(V):= -2\max\{\deg E: E\text{ a rank } n \text{ isotropic subbundle of }V\}\). To state the results, let \(\mathrm{SU}_X(2n,{\mathcal O}_X)\) denote the moduli space of semistable bundles on \(X\) with trivial determinant and write \({\mathcal M}S_X(2n)\) (respectively, \({\mathcal M}O_X(2n)\)) for the sublocus of \(\mathrm{SU}_X(2n,{\mathcal O}_X)\) of bundles admitting a symplectic (respectively, orthogonal) structure. (In fact \({\mathcal M}S_X(2n)\) is isomorphic to the moduli space of semistable symplectic bundles, but the corresponding result does not hold in the orthogonal case.) In the orthogonal case, when \(n\geq2\), there are two components, distinguished by the value of the second Stiefel-Whitney class, which are denoted by \({\mathcal M}O_X(2n)^\pm\). The invariant \(t\) induces stratifications of these spaces and we write \({\mathcal M}S_X(2n,t):=\{V\in{\mathcal M}S_X(2n,t): t(V)=t)\}\) with similar notations in the orthogonal case.NEWLINENEWLINEIn the symplectic case, the main result (Theorem 1.1) states first that \(t(V)\leq n(g-1)+1\), while, for \(V\) general, we have also \(t(V)\geq n(g-1)\). Moreover, for each even integer \(t\) with \(2\leq t\leq n(g-1)\), the corresponding stratum \({\mathcal M}S_X(2n,t)\) is non-empty and irreducible of dimension \(\frac12(n(3n+1)(g-1)+(n+1)t)\) and, for \(t<n(g-1)\), the space \(M(V)\) of maximal isotropic subbundles \(E\) of \(V\) with \(-2\deg E=t\) is a single point for general \(V\in{\mathcal M}S_X(2n,t)\). On the other hand, for a general \(V\in{\mathcal M}S_X(2n)\), \(\dim M(V)=0\) when \(n(g-1)\) is even, while \(\dim M(V)=\frac{n+1}2\) when \(n(g-1)\) is odd.NEWLINENEWLINEIn the orthogonal case, the authors first prove that a semistable orthogonal bundle \(V\) belongs to \({\mathcal M}O_X(2n)^+\) (respectively, \({\mathcal M}O_X(2n)^-\)) if and if its maximal isotropic subbundles have even (respectively, odd) degree. Then (Theorem 1.3), for \(n\geq2\), we have \(t(V)\leq n(g-1)+3\), while, for \(V\) general, we have also \(t(V)\geq n(g-1)\). Moreover, for each even integer \(t\) with \(2\leq t\leq n(g-1)\), the corresponding stratum \({\mathcal M}O_X(2n,t)\) is non-empty and irreducible of dimension \(\frac12(n(3n-1)(g-1)+(n-1)t)\) and, for \(t<n(g-1)\), the space \(M(V)\) of maximal isotropic subbundles \(E\) of \(V\) with \(-2\deg E=t\) is a single point for general \(V\in{\mathcal M}O_X(2n,t)\). The authors also compute \(\dim M(V)\) for general \(V\in{\mathcal M}O_X(2n)^\pm\) and give a more detailed description of the top strata.NEWLINENEWLINEThe methods are similar to those of Lange and Narasimhan.