Automorphisms of moduli spaces of symplectic bundles (Q2885372)

Let \(X\) be irreducible smooth projective curve of genus \(g\geq 4\), \(L\) be a holomorphic line bundle on \(X\). Let \(M_{\mathrm{Sp}}(L)\) be the moduli space of semistable symplectic bundles \((E, \varphi: E\otimes E \to L)\) on \(X\), with the symplectic form taking values in \(L\). Authors show that the automorphism group of \(M_{\mathrm{Sp}}(L)\) is generated by automorphisms of the form \(E \mapsto E\otimes M\) for line bundle \(M\) such that \(M^{\otimes 2}\cong \mathcal{O}_X\), together with automorphisms of \(X\). The automorphism group \(\mathrm{Aut}(M_{\mathrm{Sp}}(L))\) includes in a short exact sequence of groups \(e\to J(X)_2 \to \mathrm{Aut}(M_{\mathrm{Sp}}(L)) \to \mathrm{Aut}(X) \to e\) where \(J(X)_2\) is the group of line bundles of order 2 on \(X\). Also a Torelli type theorem for \(M_{\mathrm{Sp}}(L)\) is proven.