On theta characteristics of a compact Riemann surface (Q996974)

Let \(X\) denote a compact Riemann surface of genus \(g\geq 2\) and let \(K\) denote the holomorphic tangent bundle of \(X\). A theta characteristic on \(X\) is a holomorphic line bundle \(L\) of degree \(g-1\) over \(X\) such that the tensor product \(L\otimes L\) is holomorphically isomorphic to the cotangent bundle \(K\). We call a theta characteristic even if the complex dimension of \(H^{0} (X;L)\) is even. In the paper under review, the authors show that if \(\sigma\) is any non-trivial holomorphic automorphism of \(X\) which fixes pointwise the theta characteristics of \(X\), then \(X\) is a hyperelliptic surface and \(\sigma\) is the unique hyperelliptic involution. The authors then show the stronger result that if \(\sigma\) fixes pointwise all even characteristics, then the same result holds.