Tangent vectors to Hecke curves on the moduli space of rank 2 bundles over an algebraic curve
Let \(X\) be a non-singular algebraic curve defined over the complex field. Denote by \(M_i\) the moduli space of semistable vector bundles of rank 2 of a fixed determinant of degree \(i=0,1\). This moduli space \(M_i\) is a Fano variety of Picard number 1. In a series of papers N. Mok and the present author have studied this kind of Fano varieties by investigating the projective geometry of the variety of tangent directions of the minimal rational curves. The aim of the paper under review is to apply this study to \(M_i\). In this case the minimal rational curves turn out to be ``Hecke curves'' in the sense of \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. of Math., II. Ser. 101, 391-417 (1975; Zbl 0314.14004)]. Then using the results of \textit{M. S. Narasimhan} and \textit{S. Ramanan} in: ``C. P. Ramanujan. -- A tribute'', Tata Inst. fundam. Res., Stud. Math. 8, 291-345 (1978; Zbl 0427.14002) and the author's previous work [\textit{J.-M. Hwang}, Math. Ann. 312, 599-606 (1998; Zbl 0932.14025)] the author proves the following two main theorems: (1) The tangent bundle to \(M_1\) is stable; (2) Assume \(g\geq 4\) for \(i=0\) and \(g\geq 5\) for \(i=1\). Let \(Y\) be any projective manifold of dimension \(3g-3\) and let \(f:Y \to M_i\) be a surjective holomorphic map. Then \(f\) is locally rigid.
scientific article; zbMATH DE number 1425265