Tangent vectors to Hecke curves on the moduli space of rank 2 bundles over an algebraic curve
type: Item
instance of: scholarly article
title: Tangent vectors to Hecke curves on the moduli space of rank 2 bundles over an algebraic curve
zbMATH Open document ID: 0988.14013
DOI: 10.1215/S0012-7094-00-10117-2
author: Jun-Muk Hwang
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 1425265
zbMATH Keywords: tangent vectors to Hecke curves
zbMATH Keywords: moduli space
zbMATH Keywords: semistable vector bundles of rank 2
zbMATH Keywords: Fano variety
zbMATH Keywords: minimal rational curves
zbMATH Keywords: tangent bundle
reviewed by: A. Del Centina
MaRDI profile type: Publication
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full work available at URL: s0012-7094-00-10117-2
OpenAlex ID: W2048373933
description: scientific article; zbMATH DE number 1425265