Moduli of endomorphisms of semistable vector bundles over a compact Riemann surface

L. Brambila Paz
1990 Glasgow Mathematical Journal  
Introduction. Mumford and Suominen in [8] and Newstead in [11] have considered the moduli problem of classifying the endomorphisms of finite-dimensional vector spaces. Using similar ideas we consider the moduli problem for endomorphisms of indecomposable semistable vector bundles over a compact connected Riemann surface of genus In this paper we develop the 3-dimensional case, which gives an idea of how to solve the moduli problem in general. First we give the algebras which can occur as
more » ... can occur as algebras of endomorphisms of vector bundles of rank 3. Then we give necessary and sufficient conditions for a vector bundle to have a particular algebra of endomorphisms. Such conditions show that for any non-zero nilpotent endomorphism, there are extensions of vector bundles from which it can be reconstructed. Thus the problem of parametrizing endomorphisms is largely reduced to one of parametrizing extensions. We construct the corresponding universal families of extensions; unfortunately, this is not quite sufficient for us to obtain moduli spaces for endomorphisms themselves. However, in some cases we can obtain local universal families of endomorphisms. The algebra of endomorphisms depends on how the extensions are related. In Section 1 we state the moduli problem for endomorphisms of vector bundles. In Section 2 we recall from [3] the relations between the extensions and the algebras of endomorphisms. Section 3 contains the constructions of the universal families of extensions which partially solve the moduli problem. 1. Moduli of endomorphisms. Throughout this paper X will denote a compact connected Riemann surface of genus greater than 1, and S(n, d) the set of (isomorphism classes of) indecomposable semistable non-simple vector bundles of rank n and slope d over X. Let P(n, d) be the set of pairs (E, (j>) where E is in S(n, d) and (t>:E-*E is an endomorphism of vector bundles. We say that two pairs (E, ) and (F, V) are equivalent, written (£, )~ (F, xp), if there exists an isomorphism a\E-*Fsuch that ip°a = a°), where £ is a vector bundle over X xS and 3> an endomorphism of E, such that for each s eS the restriction (E, )xx s and (F, 1 / ) A -XJ are equivalent for each s e S. Given a family of endomorphisms (E, <&) parametrized by S
doi:10.1017/s0017089500009010 fatcat:s3o5ffwomjdydhlubssc4e3hg4