Moduli of High Rank Vector Bundles over Surfaces

The purpose of this work is to apply the degeneration theory developed in [GL] to study the moduli space of stable vector bundles of arbitrary rank on any smooth algebraic surface (over C). We will show that most of the recent progress in understanding moduli of rank two vector bundles can be carried over to high rank cases. After introducing the notion of stable vector bundles, the first author constructed the moduli schemes of vector bundles on surfaces. He showed that for any smooth algebraic surface X with ample divisor H and line bundle I on X , there is a coarse moduli scheme M X (I,H) parameterizing (modulo equivalence relation) the set of all H-semistable rank r torsion free sheaves E on X with detE = I and c2(E) = d. Since then, many mathematicians have studied the geometry of this moduli space, especially for the rank two case. To cite a few, Maruyama, Taubes and the first author showed that the moduli space M X (= M r,d X (I,H)) is non-empty when d is large. Moduli spaces of vector bundles of some special surfaces have been studied also. The deep understanding of M X for arbitrary X and r = 2 begins with Donaldson’s generic smoothness result. Roughly speaking, Donaldson [Do] (later generalized by Friedman [Fr] and K. Zhu [Zh]) showed that when d is large enough, then the singular locus Sing ( M X ) of M X is a proper subset of M 2,d X and its codimension in M X increases linearly in d. This theorem indicates that the moduli M X behaves as expected when the second Chern class d is large. Later, using general deformation theory, the second author proved that M X is normal, and has local complete intersection (l.c.i.) singularities at stable sheaves provided d is large [L2]. He also showed that when X is a surface of general type satisfying some mild technical conditions, then M X is of general type for d 0 [L2]. In our paper [GL], we also proved that M X is irreducible if d is large. In this and subsequent papers, we shall show that the geometry of M X and the geometry of M X , r ≥ 3, is rather similar. The main obstacle in doing so is the lack of an analogy of the generic smoothness result in high rank case. In this paper, we will use the degeneration of moduli developed in [GL] to establish the following main technical theorem.