Moduli of high rank vector bundles over surfaces
Moduli of high rank vector bundles over surfaces
复制标题
表面上高阶向量束的模
DOI：
--
复制
发表时间：
1996
期刊：
影响因子：
--
通讯作者：
Jun Yu Li
中科院分区：
文献类型：
--
作者：
D. Gieseker;Jun Yu Li
本文的目的是应用[GL]中发展的退化理论研究C上任意光滑代数曲面上任意秩稳定向量丛的模空间。我们将表明，大多数的最新进展，在了解模秩2向量丛可以结转到高秩的情况下。在引入稳定向量丛的概念后，第一作者构造了曲面上向量丛的模概型。他表明，任何光滑的代数表面X与充足的除数H和线丛我在X，有一个粗糙的模计划M X（我，H）parameterizing（模等价关系）的一套所有H-半稳定秩r扭转自由层E在X与detE = I和C 2（E）= D。从那时起，许多数学家研究了这种模空间的几何，特别是对于秩二的情况。举几个例子，丸山，陶贝斯和第一作者表明，模空间M X（= M r，d X（I，H））是非空的，当d是大的。对一些特殊曲面的向量丛的模空间也进行了研究。对任意X和r = 2的M X的深入理解始于唐纳森的一般光滑性结果。粗略地说，唐纳森[Do]（后来由Friedman [Fr]和K. Zhu [Zh]）证明了当d足够大时，MX的奇异轨迹Sing（MX）是M2，dX的真子集，且其在MX中的余维数随d线性增加.这个定理表明，当第二个陈类d很大时，模MX的行为与预期的一样。其次，利用一般变形理论，证明了MX是正规的，且具有局部完全交（l.c.i.）在稳定层处的奇异性提供d是大的[L2]。他还表明，当X是一个曲面的一般类型满足一些温和的技术条件，那么M X是一般类型的d 0 [L2]。在文[GL]中，我们还证明了当d很大时，MX是不可约的.在本文和以后的论文中，我们将证明M X的几何和M X的几何，r ≥ 3，是相当相似的。这样做的主要障碍是缺乏高秩情况下的通用平滑结果的类比。在本文中，我们将使用[GL]中发展的模的退化来建立以下主要技术定理。
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.