Moduli spaces of vector bundles on algebraic varieties

Abstract

This thesis seeks to contribute to a deeper understanding of the moduli spaces <i>M-sub X, H (r; c1,., Cmin{r;n}</i>) of rank <i>r</I>, <i>H</i>-stable vector bundles E on an <i>n</i>-dimensional variety <i>X</I>, with fixed Chern classes <i>c-sub1(E)</I> = <i>csub1 H-super2i ( X , Z)</i> , displaying new and interesting geometric properties of <i>M-sub X, H (r; c1,., Cmin{r;n}</i>) which nicely reflect the general philosophy that moduli spaces inherit a lot of .geometrical properties of the underlying variety <i>X</i>.<br/><br/>More precisely, we consider a smooth, irreducible, n-dimensional, projective variety X defined over an algebraically closed field k of characteristic zero, H an ample divisor on X, r >/2 an integer and c-subi H-super2i(X,Z) for i = 1, .,min{r,n}. We denote by <i>M-sub X, H (r; c1,., Cmin{r;n}</i>) the moduli space of rank r, vector bundles E on X, H-stable, in the sense of Mumford-Takemoto, with fixed Chern classes c-subi(E) = c-subi for i = 1, . , min{r, n}.<br/><br/>The contents of this Thesis is the following: Chapter 1 is devoted to provide the reader with the general background that we will need in the sequel. In the first two sections, we have collected the main definitions and results concerning coherent sheaves and moduli spaces, at least, those we will need through this work.<br/><br/>The aim of Chapter 2 is to establish the enterions of rationality for moduli spaces of rank two, it-stable vector bundles on a smooth, irreducible, rational surface X that will be used as one of our tools for answering Question (1), who is that follows: "Let X be a smooth, irreducible, rational surface. Fix C-sub1 Pic(X) and 0 « c2 Z. Is there an ample divisor H on X such that M-sub X,H(2; Ci, c2) is rational?"<br/><br/>In Chapter 3 we prove that the moduli space M-sub X,H(2; Ci, c2) of rank two, H-stable, vector bundles E on a smooth, irreducible, rational surface X, with fixed Chern classes C-sub1(E) = C-sub1 Pic(X) and 0 « C-sub2«(E) Z is a smooth, irreducible, rational, quasi-projective variety (Theorem 3.3.7) which solves Question (1).<br/><br/>In Chapter 4 we study moduli spaces (M-sub X,H(2; Ci, c2)) of rank r, H-stable vector bundles on either minimal rational surfaces or on algebraic K3 surfaces.<br/><br/>In Chapter 5 we deal with moduli spaces M-sub x,l (2;Ci,C2) of rank two, L-stable vector bundles E, on P-bundles of arbitrary dimension, with fixed Chern classes.