Cohomology of Vector Bundles and Syzygies
Title | Cohomology of Vector Bundles and Syzygies PDF eBook |
Author | Jerzy Weyman |
Publisher | Cambridge University Press |
Pages | 404 |
Release | 2003-06-09 |
Genre | Mathematics |
ISBN | 9780521621977 |
The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.
Vector Bundles and Representation Theory
Title | Vector Bundles and Representation Theory PDF eBook |
Author | Steven Dale Cutkosky |
Publisher | American Mathematical Soc. |
Pages | 258 |
Release | 2003 |
Genre | Mathematics |
ISBN | 0821832646 |
This volume contains 13 papers from the conference on ``Hilbert Schemes, Vector Bundles and Their Interplay with Representation Theory''. The papers are written by leading mathematicians in algebraic geometry and representation theory and present the latest developments in the field. Among other contributions, the volume includes several very impressive and elegant theorems in representation theory by R. Friedman and J. W. Morgan, convolution on homology groups of moduli spaces of sheaves on K3 surfaces by H. Nakajima, and computation of the $S1$ fixed points in Quot-schemes and mirror principle computations for Grassmanians by S.-T. Yau, et al. The book is of interest to graduate students and researchers in algebraic geometry, representation theory, topology and their applications to high energy physics.
Connections, Curvature, and Cohomology V1
Title | Connections, Curvature, and Cohomology V1 PDF eBook |
Author | |
Publisher | Academic Press |
Pages | 467 |
Release | 1972-07-31 |
Genre | Mathematics |
ISBN | 008087360X |
Connections, Curvature, and Cohomology V1
Koszul Cohomology and Algebraic Geometry
Title | Koszul Cohomology and Algebraic Geometry PDF eBook |
Author | Marian Aprodu |
Publisher | American Mathematical Soc. |
Pages | 138 |
Release | 2010 |
Genre | Mathematics |
ISBN | 0821849646 |
The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well. This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.
Syzygies
Title | Syzygies PDF eBook |
Author | E. Graham Evans |
Publisher | Cambridge University Press |
Pages | 137 |
Release | 1985-08-15 |
Genre | Mathematics |
ISBN | 0521314119 |
This 1985 book covers from first principles the theory of Syzygies.
Group Cohomology and Algebraic Cycles
Title | Group Cohomology and Algebraic Cycles PDF eBook |
Author | Burt Totaro |
Publisher | Cambridge University Press |
Pages | 245 |
Release | 2014-06-26 |
Genre | Mathematics |
ISBN | 1107015774 |
This book presents a coherent suite of computational tools for the study of group cohomology algebraic cycles.
Grassmannians, Moduli Spaces and Vector Bundles
Title | Grassmannians, Moduli Spaces and Vector Bundles PDF eBook |
Author | David Ellwood |
Publisher | American Mathematical Soc. |
Pages | 190 |
Release | 2011 |
Genre | Mathematics |
ISBN | 0821852051 |
This collection of cutting-edge articles on vector bundles and related topics originated from a CMI workshop, held in October 2006, that brought together a community indebted to the pioneering work of P. E. Newstead, visiting the United States for the first time since the 1960s. Moduli spaces of vector bundles were then in their infancy, but are now, as demonstrated by this volume, a powerful tool in symplectic geometry, number theory, mathematical physics, and algebraic geometry. In fact, the impetus for this volume was to offer a sample of the vital convergence of techniques and fundamental progress, taking place in moduli spaces at the outset of the twenty-first century. This volume contains contributions by J. E. Andersen and N. L. Gammelgaard (Hitchin's projectively flat connection and Toeplitz operators), M. Aprodu and G. Farkas (moduli spaces), D. Arcara and A. Bertram (stability in higher dimension), L. Jeffrey (intersection cohomology), J. Kamnitzer (Langlands program), M. Lieblich (arithmetic aspects), P. E. Newstead (coherent systems), G. Pareschi and M. Popa (linear series on Abelian varieties), and M. Teixidor i Bigas (bundles over reducible curves). These articles do require a working knowledge of algebraic geometry, symplectic geometry and functional analysis, but should appeal to practitioners in a diversity of fields. No specialization should be necessary to appreciate the contributions, or possibly to be stimulated to work in the various directions opened by these path-blazing ideas; to mention a few, the Langlands program, stability criteria for vector bundles over surfaces and threefolds, linear series over abelian varieties and Brauer groups in relation to arithmetic properties of moduli spaces.