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Fourier-Mukai transforms in algebraic geometry

Fourier-Mukai transforms in algebraic geometry
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Dewey Class 516.35 (DDC 22)
Title Fourier-Mukai transforms in algebraic geometry (M) / D. Huybrechts.
Author Huybrechts, Daniel
Publication Oxford : Oxford University Press , 2007
Physical Details viii, 307 pages : ill. ; 24 cm
Series Oxford mathematical monographs
ISBN 9780199296866
Note Preface ; 1. Triangulated categories ; 2. Derived categories: a quick tour ; 3. Derived categories of coherent sheaves ; 4. Derived category and canonical bundle I ; 5. Fourier-Mukai transforms ; 6. Derived category and canonical bundle II ; 7. Equivalence criteria for Fourier-Mukai transforms ; 8. Spherical and exceptional objects ; 9. Abelian varieties ; 10. K3 surfaces ; 11. Flips and flops ; 12. Derived categories of surfaces ; 13. Where to go from here ; References ; Index
Summary Note This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe. -- This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is based on a course given at the Institut de Mathematiques de Jussieu in 2004 and 2005. Aimed at postgraduate students with a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. Including notions from other areas, e.g. singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs are given and exercises aid the reader throughout.:
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0000000044311 512.7 HUY
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