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Fourier-Mukai Transforms in Algebraic Geometry$
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D. Huybrechts

Print publication date: 2006

Print ISBN-13: 9780199296866

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780199296866.001.0001

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K3 Surfaces

K3 Surfaces

Chapter:
(p.228) 10 K3 Surfaces
Source:
Fourier-Mukai Transforms in Algebraic Geometry
Author(s):

D. Huybrechts

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199296866.003.0010

After abelian varieties, K3 surfaces are the second most interesting special class of varieties. These have a rich internal geometry and a highly interesting moduli theory. Paralleling the famous Torelli theorem, results from Mukai and Orlov show that two K3 surfaces have equivalent derived categories precisely when their cohomologies are isomorphic weighing two Hodge structures. Their techniques also give an almost complete description of the cohomological action of the group of autoequivalences of the derived category of a K3 surface. The basic definitions and fundamental facts from K3 surface theory are recalled. As moduli spaces of stable sheaves on K3 surfaces are crucial for the argument, a brief outline of their theory is presented.

Keywords:   Torelli theorem, Hodge structure, moduli space

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