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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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Derived Category and Canonical Bundle --II

Derived Category and Canonical Bundle --II

(p.136) 6 Derived Category and Canonical Bundle --II
Fourier-Mukai Transforms in Algebraic Geometry

D. Huybrechts

Oxford University Press

Based on the work of Orlov, Kawamata, and others, this chapter shows that the (numerical) Kodaira dimension and the canonical ring are preserved under derived equivalence. The same techniques can be used to derive the invariance of Hochschild cohomology under derived equivalence. Going one step further, it is shown that the nefness of the canonical bundle is detected by the derived category. The chapter also studies the relation between derived and birational (or rather K-) equivalence. The special case of a central conjecture predicts that two birational Calabi-Yau varieties have equivalent derived categories.

Keywords:   Hochschild cohomology, Kodaira dimension, nef bundle, canonical ring, D-equiavelence, K-equivalence

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