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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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Spherical and Exceptional Objects

Spherical and Exceptional Objects

(p.166) 8 Spherical and Exceptional Objects
Fourier-Mukai Transforms in Algebraic Geometry

D. Huybrechts

Oxford University Press

Spherical objects — motivated by considerations in the context of mirror symmetry — are used to construct special autoequivalences. Their action on cohomology can be described precisely, considering more than one spherical object often leads to complicated (braid) groups acting on the derived category. The results related to Beilinson are almost classical. Section 3 of this chapter gives an account of the Beilinson spectral sequence and how it is used to deduce a complete description of the derived category of the projective space. This will use the language of exceptional sequences and semi-orthogonal decompositions encountered here. The final section gives a simplified account of the work of Horja, which extends the theory of spherical objects and their associated twists to a broader geometric context.

Keywords:   braid group action, Beilinson spectral sequence, Horja

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