Duke Mathematical Journal

Braid group actions on derived categories of coherent sheaves

Paul Seidel and Richard Thomas

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Abstract

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is M. Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim $X\geq 2$, our braid group actions are always faithful.

We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

Article information

Source
Duke Math. J., Volume 108, Number 1 (2001), 37-108.

Dates
First available in Project Euclid: 5 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1091737124

Digital Object Identifier
doi:10.1215/S0012-7094-01-10812-0

Mathematical Reviews number (MathSciNet)
MR1831820

Zentralblatt MATH identifier
1092.14025

Subjects
Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 14J32: Calabi-Yau manifolds 18E30: Derived categories, triangulated categories 20F36: Braid groups; Artin groups 53D40: Floer homology and cohomology, symplectic aspects

Citation

Seidel, Paul; Thomas, Richard. Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108 (2001), no. 1, 37--108. doi:10.1215/S0012-7094-01-10812-0. https://projecteuclid.org/euclid.dmj/1091737124


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