Duke Mathematical Journal
- Duke Math. J.
- Volume 108, Number 1 (2001), 37-108.
Braid group actions on derived categories of coherent sheaves
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.
Duke Math. J., Volume 108, Number 1 (2001), 37-108.
First available in Project Euclid: 5 August 2004
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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