2000/01/31 by Paul Seidel, R. P. Thomas · 11 citations
Mathematics · Physics and Astronomy · #math.AG #hep-th #math.SG #msc:14J32 #msc:18E30 #msc:53D40 #msc:20F36
published as Duke Math. Jour. 108 (2001), 37--108. · 63 pages, 6 figures. Minor referees' changes for publication in Duke Math. Jour
arxiv created 2000/08/30 · arxiv updated 2009/11/30
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 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 ≥ 2, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities that 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.