Braid group actions on derived categories of coherent sheaves

This paper gives a construction of braid group actions on the derived categ ory of coherent sheaves on a variety X. The motivation for this is M. Konts evich's homological mirror conjecture, together with the occurrence of cert ain braid group actions in symplectic geometry. One of the main results is that when dim X greater than or equal to 2, our braid group actions are alw ays faithful. We describe conjectural mirror symmetries between smoothings and resolution s of singularities which lead us to find examples of braid group actions ar ising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves an Fano manifolds are gi ven. Moreover the case of an elliptic curve is worked out in some detail.