Braid group actions on derived categories of coherent sheaves

TL;DR

This paper constructs braid group actions on the derived category of coherent sheaves on varieties, demonstrating their faithfulness in higher dimensions and exploring connections with mirror symmetry, singularity resolutions, and the McKay correspondence.

Contribution

It introduces a new construction of braid group actions on derived categories, establishing their faithfulness for varieties of dimension at least two and linking them to mirror symmetry and singularity theory.

Findings

Braid group actions are faithful when the variety dimension is at least two.

Examples of braid group actions from crepant resolutions of singularities are provided.

Connections with the McKay correspondence and exceptional sheaves are discussed.

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 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 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.