Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

Abstract

We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to deformations of F is Ext^1(F,F), and obstructions are contained in Ext^2(F,F). The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is concentrated in nonnegative degrees with a noncommutative Cech cohomology-type functor.

Several typos corrected. 22 pages, uses xy-pic

References in corpus (5)

• Higher Topos Theory
• Deformation theory via differential graded Lie algebras
• Transferring homotopy commutative algebraic structures
• Semicosimplicial DGLAs in deformation theory
• Deformations via Simplicial Deformation Complexes

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• Connections and $L_{\infty }$ liftings of semiregularity maps