$A_{\infty}$-structures, Brill–Noether loci and the Fourier–Mukai transform

Abstract

We use the -formalism to study variation in cohomology spaces under formal deformations of coherent sheaves on projective varieties. As an application we describe formal neighborhoods of some singular points in twisted Brill–Noether loci in the moduli spaces of vector bundles on a curve. Another application is a computation of the Fourier–Mukai transform of some natural line bundles on symmetric powers of a curve.