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

Abstract

Let $E^{\ast }$ be a finite complex of locally free sheaves on a complex manifold $X$. We prove that to every connection of type $(1,0)$ on $E^{\ast }$ it is canonically associated an $L_{\infty }$ morphism $g:A_X^{0,\ast }(\mathcal{H}o{m}_{O_X}^{\ast }(E^{\ast },E^{\ast }))\to \frac{A_X^{\ast ,\ast }}{A_X^{\ge 2,\ast }}[2]$ that lifts the 1-component of Buchweitz-Flenner semiregularity map. An application to deformations of coherent sheaves on projective manifolds is given.

V3: minor changes, references updated