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

TL;DR

This paper constructs an $L_{ abla}$-morphism associated with a connection on a complex of sheaves, lifting the Buchweitz-Flenner semiregularity map and applying it to deformations of coherent sheaves.

Contribution

It introduces a canonical $L_{ abla}$-morphism lifting the semiregularity map for complexes of sheaves, connecting connections to deformation theory.

Findings

Established a canonical $L_{ abla}$-morphism for sheaf complexes.

Connected connections to semiregularity maps in deformation theory.

Applied the construction to deformations of coherent sheaves on projective manifolds.

Abstract

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