Endomorphisms of Koszul complexes: formality and application to deformation theory

TL;DR

This paper investigates the structure of endomorphisms of Koszul complexes, proving homotopy abelianity over a base ring and applying the results to obstruction theory in deformation problems of algebraic geometry.

Contribution

It demonstrates that the dg Lie algebra of endomorphisms is homotopy abelian over the base ring and applies this to derive new results in deformation theory.

Findings

Endomorphism dg Lie algebra is homotopy abelian over the base ring.

Not formal over the original algebra.

Application to obstructions in deformation of ideal sheaves.

Abstract

We study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative $K$-algebra $R$ and we prove that it is homotopy abelian over $K$, while it is generally not formal over $R$. We apply this result to prove an annihilation theorem for obstructions of (derived) deformations of locally complete intersection ideal sheaves on projective schemes.