Deformations and obstructions of pairs (X,D)

TL;DR

This paper investigates the deformation theory of pairs (X,D) where X is a smooth projective variety and D is a divisor, demonstrating unobstructedness in many cases using algebraic methods involving differential graded Lie algebras.

Contribution

It provides a purely algebraic proof of unobstructedness for deformations of pairs (X,D) in several significant cases, including log Calabi-Yau pairs.

Findings

Deformations of pairs (X,D) are unobstructed in many cases.

Algebraic methods using differential graded Lie algebras are effective.

Results include cases where D is a smooth divisor in a Calabi-Yau variety or in |-m K_X|.

Abstract

We study deformations of pairs (X,D), with X smooth projective variety and D a smooth or a normal crossing divisor, defined over an algebraically closed field of characteristic 0. Using the differential graded Lie algebras theory and the Cartan homotopy construction, we are able to prove in a completely algebraic way the unobstructedness of the deformations of the pair (X,D) in many cases, e.g., whenever (X,D) is a log Calabi-Yau pair, in the case of a smooth divisor D in a Calabi Yau variety X and when D is a smooth divisor in |-m K_X|, for some positive integer m.