Deformations and obstructions of pairs (X,D)

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.

improved exposition, added some applications

References in corpus (6)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Introduction to the log minimal model program for log canonical pairs
• Deformation theory and rational homotopy type
• Lectures on deformations of complex manifolds
• Homotopy BV algebras in Poisson geometry
• Differential Graded Lie Algebras and Deformations of Holomorphic Maps

Cited by in corpus (13)

• Nonabelian higher derived brackets
• On deformations of pairs (manifold, coherent sheaf)
• Smoothing pairs over degenerate Calabi-Yau varieties
• Deformations of holomorphic pseudo-symplectic Poisson manifolds
• A logarithmic $\bar{\partial }$-equation on a compact Kähler manifold associated to a smooth divisor
• Unobstructedness of deformations of weak Fano manifolds
• Modularity of Landau-Ginzburg models
• The MV formalism for ${\mathrm{IBL}}_{\infty }$- and ${\mathrm{BV}}_{\infty }$-algebras
• Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)
• Logarithmic $\partial \bar{\partial }$-lemma and several geometric applications (with an Appendix joint with Sheng Rao)
• Deformations of Calabi-Yau varieties with isolated log canonical singularities
• On Tian-Todorov lemma and its applications to deformation of CR-structures
• On the abstract Bogomolov-Tian-Todorov Theorem