Deformations, local freeness, and base change for higher Du Bois singularities

Abstract

We prove that strict higher Du Bois singularities are invariant under small deformations. Using this, we prove a base change theorem for the relative Du Bois complex with strict higher Du Bois fibers, answering a question of Kovács--Taji. We exhibit failures of deformation invariance and base change for $1$-Du Bois fibers, showing that the strictness condition is essentially sharp. We also generalize the local-freeness theorem of Friedman--Laza beyond the local complete intersection setting for families over an arbitrary base, and show the unobstructedness of deformations of strict-$1$-Du Bois canonical Calabi-Yau varieties.

arXiv commentv2: strengthened local freeness Theorem 1.7 over arbitrary bases and added Corollary 1.8; expanded Section 6; 38 pages