On deformations of Q-Fano threefolds II

Abstract

We investigate some coboundary map associated to a $3$-dimensional terminal singularity which is important in the study of deformations of singular $3$-folds. We prove that this map vanishes only for quotient singularities and a $A_{1,2}/4$-singularity, that is, a terminal singularity analytically isomorphic to a ${\mathbb{Z}}_4$-quotient of the singularity $(x^2+y^2+z^3+u^2=0)$. As an application, we prove that a $\mathbb{Q}$-Fano $3$-fold with terminal singularities can be deformed to one with only quotient singularities and $A_{1,2}/4$-singularities. We also treat the $\mathbb{Q}$-smoothability problem on $\mathbb{Q}$-Calabi--Yau $3$-folds.

11 pages. title changed. to appear in Crelle's journal

References in corpus (3)

• Fano 3-folds in codimension 4, Tom and Jerry, Part I
• Deforming elephants of Q-Fano threefolds
• On deformations of Q-Fano threefolds

Cited by in corpus (1)

• Deformations of weak $\mathbb{Q}$-Fano 3-folds