On deformations of Q-Fano threefolds II

TL;DR

This paper studies the deformation properties of Q-Fano threefolds with terminal singularities, identifying specific singularities where certain cohomological maps vanish, and shows these threefolds can be deformed to simpler singularity types.

Contribution

It characterizes when a certain coboundary map vanishes in threefold singularities and demonstrates that Q-Fano threefolds can be deformed to have only quotient and specific singularities.

Findings

Vanishing of the coboundary map only for quotient and A_{1,2}/4-singularities.

Q-Fano 3-folds can be deformed to have only quotient and A_{1,2}/4-singularities.

Results on the Q-smoothability of Q-Calabi–Yau 3-folds.

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.