Smoothings of Fano varieties with normal crossing singularities

TL;DR

This paper establishes criteria for smoothing Fano varieties with normal crossing singularities using logarithmic deformation theory, showing smoothability is characterized by the triviality of T^1(X) over the singular locus.

Contribution

It introduces a new criterion for smoothability of Fano varieties with normal crossing singularities based on logarithmic deformation theory.

Findings

Smoothability of X is equivalent to T^1(X)=O_D.

Uses logarithmic structures to analyze deformations.

Provides explicit conditions for smoothability.

Abstract

This paper obtains criteria for a Fano variety X with normal crossing singularities defined over an algebraically closed field of characteristic zero, to be smoothable. The difference with the original version is that the theory of logarithmic structures and deformations is used in order to prove that X is smoothable by a smooth variety, if and only if T^1(X)=O_D, where D is the singular locus of X.