Deformations of semi-smooth varieties

TL;DR

This paper computes the tangent sheaf and T^1_X sheaf for semi-smooth varieties, which have specific singularities, using explicit gluing data, aiding the understanding of their deformations and smoothability.

Contribution

It provides an explicit method to compute tangent and T^1 sheaves for semi-smooth varieties based on their gluing data, advancing deformation theory.

Findings

Explicit formulas for tangent sheaf of semi-smooth varieties.

Explicit formulas for T^1_X sheaf of semi-smooth varieties.

Enhanced understanding of deformations of semi-smooth varieties.

Abstract

For a singular variety X, an essential step to determine its smoothability and study its deformations is the understanding of the tangent sheaf and of the sheaf T^1_X:=ext^1(Omega_X,O_X). A variety is semi-smooth if its singularities are \'etale locally the product of a double crossing point (uv=0) or a pinch point (u^2-v^2w=0) with affine space; equivalently, if it can be obtained by gluing a smooth variety along a smooth divisor via an involution with smooth quotient. Our main result is the explicit computation of the tangent sheaf and the sheaf T^1_X for a semi-smooth variety X in terms of the gluing data.