Infinitesimal Deformations and Obstructions for Toric Singularities

TL;DR

The paper computes the obstruction space $T^2_Y$ and the cup product $T^1_Y\times T^1_Y\to T^2_Y$ for toric singularities, building a toric–combinatorial description of the homogeneous pieces $T^i_Y(-R)$. It introduces two equivalent complexes, $L(E^R)_\bullet$ and an alternative span$(E^R)_\bullet$, to realize $T^1_Y(-R)$ and $T^2_Y(-R)$ as cohomology, and provides explicit cup-product formulas via toric relations and coboundary adjustments. In the special case of three-dimensional toric Gorenstein singularities, the cup product reduces to a bilinear map into the edge-span, and the dimensions of $T^2_Y(-kR^*)$ are determined by the diameter of the defining polygon, yielding concrete descriptions of the versal base space via equations $\sum_i t_i^k d^i=0$ for $k\ge 1$. The results give computable tools for deformation theory of toric singularities and connect obstructions to the underlying polyhedral geometry.

Abstract

The obstruction space T^2 and the cup product T^1 x T^1 -> T^2 are computed for toric singularities.