A note on higher obstruction space for surface singularities

TL;DR

This paper investigates the higher obstruction spaces in the deformation theory of certain surface singularities, revealing their non-vanishing in general and implications for the existence of virtual fundamental classes.

Contribution

It provides explicit calculations of higher obstruction spaces for rational and minimally elliptic surface singularities, extending known results and analyzing their impact on moduli space theory.

Findings

Higher obstruction spaces do not vanish for semi-log-canonical surfaces

Explicit calculations for rational and minimally elliptic singularities

No virtual fundamental class exists for the moduli space of semi-log-canonical surfaces

Abstract

In this paper we collect some results on the obstruction spaces for rational surface singularities and minimally elliptic surface singularities. Based on the known results we calculate higher obstruction spaces for such surface singularities. The results imply that in general the higher obstruction spaces of deforming semi-log-canonical surfaces do not vanish. We apply the calculation result to show that there is no Li-Tian and Behrend-Fantechi style virtual fundamental class on such moduli space of semi-log-canonical surfaces.