A supplement to Fujino's paper: On isolated log canonical singularities with index one

TL;DR

This paper provides an elementary proof relating the dimension of the dual complex of the exceptional locus in a resolution of isolated log canonical singularities to the Hodge type of a specific cohomology group, extending Fujino's work.

Contribution

It offers a new elementary proof of Fujino's main result, connecting the dual complex dimension to Hodge theory without relying on advanced minimal model theory.

Findings

Dimension of dual complex linked to Hodge type of cohomology

Elementary proof method used instead of minimal model theory

Clarifies the structure of isolated log canonical singularities

Abstract

Let $E$ be the essential part of the exceptional locus of a good resolution of an isolated, log canonical singularity of index one. We describe the dimension of the dual complex of $E$ in terms of the Hodge type of $H^{n-1}(E, O_E)$, which is one of the main results of the paper [1] of Fujino. Our proof uses only an elementary classical method, while Fujino's argument depends on the recent development in minimal model theory.