On the Griffiths numbers for higher dimensional singularities

TL;DR

This paper investigates Griffiths numbers in higher-dimensional singularities, disproving a conjecture for certain Gorenstein singularities while confirming it for irregular cases, thus clarifying their behavior in complex singularity theory.

Contribution

It demonstrates that Yau's conjecture fails for isolated rigid Gorenstein singularities of dimension greater than 2 but holds for irregular singularities, providing new insights into singularity invariants.

Findings

Yau's conjecture does not hold for rigid Gorenstein singularities in higher dimensions.

Yau's conjecture is valid for irregular singularities.

The behavior of Griffiths numbers varies significantly with singularity type.

Abstract

We show that Yau's conjecture on the inequalities for (n-1)-th Griffiths number and (n-1)-th Hironaka number does not hold for isolated rigid Gorenstein singularities of dimension greater than 2. But his conjecture on the inequality for (n-1)-th Griffiths number is true for irregular singularities.