A counterexample to Durfee conjecture

TL;DR

This paper presents a counterexample to Durfee's 1978 conjecture relating invariants of complex surface singularities, proposes a weaker valid inequality, and verifies it in specific cases.

Contribution

It introduces a counterexample to the original conjecture and formulates a new, weaker inequality that holds in higher dimensions and codimensions.

Findings

Counterexample disproves Durfee's conjecture for non-hypersurface cases

Proposed weaker inequality is asymptotically sharp

Verification of the new inequality in certain homogeneous cases

Abstract

An old conjecture of Durfee 1978 bounds the ratio of two basic invariants of complex isolated complete intersection surface singularities: the Milnor number and the singularity (or geometric) genus. We give a counterexample for the case of non-hypersurface complete intersections, and we formulate a weaker conjecture valid in arbitrary dimension and codimension. This weaker bound is asymptotically sharp. In this note we support the validity of the new proposed inequality by its verification in certain (homogeneous) cases. In our subsequent paper we will prove it for several other cases and we will provide a more comprehensive discussion.