The Milnor fibre signature is not semi-continuous

TL;DR

This paper demonstrates through examples that the signature of the Milnor fibre's homology lattice can increase during degenerations, countering previous assumptions of semi-continuity.

Contribution

It provides the first known examples where the Milnor fibre signature increases under degeneration, disproving the semi-continuity conjecture.

Findings

Counterexamples to semi-continuity of the Milnor fibre signature

Signature can increase during degenerations in Newton non-degenerate families

Answers a long-standing question by A. Durfee from 1978

Abstract

Consider the germ of an isolated surface singularity in $(\mC^3,0)$. The corresponding Milnor fibre possesses the homology lattice (the integral middle homology with a natural symmetric intersection form). An old question of A.Durfee (1978) asks: is the signature of this form non-increasing under degenerations? The present article answers negatively: We give examples of Newton non-degenerate families where the signature increases under degeneration.