Durfee-type inequality for complete intersection surface singularities

TL;DR

This paper proves an inequality relating the signature of the Milnor fiber of certain surface singularities to geometric and resolution data, confirming Durfee's weak conjecture and addressing Kerner--Némethi's conjecture.

Contribution

It establishes a new inequality for complete intersection surface singularities, confirming a longstanding conjecture and providing partial validation for another conjecture in the field.

Findings

Proves Durfee's weak conjecture for these singularities.

Provides an upper bound for the signature of the Milnor fiber.

Addresses a partial case of Kerner--Némethi's conjecture.

Abstract

We prove that the signature of the Milnor fiber of smoothings of a $2$-dimensional isolated complete intersection singularity does not exceed the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution, which implies Durfee's weak conjecture and a partial answer to Kerner--N\'emethi's conjecture.