Log-concavity from enumerative geometry of planar curve singularities

TL;DR

This paper introduces a log-concavity conjecture for BPS invariants in enumerative geometry, extending it to various geometric invariants, and proves it for specific classes of singularities.

Contribution

It formulates a new log-concavity conjecture for BPS invariants and proves it for irreducible weighted-homogeneous and ADE singularities.

Findings

Conjecture established for torus knots and ADE singularities.

Proved a multiplicative property for ruling polynomials.

Extended conjecture to ruling polynomials and E-polynomials.

Abstract

We propose a log-concavity conjecture for BPS invariants arising in the enumerative geometry of planar curve singularities, identified with the local Euler obstructions of Severi strata in their versal deformations. We further extend this conjecture to ruling polynomials of Legendrian links and to E-polynomials of character varieties. We establish these conjectures for irreducible weighted-homogeneous singularities (torus knots) and for ADE singularities, and prove a multiplicative property for ruling polynomials compatible with log-concavity.