Log-concavity from enumerative geometry of planar curve singularities

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.

Figures (2)

• Figure 1: The Legendrian link associated to the $D_n$-singularity: the rainbow closure $\bigl(\sigma_1^{\,n-2}\sigma_2\sigma_1^2\sigma_2\bigr)^>$.
• Figure 2: An illustration for the rainbow closure $(\beta_1\tilde{\gamma}\beta_2)^{>}$: in the figure, $n=4$, $m=2$, $\beta_1 = \sigma_1^2\sigma_2^2\sigma_3^2$, $\beta_2 = \sigma_3^2\sigma_2\sigma_1$$\in \mathrm{Br}_4^+$, $\gamma = \sigma_1^2\in \mathrm{Br}_2^+$, hence $\tilde{\gamma} = \sigma_4^2\in \mathrm{Br}_5^+$.

Theorems & Definitions (25)

• Conjecture 1.1: Main conjecture
• Conjecture 1.2
• Lemma 1.3
• proof
• Remark 1.4
• Conjecture 2.1
• Example 2.2
• Remark 2.3
• Remark 2.4
• Conjecture 3.1