On the mean Euler characteristic of Gorenstein toric contact manifolds

TL;DR

This paper establishes a precise relationship between the mean Euler characteristic of Gorenstein toric contact manifolds and the normalized volume of their toric diagrams, with applications to crepant toric symplectic fillings.

Contribution

It proves that the mean Euler characteristic equals half the normalized volume of the toric diagram and links it to the Euler characteristic of crepant toric symplectic fillings.

Findings

Mean Euler characteristic equals half the normalized volume.

Twice the mean Euler characteristic equals the Euler characteristic of crepant fillings.

Provides applications connecting contact and symplectic invariants.

Abstract

We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and Dais, is the following: twice the mean Euler characteristic of a Gorenstein toric contact manifold is equal to the Euler characteristic of any crepant toric symplectic filling, i.e. any toric symplectic filling with zero first Chern class.