Cylinder counts and spin refinement of area Siegel-Veech constants

TL;DR

This paper provides new methods to compute area Siegel-Veech constants for strata of abelian differentials with specific spin parity, using quasimodular forms and intersection theory, refining previous results.

Contribution

It introduces two novel approaches for calculating Siegel-Veech constants, enhancing understanding of their structure and establishing a new identity for cylinder constants.

Findings

Constants can be computed via quasimodular forms or intersection theory.

Refinement of previous theorems on Siegel-Veech constants.

New identity for cylinder Siegel-Veech constants.

Abstract

We study the area Siegel-Veech constants of components of strata of abelian differentials with even or odd spin parity. We prove that these constants may be computed using either: (I) quasimodular forms, or (II) intersection theory. These results refine the main theorems of arXiv:1606.04065 and arXiv:1901.01785 which described the area Siegel-Veech constants of the full strata. Along the proof of (II), we establish a new identity for Siegel-Veech constants of cylinders.