The Kontsevich--Zorich cocycle over Veech--McMullen family of symmetric translation surfaces

TL;DR

This paper studies the Kontsevich--Zorich cocycle over a specific family of translation surfaces, demonstrating how all monodromies of a certain type can be realized through covering constructions, advancing understanding of their geometric and algebraic properties.

Contribution

It provides a detailed description of the Kontsevich--Zorich cocycle over Veech--McMullen families and shows all $SU(p,q)$ type monodromies are realizable via covering constructions.

Findings

All $SU(p,q)$ monodromies are realized by covering constructions.

The cocycle description applies to affine invariant orbifolds from cyclic coverings.

The work connects geometric structures with algebraic monodromy types.

Abstract

We describe the Kontsevich--Zorich cocycle over an affine invariant orbifold coming from a (cyclic) covering construction inspired by works of Veech and McMullen. In particular, using the terminology in a recent paper of Filip, we show that all cases of Kontsevich--Zorich monodromies of $SU(p,q)$ type are realized by appropriate covering constructions.