Remarks on Automorphism and Cohomology of Cyclic Coverings

TL;DR

This paper investigates automorphism groups of smooth cyclic coverings over projective spaces, showing faithfulness on cohomology in most cases, and explores deformation theory and automorphisms in various characteristics using advanced algebraic geometry tools.

Contribution

It establishes conditions for automorphism faithfulness on cohomology and extends deformation and automorphism analysis to positive characteristic fields.

Findings

Automorphism groups act faithfully on cohomology in most cases.

Degeneration of Hodge-de Rham spectral sequences is proved for families of cyclic coverings.

Infinitesimal Torelli theorem is established for cyclic coverings over arbitrary fields.

Abstract

For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that the group of automorphisms acts faithfully on the cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and automorphism groups for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and Viehweg. In positive characteristics, a lifting criterion of automorphisms reduces the faithfulness problem to characteristic zero. To apply this criterion, we prove the degeneration of the Hodge-de Rham spectral sequences for a family of smooth finite cyclic coverings, and the infinitesimal Torelli theorem for finite cyclic coverings defined over an arbitrary field.