Remarks on automorphism and cohomology of finite cyclic coverings of projective 
spaces

Lyu, Renjie; Pan, Xuanyu

doi:10.4310/MRL.2021.v28.n3.a7

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Remarks on automorphism and cohomology of finite cyclic coverings of projective spaces

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Pan, Xuanyu
Max Planck Institute for Mathematics, Max Planck Society;

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https://dx.doi.org/10.4310/MRL.2021.v28.n3.a7
(Publisher version)

https://doi.org/10.48550/arXiv.1705.06618
(Preprint)

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Citation

Lyu, R., & Pan, X. (2021). Remarks on automorphism and cohomology of finite cyclic coverings of projective spaces. Mathematical Research Letters, 28(3), 785-822. doi:10.4310/MRL.2021.v28.n3.a7.

Cite as: https://hdl.handle.net/21.11116/0000-0008-AA9F-6

Abstract

For a smooth finite cyclic covering over a projective space of dimension
greater than one, we show that its group of automorphisms faithfully acts on
its cohomology except for a few cases. In characteristic zero, we study the
equivariant deformation theory and groups of automorphisms for complex cyclic
coverings. The proof uses the decomposition of the sheaf of differential forms
due to Esnault and Viehweg. In positive characteristic, a lifting criterion of
automorphisms reduce 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 cfinite yclic coverings, and the infinitesimal
Torelli theorem for finite cyclic coverings defined over an arbitrary field.