Vector fields on canonically polarized surfaces

Tziolas, Nikolaos

doi:10.1007/s00209-021-02898-1

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Vector fields on canonically polarized surfaces

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Tziolas, Nikolaos
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s00209-021-02898-1
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Tziolas, N. (in press). Vector fields on canonically polarized surfaces. Mathematische Zeitschrift, Published Online - Print pending. doi:10.1007/s00209-021-02898-1.

Cite as: https://hdl.handle.net/21.11116/0000-0009-D07E-F

Abstract

This paper investigates the geometry of canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces. In particular, an explicit integer valued function f(x) is obtained with the following properties. If X is a canonically polarized surface defined over an algebraically closed field of characteristic p>0 such that p>f(K2X) and X has a nontrivial global vector field, then X is unirational and the algebraic fundamental group is trivial. As a consequence of this result, large classes of canonically polarized surfaces are identified whose moduli stack is Deligne-Mumford, a property that does not hold in general in positive characteristic.