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Vector fields on canonically polarized surfaces

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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 \(p>f(K_X^2) \) 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.

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  1. Department of Mathematics, University of Cyprus, P.O. Box 20537, Nicosia, 1678, Cyprus

    Nikolaos Tziolas

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  1. Nikolaos Tziolas
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To my little daughter Eleonora.

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Part of this paper was written during the author’s stay at the Max Planck Institute for Mathematics in Bonn, from 01.02.2019 to 31.07.2019.

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Tziolas, N. Vector fields on canonically polarized surfaces. Math. Z. 300, 2837–2883 (2022). https://doi.org/10.1007/s00209-021-02898-1

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