Symmetric Differentials on K3 Surfaces

Frank Gounelas; Christian Liedtke

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arXiv · 2608.17953

Symmetric Differentials on K3 Surfaces

Frank Gounelas·Christian Liedtke

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Abstract

We prove that a K3 surface over an algebraically closed field admits a nonzero global symmetric differential of positive degree if and only if the characteristic is $p=2$, and it is supersingular of Artin invariant $\sigma_0=1$. Vanishing was previously only known in characteristic zero by a result of Kobayashi. For the exceptional case we show that there is a unique (up to scaling) nontrivial global symmetric differential in every positive even degree. Along the way, we extend a theorem of Jang and show that a supersingular K3 surface in characteristic $p>0$ is isomorphic to a smooth quartic surface if and only if $p\geq3$ or $p=2$ and it is of Artin invariant $\sigma_0\geq3$.

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BibTeX RIS

Frank Gounelas, Christian Liedtke. 2026-08-18. Symmetric Differentials on K3 Surfaces. https://arxiv.org/abs/2608.17953

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