Symmetric Differentials on K3 Surfaces

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 $σ_0=1$. 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 $σ_0\geq3$.

Disclosure

“that the ERC Synergy Grant HyperK (ID 854361) for its support. Artificial Intelligence: Some of the results of this paper were obtained with the assistance of the Danus system, an automated proof-search and verification system, along with OpenAI’s GPT-5.6-Sol model. Any errors or inaccuracies of this article are, of course, the authors’ sole responsibility. Notation and conventions A K3 surface X over a field k is a geometrically connected, smooth, proper surf”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:22:04.818716Z

Structural counts