Symmetric Differentials on K3 Surfaces

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 $p\ge 3$ or $p=2$ and it is of Artin invariant ${\sigma }_0\ge 3$.