AI 中文总结
该研究刻画了代数闭域上K3曲面存在非零正次数整体对称微分的充要条件，推广了Jang定理，明确了特征2超奇异K3曲面的相关结构性质。
AI 中文摘要
我们证明，代数闭域上的K3曲面存在非零的正次数整体对称微分当且仅当特征为$p=2$且该曲面是Artin不变量$σ_0=1$的超奇异K3曲面。此前，Kobayashi的结果仅证明了特征零情形下的零化结论。针对例外情形，我们证明每个正偶次数下都存在唯一的（相差标量倍意义下）非平凡整体对称微分。研究过程中，我们推广了Jang的一个定理，证明特征$p>0$的超奇异K3曲面同构于光滑四次曲面当且仅当$p\geq3$，或$p=2$且其Artin不变量$σ_0\geq3$。
英文摘要
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 $σ_0\geq3$.
