AI 中文总结
研究具有平凡典范丛的拟\(F -\)分裂簇，通过多种方法证明正则拟\(F^{\infty}-\)分裂簇性质、解答相关曲面问题、证明正规簇几何正规性，以及在奇点解消假设下证明特定拟\(F^e -\)纯正规簇是对数典范的。
AI 中文摘要
我们解决了与具有平凡典范丛的拟\(F -\)分裂簇的几何和奇点相关的某些问题。首先，证明正则拟\(F^{\infty}-\)分裂簇不是几何单有理的且具有几何典范奇点。其次，表明存在具有平凡典范丛的拟\(F -\)分裂曲面不是拟\(F^{\infty}-\)分裂的。第三，证明具有平凡典范丛的正规拟\(F -\)分裂簇是几何正规的。最后，在奇点解消假设下，证明使得\(mp^eK_X\)对与\(p\)互素的\(m\)是卡蒂尔除子的拟\(F^e -\)纯正规簇\(X\)是对数典范的。
英文摘要
We solve certain questions related to the geometry and singularities of quasi-$F$-split varieties with trivial canonical bundle. First, we prove that regular quasi-$F^{\infty}$-split varieties are not geometrically uniruled (this generalizes and significantly simplifies the earlier results of Patakfalvi and Zdanowicz) and have geometrically canonical singularities. Second, we show that there exist quasi-$F$-split surfaces with trivial canonical bundle which are not quasi-$F^{\infty}$-split, answering negatively a question raised by Kawakami, Takamatsu, Tanaka, Witaszek, Yobuko and Yoshikawa. Third, we show that normal quasi-$F$-split varieties with trivial canonical bundle are geometrically normal (this extends a result of Kawakami, Takamatsu and Yoshikawa), and finally we prove that quasi-$F^e$-pure normal varieties $X$ such that $mp^eK_X$ is Cartier for $m$ coprime to $p$ are log canonical, under a resolution of singularities hypothesis.
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