Oort猜想的一些推广
Some generalizations of Oort's conjecture
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中文总结 AI 辅助
该研究针对素数p≥5，推广Oort猜想，证明g≥4时主极化超奇异阿贝尔簇模空间中自同构群非$\boldsymbol{\boldsymbol{\times1}}$的补集余维数至少为2，通过仿射Deligne--Lusztig簇相关命题完成证明。
中文摘要 AI 辅助
对于素数p≥5，设$\boldsymbol{\tilde{\boldsymbol{S}}}_g$是$\boldsymbol{\bar{\boldsymbol{F}}}_p$上g维主极化超奇异阿贝尔簇的模空间。我们证明以下每个轨迹都包含一个开稠密子概形，其上的主极化阿贝尔簇的自同构群为$\boldsymbol{\boldsymbol{\times1}}$：(i)当g为偶数时的某些超奇异Ekedahl--Oort层；(ii)当g≥3时，$\boldsymbol{\tilde{\boldsymbol{S}}}_g$中具有正Coxeter型非超奇异Ekedahl--Oort不变量的轨迹；(iii)当g≥4时，$\boldsymbol{\tilde{\boldsymbol{S}}}_g$中a-数至少为2的轨迹。因此，对于g≥4，$\boldsymbol{\tilde{\boldsymbol{S}}}_g$中自同构群为$\boldsymbol{\boldsymbol{\times1}}$的开轨迹的补集的余维数至少为2。这些结果确认了p≥5时的Oort猜想。我们将其归结为关于$\boldsymbol{\text{GSp}}_{2g}$的仿射Deligne--Lusztig簇的命题，并证明了$\text{GL}_{2g}$和$\text{GSO}_{4m}$的类似结果。
英文摘要
For a prime $p\geq 5$, let $\mathscr S_g$ be the moduli space over $\overline{\mathbb F}_p$ of $g$-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group $\{\pm1\}$: (i) certain supersingular Ekedahl--Oort strata when $g$ is even, (ii) the loci in $\mathscr S_g$ with non-supersingular Ekedahl--Oort invariants of positive Coxeter type when $g\geq 3$, and (iii) the locus in $\mathscr S_g$ with $a$-number at least $2$ when $g\geq 4$. Consequently, for $g\geq 4$, the complement in $\mathscr S_g$ of the open locus where the automorphism group is $\{\pm1\}$ has codimension at least $2$. These results confirm Oort's conjecture for $p\geq 5$. We reduce them to statements about affine Deligne--Lusztig varieties for $\operatorname{GSp}_{2g}$ and prove analogues of (ii) for $\operatorname{GL}_{2g}$ and $\operatorname{GSO}_{4m}$.
