奇特征下超奇异阿贝尔簇的Oort猜想
Oort's conjecture on supersingular abelian varieties in odd characteristic
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中文总结 AI 辅助
针对奇特征p>2，构造含a不变量g-2开稠密集的闭子簇，为Oort猜想提供独立证明，补充了Viehmann对a=1轨迹的研究。
中文摘要 AI 辅助
Oort猜想断言：对任意整数g≥2和任意素数p，超奇异轨迹$\u0394_g$中的任意几何一般成员的自同构群为{$\u00b11$}。这一结论最近由Viehmann在一般情形下证明，此前已知当p=2且g=2、3时存在反例。我们对任意g≥3构造了一个维数为g-1的闭子簇，其包含一个a不变量为g-2的开稠密子集$\u03a5$，使得$\u03a5$与$\u0394_g$的每个不可约分支相交，且当p>2时，$\u03a5$中的每个几何点的自同构群为{$\u00b11$}。这为p>2情形下的Oort猜想提供了独立证明。当g>3时，即a=g-2>1时，这补充了Viehmann研究中a=1轨迹的相关信息，阐明了自同构群与几何结构的相互作用。
英文摘要
Oort's conjecture asserts that for any $g \geq 2$ and any prime $p$, every geometric generic member in the supersingular locus $\mathcal{S}_g$ has automorphism group $\{ \pm 1 \}$. This has been proved very recently by Viehmann in full generality, with previously known counterexamples for $p=2$ and $g=2,3$. We construct, for any $g \geq 3$, a closed subvariety of dimension $g-1$ which contains an open dense subset $\mathcal{U}$ of $a$-invariant $g-2$ such that $\mathcal{U}$ meets every irreducible component of $\mathcal{S}_g$ and every geometric point in $\mathcal{U}$ has automorphism group $\{ \pm 1 \}$ if $p >2$. This gives an independent proof of Oort's conjecture for $p>2$. When $g>3$, so $a = g-2 > 1$, this provides complementary information to the $a=1$ locus investigated in Viehmann's work on how the automorphism groups interact with the geometry.
