AI 中文总结
该研究建立了极大非奇异实射影簇希尔伯特平方的Smith-Thom亏格下界，证明了维数≥2的实阿贝尔簇的希尔伯特平方非Smith-Thom极大，且亏格随维数指数增长，还得到了部分笛卡尔积希尔伯特平方的非极大性结果。
AI 中文摘要
我们建立了极大非奇异实射影簇的希尔伯特平方的Smith-Thom亏格下界，这些下界重现了已知的曲面情形，并为所有维数至少为2的情形提供了Smith-Thom极大性的新阻碍。作为应用，我们证明了任何维数至少为2的实阿贝尔簇的希尔伯特平方都不是Smith-Thom极大的，还表明极大实阿贝尔簇的希尔伯特平方的亏格随维数指数增长，同时对某些笛卡尔积的希尔伯特平方也得到了非极大性结果。
英文摘要
We establish lower bounds for the Smith--Thom deficiency of the Hilbert square of a maximal nonsingular real projective variety. These bounds recover the known surface case and provide new obstructions to Smith-Thom maximality in every dimension at least two. As applications, we prove that the Hilbert square of any real abelian variety of dimension at least two is not Smith-Thom maximal. We further show that the deficiencies of the Hilbert squares of maximal real abelian varieties grow exponentially with the dimension. We also obtain nonmaximality results for the Hilbert squares of certain Cartesian products.
Comments23 pages, 1 figure
