极大代数潜无理三次四重
Maximally Algebraic potentially irrational Cubic Fourfolds
摘要：
一个著名的猜想断言，一个三次四重$X$是无理的，它的超越上同调$T_X$不能实现为一个$K3$曲面的超越上同调.由于几何的三次fourfolds是错综复杂的关系存在代数$2$-循环对他们来说，这是很自然的要求最代数三次fourfolds $X$，这一猜想仍然适用。在本文中，我们表明，对于一个适当的"代数性指数" $κ_X$，存在一个独特的三次类最大化$κ_X$，没有相关的$K3$表面，即，三次四倍与Eckardt点（以前研究[LPZ 17]）。可以说，它们是最代数的潜在无理三次四重，因此是哈里斯，哈西特，库兹涅佐夫代数的一个很好的试验场。
英文摘要：
A well known conjecture asserts that a cubic fourfold $X$ whose transcendental cohomology $T_X$ can not be realized as the transcendental cohomology of a $K3$ surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic $2$-cycles on them, it is natural to ask for the most algebraic cubic fourfolds $X$ to which this conjecture is still applicable. In this paper, we show that for an appropriate `algebraicity index' $κ_X$, there exists a unique class of cubics maximizing $κ_X$, not having an associated $K3$ surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in [LPZ17]). Arguably, they are the most algebraic potentially irrational cubic fourfolds, and thus a good testing ground for the Harris, Hassett, Kuznetsov conjectures.
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