关于七维和八维三次超曲面
On cubic hypersurfaces of dimension seven and eight
摘要：
三次七重是卡-丘型的法诺流形的例子。我们研究了它们与Cartan立方、$\PP^{26}$中的$E_6$不变立方的关系.我们表明，一个通用的三次七倍$X$可以被描述为一个线性部分的Cartan三次，在许多方面。对于每个这样的“Cartan表示”，我们将$X$上的秩9向量丛与非常特殊的上同调性质相关联。特别是它允许定义自等价的非交换卡-丘三重相关的$X$由库兹涅佐夫。最后证明了Cartan三次曲面的一般八维截面是有理的。
英文摘要：
Cubic sevenfolds are examples of Fano manifolds of Calabi-Yau type. We study them in relation with the Cartan cubic, the $E_6$-invariant cubic in $\PP^{26}$. We show that a generic cubic sevenfold $X$ can be described as a linear section of the Cartan cubic, in finitely many ways. To each such "Cartan representation" we associate a rank nine vector bundle on $X$ with very special cohomological properties. In particular it allows to define auto-equivalences of the non-commutative Calabi-Yau threefold associated to $X$ by Kuznetsov. Finally we show that the generic eight dimensional section of the Cartan cubic is rational.
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