A theorem of Clemmens and Griffiths states that a smooth hypesurface in $\mathbb CP^4$ of degree three is not rational. I would like to know if nevertheless it is diffeomorphic (as a smooth real $6$-dimensional manifold) to a rational complex three-dimensional variety?
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Let me work out the idea of Ulrich, and deduce that the answer to the question is negative, namely a cubic three-fold is not diffeomorphic to any rational variety. The idea of Urlich is that if a rational threefold is diffeomorpic to a cubic then it is a Fano with $Pic=\mathbb Z$. So we just have to check that cubic is not diffeomorphic to any other type of three dimensional Fanos with $Pic=\mathbb Z$. There exist $17$ families of Fano threefolds with $Pic=\mathbb Z$ ($\mathbb CP^3$, quadric, five Fanos of index two including cubics, and $10$ Fanos of index $1$). The description can be found here one page two: http://www.math.u-psud.fr/~amerik/articles/obzor-fv.pdf Let $H$ be the hyperplane section of the cubic, then $H^3$ equals $3$, i.e., the degree of the cubic. So the cubic has the following property: if we take all classes in integral second cohomology of the cubic then their cubes are of the form $3\cdot n^3$ where $n\in \mathbb Z$. Checking the list of $17$ Fanos we see that the cubic is the only one with this property. So it is not diffeomorphic to any other Fano three fold. |
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