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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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Such a rational complex three dimensional variety $X$ would be Fano since the rank of $H^2(X,\mathbb{Z})$, which is obviously invariant under diffeomorphisms, would be $1$. There is a classification of smooth three dimensional Fano varieties due to Mori and Mukai; you can probably find the answer to your question by consulting this list and comparing the topological invariants of the cubic threefold with those of the other varieties in the list. – ulrich Dec 7 '12 at 8:13
Ulrich, that is a great idea! I'll work this out now. – aglearner Dec 7 '12 at 10:43
up vote 6 down vote accepted

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:

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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The number of families of prime Fano threefolds is 17. – Sasha Dec 7 '12 at 12:41
Sasha, thank you, there was indeed a misprint in the paper I cited. – aglearner Dec 7 '12 at 16:34

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