Let $X$ be the smooth complex projective variety of complex dimension $3$. I need the examples of $X$ which have the same Betti numbers as $\mathbb{C}\mbox{P}^{3}$ but different cohomology rings over the rational numbers. Moreover, is there any complete classification of $X?$

  • $\begingroup$ This is just a remark that any threefold with same Betti numbers as $\mathbb P^3$ necessarily has the same rational cohomology ring with $\mathbb P^3$ simply because there is only one nondegenerate pairing $H^2\times H^4\to H^6$ over $\mathbb Q$. However, the pairing does not need to be unimodular (over $\mathbb Z$), for example, when $X$ is the quadric threefold, $H^4$ is generated by a line $L$ while $H^2$ is generated by a quadric surface $Q$ (hyperplane section), and $L\cdot Q=2$. What OP actually want to ask is "different cohomology ring over $\mathbb Z$". $\endgroup$
    – AG learner
    Apr 4 '20 at 3:21
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    $\begingroup$ @AGlearner: For a quadric $L \cdot Q = 1$ (because $Q$ is a hyperplane section), but $Q^2 = 2L$. $\endgroup$
    – Sasha
    Apr 4 '20 at 5:36
  • $\begingroup$ @Sasha: I see, thanks for the correction. $H^2_{\mathbb Z}\times H^4_{\mathbb Z}\to \mathbb Z$ is always unimodular by Poincare duality. It should be $H^2_{\mathbb Z}\times H^2_{\mathbb Z}\to \mathbb Z$ by cupping with Lefschetz class not unimodular, because $Q^3=2L\cdot Q=2$. $\endgroup$
    – AG learner
    Apr 4 '20 at 14:29

There are three examples besides $\mathbb{P}^3$:

  • a smooth quadric $Q^3$;
  • a smooth quintic del Pezzo threefold $V^3_5 = \mathrm{Gr}(2,5) \cap \mathbb{P}^6$;
  • a smooth prime Fano threefold of genus 12 (and degree 22) $V^{3}_{22}$.

The first two are rigid (do not deform); the last deforms in a 6-dimensional family.

EDIT: See Wilson, "ON PROJECTIVE MANIFOLDS WITH THE SAME RATIONAL COHOMOLOGY AS $\mathbb{P}^4$" for the classification.

  • $\begingroup$ Dear @Sasha, thanks for your answer, but how to show $H^3(V^3_5)=0$? And is the classification a consequence of classification of Fano threefolds plus computation on $H^3$? $\endgroup$
    – AG learner
    Apr 3 '20 at 17:57
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    $\begingroup$ @AGlearner: You can construct a birational transformation to $\mathbb{P}^3$ and check how the cohomology changes in the process. $\endgroup$
    – Sasha
    Apr 3 '20 at 20:29

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