# Smooth complex threefolds with the same Betti numbers as $\Bbb CP^3$ but different rational cohomology rings

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 $$X?$$

• 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 $\mathbb Z$". Apr 4 '20 at 3:21
• @AGlearner: For a quadric $L \cdot Q = 1$ (because $Q$ is a hyperplane section), but $Q^2 = 2L$. Apr 4 '20 at 5:36
• @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 $Q^3=2L\cdot Q=2$. 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) $$V^{3}_{22}$$.
EDIT: See Wilson, "ON PROJECTIVE MANIFOLDS WITH THE SAME RATIONAL COHOMOLOGY AS $$\mathbb{P}^4$$" for the classification.
• 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 $H^3$? Apr 3 '20 at 17:57
• @AGlearner: You can construct a birational transformation to $\mathbb{P}^3$ and check how the cohomology changes in the process. Apr 3 '20 at 20:29