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 $X?$

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.
