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

Question

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 $X?$

Answer 1

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.