Fano varieties with torsion in the third cohomology group (Q6566113)

A complex projective manifold \(X\) is stably rational if for some \(d \in \mathbb N\) the product \(X \times \mathbb P^d\) is birational to a projective space. Given a Fano manifold \(X\) it is usually very difficulty to decide whether it is stably rational, but the torsion part of the integral cohomology group \(H^3(X, \mathbb Z)\) gives an obstruction. Somewhat surprisingly, the group \(H^3(X, \mathbb Z)\) is torsion-free for every smooth Fano threefold, and Beauville asked if this also holds in higher dimension. In this paper, the authors answer Beauville's question to the negative by constructing for every \(d \geq 4\) a Fano manifold of dimension \(d\) and Picard number one such that \(H^3(X, \mathbb Z) \simeq \mathbb Z/2 \mathbb Z\). In fact, they obtain these varieties by a rather explicit determinantal construction, allowing them to show that the Brauer group of \(X\) is isomorphic to \(\mathbb Z/2 \mathbb Z\). In the appendix, János Kollár gives an alternative proof of the result in dimension four by showing that \(H_2(X, \mathbb Z) = \mathbb Z + \mathbb Z/2 \mathbb Z\); this proof also gives a geometric interpretation for the two-torsion element.\N\NThese Fano manifolds also shed a new light on the two coniveau filtrations introduced by \textit{O. Benoist} and \textit{J. C. Ottem} [Duke Math. J. 170, No. 12, 2719--2753 (2021; Zbl 1478.14022)]: the group \(N^1 H^3(X, \mathbb Z) \subset H^3(X, \mathbb Z)\) consists of classes supported on some divisor of \(X\), while \(\tilde N^1 H^3(X, \mathbb Z) \subset H^3(X, \mathbb Z)\) are classes obtained as push-forward of classes via \(f: S \rightarrow X\) with \(S\) a {\em smooth} proper variety of dimension \(\dim X-1\). The quotient \N\[\NN^1 H^3(X, \mathbb Z) / \tilde N^1 H^3(X, \mathbb Z) \N\]\Nis a stable birational invariant, for the Fanos constructed in this paper it is isomorphic to \(\mathbb Z/2 \mathbb Z\).