The question of rationality of smooth cubic fourfolds is a long standing open problem. At the time of writing this review, there are some known examples of cubic fourfolds which are rational but not a single example of a cubic fourfold which is non-rational. The widely shared expectation is that very general cubic fourfolds are non-rational and rational cubic fourfolds all have a \(K3\) surface associated to them in some meaningful way. There are two concrete conjectures making this expectation precise.
The first one is of Hodge theoretic nature. B. Hassett [Compos. Math. 120, No. 1, 1–23 (2000; Zbl 0956.14031)] showed that the cubic fourfolds \(X\) possessing an integral (2,2)-class \(T\in H^{2,2}(X, \mathbb Z)\) together with a Hodge isometry \[ H^2_{\mathrm{prim}}(S,\mathbb Z)(-1)\cong \langle h^2,T\rangle ^\perp\subset H^4(X,\mathbb Z) \] for some \(K3\) surface \(S\), form a countable union of irreducible divisors in the moduli space of cubic fourfolds; here \(H^2_{\mathrm{prim}}(S,\mathbb Z)(-1)\) denotes the Tate twist of the primitive cohomology of the \(K3\) surface and \(h\) is the hyperplane class on \(X\). We refer to this subset of the moduli space as the Hassett locus. In [loc. cit.] it was also shown that many of the cubic fourfolds in the Hassett locus are rational and it was asked weather the Hassett locus coincides with the locus of rational cubic fourfolds; though this was originally formulated as a question, we will refer to this as the Hassett conjecture in the following for simplicity.
The second conjecture, which is due to Kuznetsov, involves derived categories. For every cubic fourfold, there is an exceptional sequence \(\mathcal O_X\), \(\mathcal O_X(1)\), \(\mathcal O_X(2)\) in \(D(X):=D^b(\text{Coh}(X))\). Its right-orthogonal is denoted by \(\mathcal A_X\) so that there is the semi-orthogonal decomposition \[ D(X)=\langle \mathcal A_X, \mathcal O_X, \mathcal O_X(1), \mathcal O_X(2)\rangle. \] In [A. Kuznetsov, Prog. Math. 282, 219–243 (2010; Zbl 1202.14012)] it was conjectured that \(X\) is rational if and only if there is a \(K3\) surface \(S\) together with an equivalence \(\mathcal A_X\cong D(S)\); if the latter is the case, \(\mathcal A_X\) is said to be geometric.
In the paper under review, it is shown that the two conjectures are generically equivalent. The precise statement is the following. Every cubic fourfold with the property that \(\mathcal A_X\) is geometric lies in the Hassett locus. Conversely, every irreducible component of the Hassett locus contains a Zariski open dense subset consisting of cubic fourfolds with the property that \(\mathcal A_X\) is geometric. A very useful outline of the difficult proof is given in Section 1.2 of the introduction.
The authors also provide results on the algebraicity of the isometries \(H^2_{\mathrm{prim}}(S,\mathbb Z)(-1)\cong \langle h^2,T\rangle ^\perp\) for \(X\) contained in the Hassett locus as well as for Hodge isometries \(T(S)(-1)\cong T(X)\) between the transcendental lattices of general projective \(K3\) surfaces and general cubic fourfolds.