22 problems

Kuznetsov's conjecture. The fourfold $X$ is rational if and only if … for an ordinary, untwisted projective K3 surface $S$.

Let $X$ be a smooth cubic fourfold, and let $\mathcal{C}_d$ denote the Hassett divisor parametrizing special cubic fourfolds with a labelling of discriminant $d$. The numerical con…

Let $X \subset \mathbb{P}^5$ be a smooth cubic fourfold with Kuznetsov component $\mathcal{A}_X$, and let $\widetilde{X} \to X$ be a birational modification obtained by a sequence…

A variety is said to be of CM type when it has the complex multiplication property described in the surrounding theory. Piatetski-Shapiro–Shafarevich conjecture. Varieties of CM ty…

Let $X$ and $Y$ be cubic fourfolds over the complex numbers. Their Kuznetsov components are the admissible subcategories $\mathcal{A}_X$ and $\mathcal{A}_Y$ in the semiorthogonal d…

Let $Y$ and $Y'$ be cubic fourfolds. Write $\mathcal{A}_Y$ for the Kuznetsov component of $Y$, defined by … The cubic fourfolds are FM partners if there is a Fourier–Mukai equivale…

Let $X$ and $Y$ be smooth cubic fourfolds, and write $\widetilde{\mathrm{H}}(X)$ and $\widetilde{\mathrm{H}}(Y)$ for the Addington–Thomas Hodge structures associated with their Kuz…

Let $\mathcal{C}$ be the moduli space of cubic fourfolds, and let $\mathcal{C}_d$ denote a special divisor, consisting of cubic fourfolds with discriminant $d$. A discriminant $d$…

Let $X$ and $X'$ be smooth cubic fourfolds, and let $F(X)$ and $F(X')$ be their Fano varieties of lines. Birational Fano varieties conjecture. If … then $X$ and $X'$ are birational…

Let $X$ and $X'$ be smooth cubic fourfolds, and let $\mathcal{A}_X$ and $\mathcal{A}_{X'}$ denote their Kuznetsov components. Huybrechts's conjecture. If … then $X$ and $X'$ are bi…

Let $X\subset\operatorname{Gr}(2,V_6)$ be the Fano variety of lines of a smooth cubic fourfold. For a partition $\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$, let…

Rationality conjecture. A smooth cubic fourfold $X$ is rational if and only if $X\in\mathcal{C}_d$ for some $d$ satisfying ().

Let $X,X'\subset\mathbb{P}^5$ be smooth cubic fourfolds, let $\mathcal{A}_X$ and $\mathcal{A}_{X'}$ be their K3 categories, and let $\widetilde H(X,\mathbb{Z})$ and…

Let $X\subset\mathbb{P}^5$ be a smooth cubic fourfold, let $\mathcal{A}_X$ be its K3 category, and let $(S,\alpha)$ be a twisted K3 surface, with $\alpha\in\operatorname{Br}(S)$. T…

Let $X$ be a smooth cubic fourfold, let $F(X)$ be its Fano variety of lines, and let $S^{[2]}$ be the Hilbert scheme of length-two subschemes of a K3 surface $S$. Galkin–Shinder's…

Castravet's conjecture. The maximally rationally connected quotient of ${\mathcal C}$ is birationally equivalent to the twisted intermediate Jacobian of $Y$.

Let $Y$ be a cubic fourfold, let $Z$ be its associated symplectic eightfold, and suppose that $Y\in\mathcal C_d$ for some $d$ satisfying the relevant condition. Then $Z$ is isomorp…

Optimal filtration bound conjecture. For every nonsingular connected rational curve $C\subset Y$ of degree $e\geq5$,

Let $\mathcal{C}_d$ be the special cubic-fourfold divisor whose discriminant $d$ satisfies … For a general cubic fourfold $X\in\mathcal{C}_d$, say that its motive is associated to…

Countability conjecture. There are at most countably many integral polarized Hodge structures $\mathcal{T}_{S_b}$, arising from the fibers $S_b$ of $f$, which are decomposable.

Let $X\subset\mathbb P^5$ be a smooth cubic fourfold, let $\mathcal A_X$ be its K3 category, and let $\operatorname{Aut}_{\rm s}(\mathcal A_X)$ denote the group of symplectic autoe…

Let $S$ be a smooth projective surface over the complex numbers. Write $T_S$ for the integral polarized Hodge structure on the transcendental part of $H^2(S,\mathbb Z)$. Nondecompo…