26 problems

Let $X$ be a smooth proper $\mathbb{C}$-scheme. Let $J^p(X)$ be its intermediate Jacobian, let $A^p(X)\subseteq J^p(X)$ be the largest abelian subvariety, and let…

Debarre–Iliev–Manivel conjecture. The general fiber of the classical period map is the disjoint union of $\mathcal{C}_m(X)$ and $M_G^X(2,1,5)$, both quotiented by involutions.

Let $X=E_1\times E_2\times E_3$ be a product of three Tate elliptic curves with parameters $q_i$ for $i=1,2,3$. Let $r$ be the rank of the space of integer triples $(n_1,n_2,n_3)$…

Let $W$ be the fourfold appearing in the degeneration $X=W_1\cup_Z W_2$, and let $J_W$ denote its intermediate Jacobian. Consider $E_8$ bundles whose restriction to each fiber is f…

Griffiths's conjecture on incidentally trivial cycles. For every $p\in\{1,\ldots,d\}$ and every $z\in I^p(X)$, there is a positive integer $N$ such that

Let $S$ be a scheme over $\mathbb{C}$, and let $\cC/S$ be a smooth and proper $S$-linear triangulated category. Assume that for every point $s\in S$, the fiber $\cC_s$ is equivalen…

Let $X$ be an index one prime Fano threefold of genus $g=6$, equivalently a Gushel–Mukai threefold. Denote its intermediate Jacobian by $J(X)$ and its Kuznetsov component by…

Jacobi-type theorem. There is some positive integer $k$ such that the topological Abel--Jacobi mapping

Let $Y$ be a quartic double solid and let $X$ be a GM threefold. Let $J(Y)$ and $J(X)$ denote their intermediate Jacobians, each with its principal polarization. Intermediate-Jacob…

Let $Y$ be a quartic double solid and let $X$ be a GM threefold. Non-birationality conjecture. Then $Y$ is not birational to $X$. The conjecture is motivated by the heuristic that…

Let $X$ be the fourfold considered in the paper, let $J^2(X)$ be its intermediate Jacobian, and let ${\mathsf L}:J^2(X)\to J^3(X)$ be induced by intersection with an ample cycle. D…

Let $X$ and $X'$ be prime Fano threefolds of index one or two, with intermediate Jacobians $J(X)$ and $J(X')$ and Kuznetsov components $\mathcal{K}u(X)$ and $\mathcal{K}u(X')$. Cat…

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

Let $X$ be a threefold satisfying the stated properties, including a decomposition of its intermediate Jacobian … where the $S_i$ are smooth projective curves of positive genus and…

Let $X$ be an $n$-dimensional connected scheme smooth and projective over ${\bf C}$, with $n\geq 2$. Write $CH_{Jac,\mathbf Q}^2(X)$ for the kernel of the Abel–Jacobi map on homolo…

Let $X$ be an $n$-dimensional connected scheme smooth and projective over ${\bf C}$. Let $M_2(X)^*$ and $i_{2n-2}:M_2(X)^*\otimes {\bf L}^{n-2}\to M(X)$ be as obtained from the dec…

Let $X$ be a general cubic fourfold, let $B=(\mathbb{P}^5)^\vee$ parametrize its hyperplane sections, and let $U=B\setminus X^\vee$ be the locus of smooth hyperplane sections. Writ…

Deligne's conjecture. The intermediate Jacobian of $X$ may also be defined over any field of positive characteristic.

Yui's conjecture. Then

Let $X$ be a smooth projective variety with $H^3(X,\mathcal{O}_X)=0$, and let its intermediate Jacobian be … Write ${\rm CH}^2(X)_{hom}$ for codimension-two cycles on $X$ that are…

Let $X$ and $Y$ be schemes as in Example (E8), and let $J(X)$ and $J(Y)$ denote their intermediate Jacobians. The E8 verepresentability and Torelli conjecture. The scheme $Y$ is ve…

Let $(J(T),\Xi)$ be the intermediate Jacobian of a general Verra threefold $T$, let $\Xi$ be its theta divisor, and let $e$ be the canonical extension class; write $\Xi_e$ for the…

Let $X'$ be a general Fano threefold of type $\mathcal{X}_{10}$, and let $(J(X'),\Xi)$ be its intermediate Jacobian with theta divisor $\Xi$. Singular-locus conjecture. The singula…

Let $\mathcal{X}_{10}$ be the moduli space of Fano threefolds of degree $10$, let $\mathcal{A}_{10}$ be the moduli space of principally polarized abelian varieties of dimension…

Let $X_{10}$ be a general Fano threefold of degree $10$, and let a quartic double solid be a smooth double cover of $\mathbb{P}^3$ branched over a quartic surface. Tyurin's conject…