14 problems

Defect-condition conjecture. The defect condition holds in all mixed Shimura varieties. The paper notes that this condition is known for products of a pure Shimura variety and an a…

Let $M$ be a mixed Shimura variety, let $Z\subset M$ be a subvariety, and let $\delta(A)=\dim\langle A\rangle-\dim A$ denote the defect of a subvariety $A$, where…

Mixed hybrid conjecture. (1) The subvariety $S\subseteq {\mathbb A}_\Gamma$ is weakly special. (2) If, for some $s\in S$, one has $V\cap (A_s)_{\operatorname{tors}}\neq\varnothing$…

Let $X \stackrel{\iota}{\hookrightarrow} {\mathcal S}$ be a closed irreducible algebraic subvariety of a mixed Shimura variety ${\mathcal S}$. Let…

Revised relative unlikely-intersection conjecture. The intersection $Z\cap\mathcal{G}^{[>d]}$ is not Zariski dense in $Z$.

Pink's original relative Manin–Mumford conjecture. The intersection $Z\cap\mathcal{G}^{[>d]}$ is not Zariski dense in $Z$.

Relative Manin-Mumford conjecture. The stated alternative holds.

Let $Y\subset {\rm Sh}_K^0({\mathbf P},X)$ be a closed irreducible algebraic subvariety of a connected mixed Shimura variety. Classical André–Oort conjecture. If $Y$ contains a Zar…

Let $S$ be a connected mixed Shimura variety and let $Y$ be an irreducible subvariety of $S$. Mixed André–Oort conjecture. If the special points of $S$ contained in $Y$ are Zariski…

Let $S$ be a connected mixed Shimura variety associated with $(P,\mathcal{X}^+)$ and let $\mathrm{unif}\colon\mathcal{X}^+\to S$ be its uniformization. Let $\mathbscr{Z}$ be the gr…

Let $X$ be a mixed Shimura variety or a semi-abelian variety defined over $\mathbb{C}$. Let $\langle A\rangle$ denote the smallest special subvariety containing a subvariety $A$, a…

Let $\mathfrak{A}_g$ be the universal family of principally polarized abelian varieties, and let $Y$ be a subvariety of $\mathfrak{A}_g$. For $s\in\mathfrak{A}_g$, let $\Sigma$ be…

A mixed Shimura variety is an algebraic variety arising from a mixed Shimura datum; its special points and special subvarieties are the distinguished points and subvarieties define…

André–Oort–Pink conjecture. The set $\Sigma_{\mathrm{max}}(Z)$ is finite.