11 problems

Strong Zilber–Pink equivalence. The following conditions are equivalent: (a) the atypical Hodge locus is a finite union of maximal atypical special subvarieties; (b) it is a strict…

Algebraicity conjecture in high level. If ${\mathbb V}$ has level at least $3$, then $\operatorname{HL}(S, {\mathbb V}^{\otimes})$ is algebraic.

Algebraicity-or-density conjecture. If $\operatorname{HL}(S, {\mathbb V}^{\otimes})_{\operatorname{typ}}$ is empty, then $\operatorname{HL}(S, {\mathbb V}^{\otimes})$ is algebraic;…

Density conjecture for the typical Hodge locus. If $\operatorname{HL}(S, {\mathbb V}^{\otimes})_{\operatorname{typ}}$ is nonempty, then it is dense in $S$ for the analytic topology…

Let $\mathcal{H}^n$ be a weight $n$ variation of Hodge structures with thin monodromy over a hyperbolic Riemann surface $C$ with singular locus $S$. Let…

Let ${\mathbb V}$ be a ${\mathbb Z}$-variation of Hodge structure on a smooth irreducible complex quasi-projective variety $S$. A special subvariety for $(S,{\mathbb V})$ is positi…

Let ${\mathbb V}$ be a ${\mathbb Z}$-variation of Hodge structure on a smooth irreducible complex quasi-projective variety $S$. A point of $S$ is a CM point for $(S,{\mathbb V})$,…

Let ${\mathbb V}$ be a ${\mathbb Z}$-variation of Hodge structure on a smooth irreducible complex quasi-projective variety $S$, with generic Hodge datum $({\pmb G},{\mathcal D})$.…

Let $U$ be an algebraic variety admitting an integral variation of Hodge structures, abbreviated $bZ$-VHS, with quasi-finite period map. Javanpeykar–Kucharczyk's conjecture. The va…

Mirror conjecture. The Laplace transform of the $G$-series for $Y$ is the solution of the Picard--Fuchs differential equation for some $1$-parameter family of $K3$ surfaces called…

Logarithmic Riemann–Hilbert correspondence conjecture. The two tensor functors from the category of geometric $p$-adic étale local systems on $X$ to the category of regular integra…