15 problems

Density conjecture for the typical Hodge locus. If $\operatorname{HL}(S, {\mathbb V}^{\otimes})_{\operatorname{typ}}$ is not empty, then it is dense in $S$ for the analytic topolog…

Let $f:\mathcal{X}\to U_{n,d}$ be the universal family of smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$, and let $\mathbb{V}=(R^{n-1}f_*\mathbb{Z})_{\text{prim}}$ be the pol…

Let $(\mathbb{V},\mathscr{F}^{\bullet})$ be an integral polarized variation of Hodge structures on a smooth connected quasi-projective variety $S$. The Baldi–Klingler–Ullmo conject…

BKU atypical Hodge locus conjecture. The following conditions are equivalent: $\mathrm{HL}(S,\mathbb V)_{\mathrm{atyp}}$ is a finite union of maximal atypical special subvarieties;…

Let $k\geq 2$ be an integer. Let $X$ be a hypersurface of degree $3$ in $\mathbf{P}^{2k+1}$ containing two $k$-planes $\Pi_1$ and $\Pi_2$ whose intersection has dimension $k-3$. Fo…

Let $g\geqslant4$ be an integer, let $S\subset\mathcal A_g$ be a closed Hodge generic subvariety of dimension $q$, and let $\mathbb V$ be the induced polarizable $\mathbb Z$-VHS on…

Let $\mathbb V$ be as in Theorem mainqsimple, and let $(\mathbf M,D_M)\subsetneq(\mathbf G,D)$ be a strict Hodge sub-datum satisfying … The expected-dimension typical-point conject…

Let $\mathbb V$ be a polarizable $\mathbb Z$-VHS on a smooth, irreducible and quasi-projective variety $S$ over $\mathbb C$. The typical Hodge-locus density conjecture. If…

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 $S$ be a curve in an orthogonal modular variety, let $\mathcal{Z}(\gamma,-n)$ denote the corresponding special cycle for $\gamma\in V^{\vee}/V$ and $n\in -Q(\gamma)+\mathbb{Z}$…

Let $V$ be the parameter space of smooth hypersurfaces of degree $d\geq 2\frac{{\sf m}+2}{{\sf m}-2}$ in $\mathbb{P}^{{\sf m}+1}$, with ${\sf m}\geq 4$. A Hodge class of $Y_t$ is c…

Let $f:{\mathcal Y}\to X$ be defined over $k_0=\mathbf Q$, let $V$ be the corresponding vector bundle, and let $\alpha$ be a Hodge class in…