27 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 real numbers $0\leq \alpha_0\leq\cdots\leq\alpha_{n-1}<1$ and $0\leq\beta_0\leq\cdots\leq\beta_{n-1}<1$ satisfy $\alpha_j\neq\beta_k$ for all $j,k$, and suppose that the sets…

Factorization and stability conjecture. There exists a unique $i\in\{1,\ldots,s\}$ such that the Higgs field factors through

Let a ${\mathbb Z}$-VHS arise from a family of Calabi-Yau threefolds $\widetilde{\mathfrak X}$ with $h^{2,1}=1$ over ${\mathbb P}^1\setminus\{0,1,\infty\}$. Let $r$ and $t$ be the…

Let $(X,Y)$ be a geometric mirror pair. Let $H_n^{sL}(Y)^*$ denote the dual of the subspace of homology generated by special Lagrangian $n$-cycles on $Y$, and let…

Let $(X,Y)$ be a geometric mirror pair, meaning that there is a parameter space $U$ and, for each $t\in U$, a correspondence $Z_t\subset X\times Y$, sections of the projections, Ri…

Nonabelian Hodge conjecture. The points of $NL(X_s,GL(n))$, that is, the $\mathbb Z$-variations of Hodge structure, are motivic representations on $X_s$.

Nonabelian Noether–Lefschetz algebraicity conjecture. If $S$ is a quasiprojective variety, then $NL(X/S,GL(n))$ is an algebraic variety and the morphisms

Let $Y$ be a Calabi–Yau manifold with $h^{2,0}(Y)=0$, let $\pi:\mathcal{Y}\to S$ be a family of complex structures on $Y$ with everywhere-isomorphic Kodaira–Spencer map, and let…

Let $Y$ be a Calabi–Yau manifold with $h^{2,0}(Y)=0$, and let $\pi:\mathcal{Y}\to S$ be a family of complex structures on $Y$ whose Kodaira–Spencer map is an isomorphism everywhere…

Suppose that $X$ is a Calabi–Yau manifold with $h^{2,0}(X)=0$, endowed with a complex structure satisfying the cone conjecture. Let … and let…

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…

Let $L_0,\ldots,L_{2n+1}\subset \mathbb{P}^n$ be hyperplanes in general position, let $X\to\mathbb{P}^n$ be the double cover branched along their union, and let…

Let $f:X\to Y_1\times\cdots\times Y_\ell=Y$ be the product-base family, with boundary divisors $S_i$ on $Y_i$ and $S$ on $Y$, and let $(F,\tau)_Y$ be its graded logarithmic deforma…

Let $f: X \to S$ be a family of smooth projective varieties over a number field $K \subset \mathbb{C}$, and let … be the associated variation of Hodge structure. For points…

Let $S$ be a variety, let $f:S\to S_{\Gamma,G,M}$ be bi-algebraic with $S_{\Gamma,G,M}$ an arithmetic quotient, and let $S_{\mathcal M}$ denote the base change of $S$ to…

Arakelov equality characterization. The general fiber has Kodaira dimension zero, and the family is Shimura in the following sense: $Y$ parameterizes a compactified universal famil…

Let $\mathbb V$ be a $\mathbb Z$-variation of Hodge structure defined over a number field $L\subset\mathbb C$. Let…

Let $\mathbb{H}$ be a torsion-free integral variation of Hodge structure on a smooth irreducible complex variety $B$. Let $\varphi:B^{\mathrm{an}}\to\Gamma\backslash D$ be its peri…

Let $X$ be a smooth proper algebraic variety over a formal punctured disk. The VSHS associated with the derived category of coherent sheaves on $X$ and Griffiths' VSHS are the two…

For each index $\alpha$, let $G_\alpha^1$ be a de Rham stratum and let $\big(G_\alpha^1\big)^{\rm VHS}\subset G_\alpha^1$ be the subset of polarized complex variations of Hodge str…

Let $X$ be a smooth projective variety and let $D\subset X$ be an ample divisor. Let $\mathscr{H}\to D$ be a polarized variation of Hodge structure, and let its numerical invariant…

Let $C$ be a hyperbolic Riemann surface and let $H\to C$ be a local system arising from a variation of Hodge structure. The Lyapunov exponents of $H$ determine a polygon, and the H…

Let $C$ be a Teichmüller curve of genus $g$, with Lyapunov exponents $\lambda_1\geq\cdots\geq\lambda_g$ and normalized Harder–Narasimhan numbers $w_1\geq\cdots\geq w_g$. Let $k$ be…

Hodge conjecture. When $\mathcal{H}$ comes from a family of smooth projective varieties, $\operatorname{Hdg}(\mathcal{H})$ should be a countable union of algebraic varieties.