43 problems

Let $f:\mathfrak{X}\to S$ be an algebraic family of polarized algebraic manifolds over a quasi-projective manifold $S$, with period map $\Phi:S\to\Gamma\backslash D$ and lifted per…

Let $S$ be an irreducible algebraic variety with a non-trivial bi-algebraic structure. Let $U\subset \widetilde{S^{\mathrm{an}}}\times S^{\mathrm{an}}$ be an algebraic subvariety f…

Hassett's period-map conjecture. The image of the period map is the complement of two specific Noether–Lefschetz divisors.

Let $(X,D)$ be a log-smooth algebraic space, let $X\backslash D$ support a polarizable integral pure variation of Hodge structures, and let…

Period-preserving involution conjecture. The space $F_{1,1}$ admits a period-preserving birational involution $S\leftrightarrow S'$ such that

Kloosterman's conjecture. When

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 $X$ be a closed oriented $4$-manifold with $b_2^+(X)=1$. The period map sends a Riemannian metric to the self-dual line $H^+$ in the cone $\mathcal{P}$ of elements of positive…

Let $\widetilde P$ be the complex period homomorphism from the cohomology of the smooth affine base to $({{\bf C}}_t^{mer})^\times$, and let $P'$ be the homomorphism to $K^\times$…

Let $\mathcal V\to\Delta^*$ be a variation of mixed Hodge structure arising from a family of complex algebraic varieties, where $\Delta^*$ is a punctured disk. A period map is said…

Arithmetic quotients of bounded symmetric domains admit analogues of Satake–Baily–Borel compactifications with a kind of minimality property among compactifications. Griffiths's co…

Let $(X,D)$ be a log-smooth algebraic space supporting a polarizable integral pure variation of Hodge structures with unipotent local monodromy. Let $L_X$ be the Schmid extension t…

Yu–Zheng conjecture. Under suitable conditions, the period map should be an open embedding.

Griffiths conjecture. The period image $\mathlcal P$ is quasi-projective and $\varphi$ is induced by an algebraic morphism; $L$ is ample on $\mathlcal P$; and there exists a Baily–…

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 $\mathcal{M}$ be the moduli space of I-surfaces, let $\widetilde{\mathcal{M}}$ be the natural blowup of the moduli space along the boundary, let…

Let $\mathcal{M}$ be the moduli space of I-surfaces, let $\widetilde{\mathcal{M}}$ be its natural blowup along the boundary strata, and let…

Let Q-functions be horizontal sections of regular flat connections defined over $\overline{\mathbb Q}$ with quasiunipotent monodromy, and let the associated structure be the struct…

Let $\mathcal{M}$ be a smooth quasi-projective complex variety with universal covering $\mathcal{M}^u$. Let $\mathbb{D}$ be a Hermitian symmetric domain of noncompact type, and let…

Let $S$ be a parameter space with compactification $\overline{S}$, let $\Phi:S\to\wp$ be a period map with image $\wp=\Phi(S)$, and let $\Phi^*:\overline{S}\to\wp^*$ be a proper, c…

Let $D$ be a period domain and let $\Gamma$ be a discrete subgroup acting on $D$. A horizontal completion of the period map is a compactification extending the period map to the bo…

Toroidal compactification conjecture. There is a morphism

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…

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…