98 problems

de Cataldo–Hausel–Migliorini P=W conjecture. The diffeomorphism $\psi$ between the spaces respects these filtrations.

Let $X$ be a smooth Fano variety with a normal crossing anti-canonical complement $D=D_1+\ldots+D_r$, and let $U=X\setminus D$. Suppose that $U^\vee$ is a mirror dual open Calabi–Y…

Let $X$ be a separated finite type scheme over $\mathbb{C}$, and let $d$ denote its dimension. A 1-motive realization conjecture asserts that the mixed Hodge structures … and … ari…

Let $M$ be the formal moduli space of Landau–Ginzburg models, equipped with its $\mathbb{C}^*$-action, and let $V$ be the local system over $M$ arising from the cohomologies of the…

Let $X$ and $S$ be as in (0.6), and suppose $\d7eta \in \text{CH}^{2}(X,1)_{{\Bbb Q}}$ restricts to zero on the relevant fibers. The higher Abel–Jacobi map … is injective for any f…

Let $X$ be a projective nonsingular variety, and define its Griffiths group by … The source identifies this group conjecturally with a motivic extension group involving…

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…

Let $Li_{n_1,\ldots,n_k}(x_1,\ldots,x_k)$ be a multiple polylogarithm, and let…

Let $X$ be a complex algebraic manifold and let $d$ be a positive integer. Write $QH^k(X)$ for the quotient cohomology groups appearing in the exact sequence associated with iterat…

Let $X$ be a simply connected projective manifold, and let crossover mappings be the maps from products of Chow groups to the corresponding intermediate Jacobians arising from cycl…

Let $X$ be a proper scheme over $\mathbb C$, let $\pi:X_\bullet\to X$ be a smooth proper hypercovering, and let $H^{2p+i}(X)$ be Deligne's mixed $\mathbb Q$-Hodge structure on…

Let $X$ be a proper integral scheme over $\mathbb C$, and let $\nu^{i,l}$ be the natural map … where $(N^lH^t)^\bullet$ is induced by the coniveau filtration on the components of a…

Let $N$ be a positive integer. Let ${\cal Z}^{\cal H}_{\bullet}(\mu_N)$ be the commutative graded Hopf algebra generated by framed Hodge-Tate structures associated with multiple po…

Let ${\cal Z}^{\cal H}_{\bullet}$ be the graded Hopf algebra of framed Hodge-Tate structures generated by the Hodge multiple-zeta structures, and let…

Let $\widetilde A_n$ be the free abelian group generated by admissible pairs of simplices, let $\overline C_{2n}$ be the Grassmannian configuration group, and suppose the maps…

Let $f:X\to S$ be a projective morphism of smooth varieties, and let $\underline{{\cal U}}$ be a (linearized) pre-vnamhs over $X$. Relative sections conjecture. There is a (lineari…

Let $S$ be a base and let $\underline{{\cal U}}$ be a variation of nonabelian mixed Hodge structure over $S$, meaning a linearized pre-vnamhs whose fibers are nonabelian mixed Hodg…

Suppose $(Y,W(y))$ is the filtered object associated with a quadratic cone $Y$ at a point $y$. Quadratic-cone criterion conjecture. The result that $(Y,W(y))$ satisfies condition…

Let $X$ be a simply connected smooth projective variety. Universal nonabelian mixed Hodge structure conjecture. There is a universal morphism to a simply connected nonabelian mixed…

Let ${\cal V}$ and ${\cal Y}$ be nonabelian mixed Hodge structures, and suppose that ${\cal Y}$ is $1$-connected, meaning that its $\pi_0$ and $\pi_1$ are trivial. Internal Hom con…

Let $X$ be a projective variety, and let $\rho$ be a reductive representation of its fundamental group, assumed reductive after restriction to the fundamental group of the normaliz…

Let $X$ be a variety, let $X_B$ be the $n$-groupoid truncation of the homotopy type of $X^{\rm top}$, and fix a representation $\rho$. The Betti relative Malcev completion is…

Let $M$ be the relevant hyperbolic three-manifold, let $X=\tilde{X}(M)_0^{\mathrm{sm}}$, and let $P$ be an ideal point of $X$. Choose a local uniformizer $z$ at $P$ with…

Let $V(z)$ denote the polylogarithmic variation of mixed Hodge structure over $\mathbb{P}^1\backslash\{0,1,\infty\}$, with its fibers and extension data described by the polylogari…

Let $(Y,D)$ and $(Y',D')$ be $n$-dimensional, deformation-equivalent log Calabi–Yau pairs, where a log Calabi–Yau pair is a smooth projective variety together with an anticanonical…