61 problems

Let $V$ be an arbitrary variety over an arbitrary field $k$, let $n$ be an integer, and let $\bar{k}$ be an algebraic closure of $k$. Write $T_{\ell}$ and $T_{DR}$ for the $\ell$-a…

Mixed Hodge structure conjecture. The coordinate ring and Lie algebra of $G_M$ should have natural mixed Hodge structures such that the homomorphisms

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 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 $X$ 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 $M$ 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 $(Y,D)$ 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…

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…

Zeta-extension conjecture. The mixed Hodge structure on $H^{2g}_c(\mathcal{A}_g)$ has a subquotient isomorphic to such an extension whose class is given by a nonzero rational multi…

Let $M$ be a simply connected compact Kähler manifold, let ${\mathscr M}$ be the moduli space of algebraic structures on the manifold underlying $M$, and let ${\mathcal G}_M$ be th…

Let $f:X\longrightarrow C$ be a proper semi-stable degeneration over a smooth curve $C/k$, with $k\subset\mathbb C$. Let $X_t$ be a fiber at a complex point $t\neq s$, and write it…

de Cataldo–Hausel–Migliorini P=W conjecture. Under this identification, the perverse filtration associated with $\pi$ satisfies

Let $U$ and $U^\vee$ be mirror $n$-dimensional log Calabi–Yau varieties. For a nonsingular quasi-projective variety $V$, let … be its perverse-mixed Hodge polynomial. Mirror P=W co…

Degeneration and LMH conjecture. (1) This spectral sequence degenerates at $E_1$, and all $R^qf_*(\omega^p_{X/S})$ are locally free of finite rank as $\cO_S$-modules; hence…

Let $X_0$ be an odd-dimensional geometrically integral projective hypersurface of degree at least three over a number field, with a single nodal singularity. Let $Q$ be the excepti…

Let $F$ be a quadratic imaginary field, let $\phi$ be a Hecke character of odd weight $w$, and let $H_{\phi}$ be the pure Hodge structure associated with the motive attached to…