13 problems

Grothendieck's generalized Hodge conjecture. The subspace $N^{d-2}H^{2d-3}(X,\mathbb{Q})$ should be the largest rational Hodge substructure contained in

Grothendieck-Hodge conjecture. The substructure $N^iH^j(X,\mathbb{Q})$ is the largest substructure of level at most $j-2i$ contained in

Let $X$ be a variety with totally degenerate reduction, and let $N_{1,L}$ and $N'_{1,L}$ be the enriched monodromy operators defined in Section 4.3. Write…

Let $X$ be a smooth projective variety, and let $T^pH^l(X;\Q)$ be the topological filtration defined using the natural transformation from morphic cohomology to singular cohomology…

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 $Y$ be a smooth projective variety of dimension $n-c$, and let $\widetilde{N}_{c,\operatorname{cyl}}H^{n-2c}(Y,\mathbb{Q})$ denote the strong cylinder homomorphism filtration g…

Let $M$ be a strictly homotopy invariant sheaf, let $M^{(q)}$ denote its homotopy coniveau tower, and let $\ul{\pi}_0(M^{(q)})$ be the zeroth homotopy sheaf of its $q$-th stage. Co…

Let $X$ be a smooth variety, let $N^kH^{2k+1}_{\mathrm{dR}}(X)$ and $N^kH^{2k+1}(X_{\rm an},\mathbf{Z}_{\rm an}(q))_{\rm fr}$ denote the de Rham and torsion-free integral parts of…

Generalized Lefschetz standard conjecture. For every such $p,q,k$, the homomorphism $u^{q-p}_a$ is an isomorphism. The assertion is presented as a strengthened or generalized form…

Let $X$ be an abelian variety of dimension $d$ over $\mathbf{C}$, let $\widetilde{N}^1H^d(X)$ be the first niveau-filtration step, and let $G^dCH_0(X)$ be the deepest piece of the…

Let $X$ be a smooth complete variety of dimension $n$ over $\mathbb{C}$, and let $n=n_1+\cdots+n_r$ be a partition with $n_i\in\mathbb{N}_{>0}$. Property (P1) means that there exis…

Covariant weak Lefschetz conjecture. Suppose

Let $X$ be a smooth projective complex scheme, let $l\ge1$ and $p\ge0$, and define the coniveau filtration by … Here $F^pH^l(X,\mathbb C)$ is the Hodge filtration. General Hodge co…