7 problems

Let $X\subset\mathbb{P}^5$ be a smooth sextic defined by a $\sigma$-invariant equation, where $\sigma$ is the involution under consideration, and write … The invariant part…

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

Let $(A,\Theta)$ be a principally polarized abelian variety over $\mathbb C$ of dimension $g\geq 4$ with smooth theta divisor $\Theta$. Let $K(\Theta,\mathbb Q)$ be the primitive p…

Let $X$ be a projective hyper-Kähler manifold of generalized Kummer type, and let $J^3(X)$ be its intermediate Jacobian. The degree-three cohomology is of Hodge level $1$, of type…

Generalized Hodge conjecture. The two definitions agree when $k=\mathds{C}$.

Grothendieck's generalized Hodge conjecture. There exists a proper closed algebraic subset $Y\subset X$ such that the restriction map

Weakly pseudo-effective pushforward conjecture. Any weakly pseudo-effective class $\alpha\in H^{2l}(X,\mathbf{R})$ such that $\varphi_*\alpha=0$ belongs to the real vector space sp…