9 problems

Let $X$ and $Y$ be smooth varieties, let $\pi:X\longrightarrow Y$ be a morphism, and fix a dimension $k$. Consider the $\mathbb{R}$-vector space generated by pseudo-effective $k$-d…

Let $X$ be a projective irreducible holomorphic symplectic (IHS) manifold with Picard number greater than $3$. Let $\operatorname{Neg}(X)$ denote the set of negative divisors used…

Let $X$ be a projective irreducible holomorphic symplectic (IHS) manifold and let $D$ be an integral divisor on $X$ that is isotropic with respect to the Beauville–Bogomolov–Fujiki…

Fourfold Strong Conjecture remaining-case statement. This is the only unsettled case of the Strong Conjecture in dimension $4$ over $\mathbb{C}$. The claim identifies, rather than…

Let $X$ be a projective variety, let $\alpha$ be a cycle class admitting a Zariski decomposition, and write … Here $P(\alpha)$ and $N(\alpha)$ are respectively the positive and neg…

Let $X$ be a projective variety of dimension $n$. For a class in $N_k(X)$, let $\operatorname{mob}$ denote its mobility, and let $\operatorname{\overline{Eff}}_k(X)$ be the pseudo-…

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…

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

Pseudo-effective pushforward conjecture. Any pseudo-effective class $\alpha\in H_{2k}(X,\mathbf{R})$ such that $\varphi_*\alpha=0$ in $H_{2k}(Y,\mathbf{R})$ belongs to the real vec…