16 problems

Ito–Lozovanu's conjecture. If

Let $(C,S)$ be a pair consisting of a nonsingular projective curve $C$ and a finite set $S\subset C$, and let $h(\nu)$ be a fixed Hilbert polynomial. Fix a type $K$ of smooth proje…

Strong saturation conjecture. The complex analytic variety $V({\bf C})$ is ${\cal L}$-primitive.

Existence-of-fibrations conjecture. There exist a resolution of singularities

Polyhedrality conjecture. The cone $\Lambda_{\rm eff}(V,{\cal L})$ is a rational finitely generated polyhedral cone.

Rationality conjecture. If

Weak saturation conjecture. The complex polarized variety $V({\bf C})$ admits an ${\cal L}$-primitive fibration having infinitely many fibers $W$ defined over $F$ which form an asy…

Let $(A,L)$ be a polarized abelian variety of dimension $g$, with degree … Here, “similar results” refers to the established singularity properties for polarized abelian varieties…

A polarized normal variety is a pair $(X,L)$ consisting of a normal variety $X$ and a polarization $L$. K-stability and uniform K-stability are the corresponding notions for polari…

Uniformizability conjecture. The stack $\mathcal{M}_h$ is uniformizable by a quasi-projective scheme over $\mathbb{Q}$. Uniformizability would allow finiteness results proved for s…

Let $(X,L)$ be a polarized manifold in the setting of Conjecture $C(3,2)$, and let a $(-1)$-plane mean the type of subvariety excluded in the source. Conjecture C(3,2). If $(X,L)$…

Let $(X,L)$ be a polarized manifold of dimension $n egeq 3$ with $operatorname{rk}langle KX,Lrangle=2$. Its Hilbert curve $Gamma$ is associated with the polarized manifold $(X,L)$,…

Let $(X,L)$ be a quasi polarized manifold of dimension $n$, meaning that $X$ is a projective manifold and $L$ is a nef and big Cartier divisor. Let $s>0$. Effective adjunction thre…

Let $(X,L)$ be a projective variety of dimension $n$ with at most terminal $\mathbb{Q}$-factorial singularities, where $L$ is a nef and big Cartier divisor; such a pair $(X,L)$ is…

Fujita–Sommese conjecture. If

Let $(X,\mathcal{L})$ be a polarized manifold of dimension $n\geq 3$. For each integer $i$ with $2\leq i\leq n-1$, let $K_X$ denote the canonical divisor, let $g_i(X,\mathcal{L})$…