9 problems

Andreatta–Wiśniewski's equality conjecture. If

Let $\varphi:X\rightarrow Z$ be a Fano–Mori contraction of a manifold of dimension at most $4$. Andreatta–Wiśniewski's conjecture. All fibers are normal, with the exception describ…

Let $X$ be a Fano manifold, let $n=\dim X$, and define the $(-K_X)$-slope of a subsheaf $\mathcal{E}\subseteq T_X$ by … The tangent bundle has $(-K_X)$-slope $\mu(T_X)=(-K_X)^n/n$.…

F-conjecture. The cone $\overline{\text{NE}}_1(\overline{\rm{M}}_{g,n})$ is generated by the cycle classes of F-curves. Equivalently, a line bundle on $\overline{\rm{M}}_{g,n}$ is…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration. Let $A$ be the affine space of curve classes having intersection $1$ with a general fiber, and let $\mathcal{P}$ be the convex hull…

Let $(X,\Delta)$ be a Kawamata log-terminal pair, and let $e^{\rm str}_{\rm alg}(X,\Delta)$ denote its algebraic stringy Euler number. Monotonicity conjecture. The algebraic string…

Let $g\colon X\dashrightarrow X^+$ be a birational morphism between $[200~\mathbb{Q}$-Gorenstein varieties, together with birational morphisms $f\colon X\to Z$ and…

Let $n>0$ be an integer and let $0<\epsilon<1$. A normal projective variety is of $\epsilon$-Fano type if there exists an effective $\mathbb{Q}$-divisor $B$ such that $(X,B)$ is an…

Width-decreasing conjecture. Let $X$ be a CP-manifold which is not a product of positive-dimensional varieties. If $\tau(X)>0$, then there exists a surjective morphism $f:X'\to X$…