57 problems

Let $X$ be a compact Kähler manifold and $D=\sum_{i=1}^{N}D_i$ be a simple-normal-crossing (snc) divisor on $X$. Let $F$ be a semi-positive line bundle on $X$, meaning that it admi…

Broustet–Gongyo's conjecture. Every normal projective variety admitting a polarized endomorphism is of Calabi–Yau type. This has been proved for surfaces, smooth threefolds, and ra…

DCC of invariant Iitaka volumes. This set is a DCC set.

Good Minimal Model Conjecture. The pair $(X/Z,B)$ has a good minimal model $(Y/Z,B_Y)$ over $Z$, meaning that $(Y,B_Y)$ is a minimal model of $(X,B)$ over $Z$ and $K_Y+B_Y$ is semi…

Let $X$ be a normal $\mathbb{Q}$-factorial algebraic variety with a $\mathbb{Q}$-boundary divisor $D=\sum d_iD_i$, where $0\leq d_i\leq 1$ and the $D_i$ are prime Weil divisors. As…

Log canonical Fano Mori dream space conjecture. Then $X$ is a Mori dream space.

Gongyo's conjecture. After replacing $f$ by an iterate, the pair

Almost abelianity conjecture. The fundamental group

Effective log canonical boundary conjecture. There exists an effective $mathbb Q$-divisor $Delta$ on $X$ such that $(X,\Delta)$ is log canonical.

Let $(X,D_X)$ be a log canonical pair with standard coefficients, where … with each $D_i$ a prime divisor and $m_i\in\mathbb{Z}_{>0}\cup\{\infty\}$. The orbifold fundamental group…

Let $X$ be a normal projective variety, let $f\colon X\to X$ be a $q$-polarized endomorphism, and let $\Delta$ be an effective $\mathbb{Q}$-divisor such that $(X,\Delta)$ is an $f$…

Neighborhood nefness conjecture. Then $K_X+\Delta$ is $\pi$-nef over some open neighborhood of $W$.

Abundance conjecture. Then $K_X+\Delta$ is semiample.

Non-vanishing conjecture. Then

Analytic MMP termination conjecture. After shrinking $Y$ around $W$, there exists a finite sequence of steps of a $(K_X+\Delta)$-minimal model program over $Y$ around…

Work over the complex numbers. Let $(X,B)$ be a projective log canonical pair, where $X$ is a normal projective variety and $B$ is a boundary divisor, and let $K_X+B$ be its log ca…

DCC of Iitaka volumes. The set $\operatorname{Ivol}_\mathrm{lc}^\Gamma(d,\kappa)$ is a DCC set.

Let $(X,B)$ be a projective log canonical pair of dimension $n$, where $B$ is a nonzero reduced divisor and $K_X+B$ is ample. The volume of $K_X+B$ is the top self-intersection num…

Let $(X,\Delta)$ be a log canonical pair such that $U:= X \mathbin{\rule[0.2em]{0.67em}{0.12em}} \Delta$ is a subvariety of a torus. Let … be the tropicalization map, and let…

Finite generation conjecture. The graded $\mathcal O_Y$-algebra $R(X/Y,K_X+\Delta)$ is locally finitely generated.

Non-vanishing conjecture. If $K_X+B$ is pseudo-effective over $Z$, then

Bounded discrepancy conjecture. The divisor computing the minimal log discrepancy can be chosen with discrepancy for $(X,0)$ bounded uniformly in terms of $d$ and $\Gamma$. The cla…

Openness conjecture. The subset $\mathfrak N$ is open. The source explains that this would follow from a minimal model program for projective morphisms between complex analytic spa…

Bigness conjecture. The $\mathbb Q$-Cartier divisor $K_X+\Delta+tL$ is big for $t>2n$.

Sharp cone-length conjecture. It is conjectured that