40 problems

Adjunction and effective adjunction conjectures. One can choose an effective representative $M_Z\geq 0$ in its $\mathbb{R}$-linear equivalence class such that $(Z,D_Z+M_Z)$ is log…

Let $k$ be a perfect field of characteristic $p\geq 7$, and let $(X,\emptyset)$ be a log canonical pair over $k$ of dimension at most $p-2$. Logarithmic Extension Theorem conjectur…

Non-vanishing conjecture. There exists an effective $\mathbb{R}$-divisor $E\geq 0$ such that

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 $f$…

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

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…

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

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…

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

Existence of Complements Conjecture. There exists a positive integer $n$, depending only on $d$ and $\Gamma$, such that every such pair $(X/Z\ni z,B)$ has an $n$-complement…

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 $X$ 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…

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.

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…

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

Log version of Serrano's conjecture. The $\mathbb Q$-Cartier divisor $K_X+\Delta+tL$ is ample for $t>2n$.

Let $f\colon (X, B_X) \to (Y,D)$ be a very basic slc-trivial fibration with structure decomposition … Here $B_Y$ is the discriminant part and $M_Y$ is the moduli part. The lc-conje…

Let $d$ be a natural number and let $\mathfrak{R}\subset [0,1]$ be a finite set of rational numbers. For a finite set $\mathfrak{R}$, write $\Phi(\mathfrak{R})$ for the associated…

Soit $(X,\Delta)$ une paire projective à singularités log canoniques, et soit son algèbre canonique … où $\left\lfloor l\Delta\right\rfloor$ est le diviseur de Weil obtenu en arron…

Let $d$ be a natural number and let $\mathfrak{R}\subset [0,1]$ be a finite set of rational numbers. For a finite set $\mathfrak{R}$, write … An $n$ complement of $K_X+B$ is a divi…

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

Non-vanishing conjecture. The stated effective representative exists.