1,046 problems

DK hypothesis. K-equivalence should imply an equivalence … More generally, if $p^*K_X\geq q^*K_Y$, one expects a fully faithful embedding from $D^b\operatorname{Coh}(Y)$ into…

Let $X$ be a smooth, projective variety over $\mathbb{C}$, and write $h^{p,0}(X)$ for the Hodge number measuring global holomorphic $p$-forms. Bloch's conjecture. If … for every…

Let $(X,\Delta)$ be a Klt pair, and let $f:X\to Y$ be an algebraic fiber space, where $X$ and $Y$ are smooth projective varieties of dimensions $n$ and $m$, respectively. Let $F$ b…

Voisin's conjecture. If $\phi$ is symplectic, then

Let $(X/Z,\Delta)$ be a klt log Calabi–Yau pair over $Z$, where $X$ is $\mathbb{Q}$-factorial and $\Delta$ is an $\mathbb{R}$-boundary. Let $\mathcal{A}^e(X/Z)$ and…

Let the Calabi–Yau threefolds under consideration be those represented within the Kreuzer–Skarke database. Reid's fantasy. All Calabi–Yau threefolds in this database are birational…

Let $X$ be a Calabi–Yau variety in one of the cases under consideration, and let $\operatorname{Aut}(X)$ act on its Kähler cone. A Kawamata–Morrison conjecture. The action of…

Let $\operatorname{\overline{M}}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$. A divisor $L$ is F-nef if it has non-negative intersection with every F-curve…

Kawamata's K-equivalence conjecture. If $\phi$ is K-equivalence, then

A sextic double solid is a double cover of $\mathbb{P}^3$ branched over a sextic surface; it is factorial when its local divisor class groups are generated by Cartier divisors, and…

Let $d$ be a positive integer and let $\boldsymbol{\mathcal{X}_{d,\varepsilon}}$ denote the class of $\varepsilon$-klt Fano varieties of dimension $d$, where $\varepsilon>0$ is rat…

BAB conjecture. The set of $\epsilon$-lc Fano varieties of dimension $d$ forms a bounded family.

Coolidge–Nagata conjecture. Every rational cuspidal curve can be mapped onto a line by a Cremona transformation of the plane.

Ambro's lower semi-continuity conjecture. This function is lower semi-continuous. The conjecture concerns variation of minimal log discrepancies in families of singularities and, t…

Ambro–Kawamata conjecture. One has

A smooth projective variety is a variety that is smooth and projective. A line bundle is nef if its degree on every curve is nonnegative, and it is adjoint to an ample line bundle…

Birational invariance conjecture. The set-theoretic image $\operatorname{im}(h^r_X)$ of the Hitchin morphism is invariant under smooth birational modifications.

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let its boundary divisors be the irreducible components of…

Artin's conjecture. Birationally, there is a short list of noncommutative projective surfaces, with the generic case being a Sklyanin algebra.

Let $f:X\to Z$ be a $K$-negative contraction of terminal threefolds. A general member of the anticanonical system should satisfy an analog of the result that, for a three-dimension…

Let $X$ be a normal algebraic variety with two crepant resolutions … Here $\operatorname{D}^{\operatorname{b}}(\operatorname{coh}Y_i)$ denotes the bounded derived category of coher…

Let $X$ and $Y$ be birational Calabi-Yau manifolds, meaning smooth projective varieties with trivial first Chern class. Their quantum cohomology includes the big quantum cohomology…

Let $X$ be a fixed normal variety of dimension $n$, and let the point vary among its closed points. The minimal log discrepancy (mld) assigns an invariant to each such point. Lower…

Fix a dimension, and let the minimal log discrepancy be the invariant denoted by $\operatorname{mld}$ for singularities in that dimension. Shokurov's conjecture. The minimal log di…

Strictness conjecture. All the inclusions in this sequence are strict.