38 problems

Let $X$ be a compact complex manifold of complex dimension $n$, and let $L$ be an ample holomorphic line bundle on $X$. Fujita's conjecture. The line bundle…

Let $A$ be an abelian variety defined over an algebraically closed field ${\mathbb K}$. Given an integer $p \geq 0$, an ample line bundle $L$ on $A$, and an integer $m$, the Green–…

Hyperkähler SYZ conjecture. The line bundle $L$ is semiample.

Lagrangian conjecture. If

Rational Lagrangian fibration conjecture. One has

Let $X$ be a fake projective plane, and let $L$ be a cubic root of its canonical bundle, so that $L^{\otimes 3}\cong K_X$, assuming such a root exists. Acyclicity conjecture. The l…

Let $X$ be a smooth complex variety of dimension $n\ge 2$. Let $L$ be an ample line bundle on $X$, spanned by a subspace $V\subseteq H^0(L)$ with $\dim(|V|)\ge n+1$. Let…

Let $L$ be the line bundle in Grauert's example, a projective surface example discussed by Filip and Tosatti. Filip–Tosatti's conjecture. The line bundle $L$ is nef and big, althou…

Let $p:\mathcal X\to\mathbb D$ be a nonsingular, proper family of Calabi-Yau Kähler manifolds with $K_X\equiv 0$, where $X$ is the central fiber. Let $L\to\mathcal X$ be a holomorp…

Let $p:\mathcal X\to \mathbb D$ be a smooth, proper family of Calabi-Yau manifolds, and let $L\to\mathcal X$ be a line bundle whose restriction to the central fiber $X$ is globally…

Existence conjecture. Such a line bundle $\mathcal{J}$ always exists for every $\mathcal{C}_{0,n}$, although it need not be unique.

Let $(X,D)$ be a log-smooth algebraic space supporting a polarizable integral pure variation of Hodge structures with unipotent local monodromy. Let $L_X$ be the Schmid extension t…

Let $\alpha\colon C\to\mathbb{P}^1$ be a degree $k$ primitive cover with splitting type $\vec{a}$, and let $L$ be a line bundle on $C$ with splitting type $\vec{e}$. O-L conjecture…

Let $X$ be a projective Calabi–Yau manifold, meaning a compact Kähler manifold with $c_1(K_X)=0$ in $H^{1,1}(X,\mathbb{R})$, and let $L\to X$ be a nef line bundle. Numerical abunda…

Let $X$ be a projective manifold with $K_X$ pseudoeffective, and let $L\to X$ be a nef line bundle such that … is nef for some $m\geqslant 1$. Lazić–Peternell's generalized abundan…

Let $X$ be the compact Kähler manifold under consideration, and let $L$ be a holomorphic line bundle on $X$. A metric solving the deformed Hermitian–Yang–Mills equation is a metric…

Continuity of envelopes conjecture. Continuity of envelopes holds for every semiample $\mathbb{Q}$-line bundle $L$ on $X$: for every $\varphi\in C^0(L)$, $P(\varphi)=Q(\varphi)$ is…

Let $X$ be the underlying projective scheme and let $L$ be an invertible $\mathcal O_X$-module. Assume that $L$ is big. Slope-boundedness conjecture. Any such $L$ is slope-bounded.…

Ein–Lazarsfeld conjecture. There exists a countable union $\mathcal{B}\varsubsetneq X$ of proper subvarieties such that

Pareschi's syzygy conjecture. For every $n\geqslant p+3-m$, the line bundle $A^n$ satisfies property $(N_p)$.

Few-zeros conjecture. For “most” ample line bundles $L$, every section of $L^m$ has at least $g(C)$ zeros for every $m\geq 1$.

Sufficient positivity conjecture. Every sufficiently ample line bundle on a projective scheme satisfies property $\mathsf{QR}(3)$.

The H-trivial line-bundle and Picard-region conjecture. There are infinitely many H-trivial line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ if and only if there exists a nonzero ele…

The H-trivial line-bundle and piecewise-linear-function conjecture. There are infinitely many H-trivial line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ if and only if there is a…

Let $\pi:\mathcal{X}\rightarrow\Delta$ be the holomorphic family under consideration, with fibers $X_t$. A holomorphic line bundle on a compact complex manifold is big when its Iit…