17 problems

Let $X$ be a complex manifold, let $L$ be a semi-positive holomorphic line bundle on $X$, and let $Y$ be a compact Kähler submanifold such that $L|_Y$ is topologically trivial. Let…

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 $X$ 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…

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

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…

The dimension-three H-triviality conjecture. There are infinitely many $\mathrm{H}$-trivial line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ if and only if there exists…

Rational Lagrangian fibration conjecture. One has

Let $X$ be a projective manifold with torsion canonical bundle, and let $L$ be a nef line bundle on $X$. Semi-ampleness conjecture. The line bundle $L$ is numerically equivalent to…

Abundance conjecture. $\mathcal L$ is semiample, i.e. there exists a positive integer $m$ such that $\mathcal L^{\otimes m}$ is generated by global sections.

Equality of the Chern-class slant-product homomorphisms. The homomorphisms $c_1(\eta)/$ and $c_1(\xi)/$ coincide. This asserts that the analytically constructed line bundle and the…

Folk conjecture. If $(c_1(L),H^2(X;\mathbb Z))_X=\mathbb Z$, then either $L$ or $L^{-1}$ has a nonzero section. If $(c_1(L),H^2(X;\mathbb Z))_X=2\mathbb Z$, then either $L^2$ or…

Base-point-free conjecture. If $L$ is nef, then $L$ is semiample.