42 problems

gl(N) quasi-limit conjecture. For every compact quantizable Kähler manifold, the Poisson algebra of functions is a $gl(N)$ quasi-limit.

Cone duality conjecture. The dual cone of the pseudoeffective cone $\mathcal{E}$ equals the movable cone $\mathcal{M}$:

Minimal-model conjecture. Every compact Kähler manifold of non-negative Kodaira dimension should have a minimal model.

Conjecture of Tits type. One of the following assertions holds: either $G$ contains a subgroup isomorphic to $\mathbf Z*\mathbf Z$, or there is a finite-index subgroup $G_1$ of $G$…

Douglas–Reinbacher–Yau strengthened Bogomolov conjecture. The Chern classes of $E$ satisfy

Generalized connection conjecture. The conclusion of Corollary 1.6 should hold for $M$: both $E$ and its holomorphic linear connection are trivial.

Let $\pi:X\rightarrow\Delta$ be a holomorphic family of compact Kähler manifolds with fiber $X_t$, where $\Delta=\{t\in\mathbf{C}\mid |t|<1\}$. For a compact complex manifold $Y$,…

Extension conjecture. The results proved for abelian coverings of projective manifolds are also true for abelian coverings of an arbitrary compact Kähler manifold.

Let $M$ be a Kähler manifold equipped with a pre-quantum line bundle $E$, and choose a proper moment map for the relevant $S^1$-action. Let $M_{+}$, $M_{-}$, and $M_{\mathrm{red}}$…

Let $X$ be a complete Kähler manifold that is elliptically connected, meaning that any two points of $X$ can be connected by a finite chain of holomorphic images of non-hyperbolic…

Classification conjecture. The manifold $X$ should be either a complex torus or an irreducible hyperkähler manifold. In particular, $K_X$ should be trivial, and the two cases shoul…

Campana–Demailly–Verbitsky conjecture. Either $X$ has a finite étale cover which is bimeromorphic to a complex torus, or $H^0(X,\Omega_X^2)$ is generated by a holomorphic $2$-form…

Let $n\geq 2$ be an integer. A compact Kähler manifold is a compact Kähler manifold with its holomorphic sectional curvature nonnegative, and an ample anti-canonical line bundle me…

Let $X$ be a $k$-coframed projective manifold, let $q=h^0(\Omega^1_X)$, and suppose that $q=k$. Assume also that the canonical divisor $K_X$ is nef. Pseudo-Abelian variety conjectu…

Let $X$ be a compact Kähler space, and let $\sigma\in\operatorname{Aut}(X)$ be a wild automorphism. Zero-entropy conjecture. The automorphism $\sigma$ has zero entropy. The stateme…

Let $X$ be a compact Kähler space. An automorphism $\sigma\in\operatorname{Aut}(X)$ is wild if every non-empty analytic subset $Z\subseteq X$ satisfying $\sigma(Z)=Z$ is equal to…

Analytic canonical-extension nefness conjecture. If the canonical extension is Stein for some Kähler class, then $T_M$ is nef.

Stronger Kotschick conjecture. The following conditions are equivalent: $X$ admits a holomorphic one-form $w$ without zeros such that

Arapura–Wang's conjecture.

Campana–Winkelmann conjecture. If $Y$ satisfies BOP, then $Y$ is special and every holomorphic map from $Y$ to a Brody hyperbolic Kähler manifold is constant.

Let $(M,\omega)$ be a six-dimensional closed monotone symplectic manifold equipped with an effective Hamiltonian circle action. A Kähler realization conjecture asserts that…

Kählerness conjecture. Then $(M,\omega)$ is $S^1$-equivariantly symplectomorphic to some Kähler manifold $(X,\omega_X,J)$ with some holomorphic Hamiltonian $S^1$-action.

Let $X$ and $Y$ be connected, compact Kähler manifolds that are symplectically deformation equivalent. A complex manifold is rationally connected if any two general points can be j…

Yau's ampleness conjecture. The canonical line bundle $K_X$ is ample.

Bimeromorphic algebraic approximation conjecture. There exists a bimeromorphic map