23 problems

Mixed Hodge structure conjecture. The coordinate ring and Lie algebra of $G_M$ should have natural mixed Hodge structures such that the homomorphisms

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

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…

Popovici's conjecture. A limit of Kähler manifolds is of Fujiki class $\mathcal C$, that is, it is bimeromorphic to a Kähler manifold.

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…

Arapura–Wang's conjecture.

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.

Bimeromorphic algebraic approximation conjecture. There exists a bimeromorphic map

Projectivity conjecture. The fundamental group $cpi_1(X)$ is projective, meaning that there exists a projective manifold $M$ such that

Demailly–Peternell–Schneider conjecture. The Albanese morphism of $X$ is surjective.

Let $X$ be a compact Kähler manifold of complex dimension $n$, and write $H^*(X,\mathbb{Q})$ for its rational cohomology algebra. The rational cohomology conjecture for complex tor…

Core-map decomposition conjecture. For any such $X$,

Conjecture S. Some finite étale cover of $X$ is bimeromorphic to a complex torus.

The $CZ_n$ conjecture. Up to finite étale cover, $X$ is bimeromorphic to a torus submersion over a variety of general type.

Let $X$ be a compact Kähler manifold of dimension $n$ with generically large fundamental group, meaning that the fundamental-group image of every subvariety through a very general…

Cohomological Carlson–Toledo conjecture. This natural map is not injective.

Let $M$ be a compact Kähler manifold with infinite fundamental group $\Gamma$. Strong Carlson–Toledo conjecture. There exists a finite étale cover $M'$ of $M$ such that the Hurewic…

Let $M$ be a compact Kähler manifold of complex dimension $n$. A complex immersed solenoid in $M$ is a solenoid whose leaves are mapped to complex immersed submanifolds; such a sol…

Let $(X,\omega_X)$ be a polarized compact connected Kähler manifold of complex dimension $n$, let $(Y,\omega_Y)$ be a compact irreducible Kähler space of complex dimension…