26 problems

Wu–Zheng's conjecture. There exists a finite étale cover $X'$ of $X$ admitting a smooth fibration in complex tori

Campana's simple-manifold conjecture. The manifold $X$ is either bimeromorphic to a quotient of a compact complex torus by a finite group of automorphisms, or has even complex dime…

Let $M$ be a nef Kähler manifold of dimension $n$. Suppose there is an epimorphism … where $\Gamma$ is a torsion-free nilpotent group of rank at least $2n$. Nilpotent quotient conj…

Standard-complex-structure conjecture. Every complex structure on $T^{2m}$ is standard.

Let $\Gamma$ be a discrete subgroup of $\mathbb{C}^n$ of rank $r\leq 2n$, and consider the quotient $\mathbb{C}^n/\Gamma$. The Riemann bilinear relations are the following conditio…

Let $X$ be a torus with a geometric $N=2$ structure $(I,J)$, and let $C^{\mathrm{B}}_{I,J}(X)$ denote the category of B-branes and $\mathcal{D}\operatorname{Coh}_{I,J}(X)$ the deri…

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…

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

Characterization of torus quotient pairs. In the above setting, the following are equivalent: 1. $\mathrm{c}_{1}(K_X+\Delta)=0\in \HH2.X.{\mathbb{R}}.$ and

Greb–Kebekus conjecture. 1. $\mathrm{c}_{1}(X)=0\in \HH2.X.{\mathbb{R}}.$, and there exists a Kähler class $\alpha\in \HH2.X.{\mathbb{R}}.$ such that

Conjecture A. If $(\Omega,\Lambda)$ is a transcendental pair, then $(g_\Omega,\Lambda)$ is a Gabor frame for $L^2(\mathbb R^n)$ if and only if

Torus-covering conjecture. Such manifolds are covered by compact complex tori.

Subvariety-free-manifold conjecture. The manifold $X$ is either a simple compact complex torus or a hyperkähler manifold; in particular, $c_1(X)=0$.

For $n\geq 2$, consider the complex algebraic torus $({\mathbb C}^*)^n$ and the density property, meaning that complete holomorphic vector fields generate the relevant holomorphic…

Torus-cover conjecture. The manifold $X$ admits a finite unramified cover which is a complex torus.

Translation-invariance conjecture. Every flat holomorphic Cartan geometry on a complex torus is translation invariant.

Let $E_{\tau}:=\mathbb{C}/\Lambda_{\tau}$ be the flat torus, let $G(z\mid\tau)$ be its Green function, and define the multiple Green function … Count critical points of…

Let $X$ be a compact complex space of dimension $n\geq 2$. Let $\widetilde{c}_2(X)$ denote the second orbifold Chern class of $X$, and let a finite group action be free in codimens…

Let $M$ be a compact complex manifold homeomorphic to a torus, and suppose that $M$ admits a holomorphic $(X,G)$-structure. Complex-torus characterization conjecture. Then $M$ is b…

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…

Let $\mathbb{T}_{n}=\mathbb{C}^{n}/\Gamma$ be a complex torus, let $\pi:\mathbb{C}^{n}\to\mathbb{T}_{n}$ be the canonical projection, and let $M$ be a compact Levi flat hypersurfac…

Let $X$ be a Kähler, possibly non-projective, complex $4$-torus. Let $(X,\omega)$ be an abelian variety in the projective case, let $\beta\in{\mathcal H}_\omega$, and let $A_+$ and…

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

A compact Kähler Calabi–Yau manifold is a compact Kähler manifold with trivial canonical bundle. A holomorphic Cartan geometry is a Cartan geometry whose geometric data are holomor…