11 problems

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…

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…

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

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

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 $X$ 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 $X$ be a simple compact Kähler manifold of dimension $n$, and write $q=h^1(X,\mathcal O_X)$ and $\kappa(X)$ for its irregularity and Kodaira dimension. A holomorphic $2$-form…

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

Volume-form preservation conjecture. Every automorphism $F$ of $\mathbb{C}^*\times\mathbb{C}^*$ preserves this form: