12 problems

The arbitrary-real-Lie-group conjecture. The main theorem should be valid for arbitrary real Lie groups, including those with finitely or countably infinitely many connected compon…

Let $M$ and $M'$ be Kobayashi hyperbolic complex manifolds. Let $f:(M,K_M)\to (M',K_{M'})$ be a Kobayashi isometry. If $M$ and $M'$ are not biholomorphic to a product of complex ma…

A zigzag multiplicity is the multiplicity of a zigzag summand in the bicomplex of differential forms of a compact complex manifold, and a Chern number is a characteristic number ob…

Complex-curve criterion. $T_M$ admits complex curves if and only if the components of $a$ are not linearly independent over $\mathbb{Z}$.

Let $f$ be a holomorphic partially hyperbolic diffeomorphism on a complex $3$-manifold. Three-dimensional classification conjecture. Up to passing to a finite cover, $f$ is holomor…

Let $M$ be a closed complex manifold of real dimension $2n$, and let the total Betti number of $M$ mean the sum of all its Betti numbers. Sullivan's conjecture. The minimal total B…

Local homogeneity conjecture. For higher complex odd dimensions, holomorphic ${\rm SL}_2({\mathbb C})$-geometries on compact complex manifolds are always locally homogeneous.

Let $U\subset M$ be a domain with smooth boundary, and let $\mathcal{N}^U_k$ denote the number of simultaneous zeros in $U$ of $m$ independent Gaussian holomorphic sections of…

Spherical-shell or uniruled-subvariety conjecture. Either $X$ contains a $(q+1)$-dimensional spherical shell, or $X$ contains an uniruled compact subvariety of dimension $q$, or…

Let $X$ be a complex manifold with the density property, and suppose that $X$ is diffeomorphic to $\mathbb{C}^n$. Density-property conjecture. Then $X$ should be biholomorphic to…

Varolin–Tóth conjecture. Then $M$ is biholomorphic to $\mathbb{C}^n$.

Let $X$ be a closed smooth four-manifold. The lcK existence conjecture. If $X$ admits a complex structure, then $X$ also admits a locally conformally Kähler (lcK) structure, not ne…