7 problems

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 compact complex manifold and let $f:M\to M$ be a holomorphic Anosov diffeomorphism. A complex infra-nilmanifold is a quotient of a complex nilmanifold by a finite grou…

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…

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…