10 problems

Abelian-variety entire-curve conjecture. Every non-constant holomorphic map

Pointed Brody conjecture. If the infinitesimal Kobayashi–Royden pseudometric is degenerate on $T_xX$, then there exists a non-constant holomorphic map

Let $c_X:X\to C(X)$ be an admissible representative of the core of $X$, and let $\Delta(c_X)$ be its base orbifold. Conjecture IV'H. One should have … In addition, if $(X/\Delta)$…

Let $X\in\mathcal C$ be a compact complex manifold with core $c_X:X\to C(X)$, and let $d_{C(X)}^*$ be the induced pseudometric on the core. Conjecture IVH. The pseudometric…

Let $X\in\mathcal C$, and call $X$ $H$-special when its Kobayashi pseudometric satisfies $d_X\equiv 0$. Conjecture IIIH. The manifold $X$ is special if and only if it is $H$-specia…

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…

Let $(M,h)$ be a simply-connected complete Kähler manifold satisfying … for two positive constants $A$ and $B$. Let $\mathfrak{K}$ denote the Kobayashi metric on $M$. Greene–Wu's c…

Let $d\geq 1$, let $C\subset\operatorname{\mathbb{R}}^d$ be a convex domain, and let … be its tube domain. Assume that $(\Omega,K_\Omega)$ is Gromov hyperbolic; equivalently, $C$ i…

Continuous extension conjecture. Every complex geodesic of $\Omega$ extends continuously to $\overline{\mathbb{D}}$.

Polynomial realization conjecture. If