10 problems

Let $X$ and $Y$ be fixed smooth compact complex curves of genus at least $2$, and let the genus of $X$ be $g$. A polynomial bound conjecture. There is a polynomial function $B(g)$…

Let $X$ be a nilpotent complex algebraic variety such that $\pi_2(X)$ is a free abelian group of rank $r$. Assign to each holomorphic map $f:\mathbb{C}\mathbb{P}^1\to X$ a multiple…

Let $f: \operatorname{\mathbb{B}}^m \rightarrow \operatorname{\mathbb{B}}^M$ be a proper holomorphic map that extends to a Hölder-continuous map…

Let $f: \operatorname{\mathbb{B}}^m \rightarrow \operatorname{\mathbb{B}}^M$ be a proper holomorphic map, and let $\operatorname{\mathsf{G}}_f$ denote its symmetry group. Suppose t…

Fix cycles $c_i^X$ in the source $X=\mathbb{P}^k$ and cycles $c_i^Y$ in the target $Y=\mathbb{P}^n$, for $i=1,\ldots,l$. Let $\mathrm{Maps}_d(\mathbb{P}^k,\mathbb{P}^n)$ be the spa…

Let $D^{\mathrm{I}}_{p,q}$ denote the irreducible type-I bounded symmetric domain, and let $F:D^{\mathrm{I}}_{p,q}\to D^{\mathrm{I}}_{p',q'}$ be a proper holomorphic map, where…

Gromov's conjectures. The following assertions hold:

Let $n\ge 3$ be a positive integer. Write $K(n)$ for the largest positive integer $m$ such that … For each $k$ with $1\le k\le K(n)$, let $\mathcal{I}_k$ be the collection of integ…

Generalized Green-Griffiths-Lang conjecture. There exists an algebraic subvariety $Y\subsetneq X$ such that any non-degenerate holomorphic map

Let $f:{\mathbb C}^{p}\rightarrow X$ be a holomorphic map into a compact complex manifold $X$ of general type, meaning that its canonical bundle $K_X$ is big. Green–Griffiths conje…