28 problems

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…

Segal's conjecture. Analogues of Segal's theorem should hold for holomorphic maps from the Riemann sphere $S^2$, or more generally from any compact Riemann surface, to various comp…

Let $X$ be the image of a surjective holomorphic map from $\mathbb{P}^n$. Remmert–Van de Ven conjecture. Then $X$ is biholomorphic to $\mathbb{P}^n$. This conjecture was proved by…

CR transversality and nonvanishing Jacobian conjecture. The following are equivalent:

Connected-fibres conjecture. Connectedness of the pre-images alone should imply that $F$ is injective.

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 $B(g)$…

Conjecture. Theorem and Corollary remain true when $Y$ is a compact Kähler manifold of nonnegative Kodaira dimension.

Subelliptic submersion Oka conjecture. A holomorphic map which is both a subelliptic submersion and a topological fibration is Oka.

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…

Cohen–Jones–Segal's stability conjecture. The stability theorem should hold if and only if the evaluation map $E$ has the required rational homotopy property. The supplied statemen…

Let $n\geq4$ and $m\geq3$. Let $\operatorname{UConf}_{n}(\mathbb{C})$ be the configuration space of $n$ unordered points in $\mathbb{C}$, and let…

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…

Let $M$ and $N$ be Kobayashi hyperbolic manifolds, and let $F:M\to N$ be an isometry with respect to the Kobayashi metric or distance. Assume that $M$ and $N$ are not Cartesian pro…

Let $\operatorname{Conf}_n\mathbb{C}$ denote the unordered configuration space of $n$ distinct points in $\mathbb{C}$. A holomorphic map is considered up to affine equivalence, whe…

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 $\mathcal{F}$ be an infinite set of $p$-tuples $f=(f_1,\ldots,f_p)$ of nowhere vanishing holomorphic functions on the disc $\mathbb D$ satisfying the identity referenced in the…

Radial-extension conjecture. Under these conditions, a Denjoy–Wolff point exists in $\overline{U}$ for $f$.

Twisted-holomorphic-curve conjecture. All such exotic vortex solutions on constant-curvature backgrounds arise from flat $PSU(N_c+1)$ connections and twisted holomorphic maps into…

Gauged-holomorphic-map conjecture. All such exotic vortex solutions arise from gauged holomorphic maps into complex space forms of dimension $N$.

Let $(X,x)$ and $(Y,y)$ be complex analytic germs, and let $F:(X,x)\to(Y,y)$ be a holomorphic map germ. A map is subflat when it satisfies Huckleberry's algebraic condition referre…

Gromov's conjectures. The following assertions hold:

Rigidity conjecture. Any such map $H$ is a holomorphic isometry, and hence is equivalent to one of the maps in Theorem 1.

Let $f$ be the holomorphic self-map of $\mathbb{C}^2$ considered in the paper, with degenerate characteristic direction $[1:1]$. Domain-of-attraction conjecture. There is a non-emp…

Let $M_1subset \mathbf C^n$ and $M_2\subset \mathbf C^N$ be connected Levi non-degenerate real analytic hypersurfaces with the same signature $0<\ell\le\frac{n-1}{2}$, where…