57 problems

Let \overline{X} = \overline{{\left.\raisebox{-.2em}{Gamma}\middle\backslash\raisebox{.2em}{mathbb B^n}\right.}} be a compactification as in Theorem, with $n \geq 6$. Here…

Let $k$ be a non-Archimedean complete valued field that is nontrivially valued and algebraically closed, and let $X$ be a smooth compact boundaryless $k$-analytic space. An entire…

Let $f$ be a mapping (neural network) as defined in the paper, with sufficiently high dimension $d$. A hyperbolicity violation conjecture asserts that there exists at least one bif…

Let $f:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ be a rational map of degree $d\geqslant 2$, and let $\mathrm{Rat}_{d}$ denote the space of Möbius equivalence classes of ration…

Green–Griffiths conjecture. If $V$ is of general type, then $V$ is pseudo-hyperbolic.

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 $f:\mathscr{X}\to\mathbb{D}$ be a holomorphic proper submersion from a complex manifold to the unit disk with connected fibers, and write $X_t=f^{-1}(t)$. A fiber is pseudo-Bro…

Let $X$ be a quasi-projective normal variety. A finitely generated group is semisimple if every abelian normal subgroup is finite. A quotient $\varrho:\pi_1(X)\twoheadrightarrow G$…

Let $X$ be a smooth quasi-projective variety. A variety is pseudo-Picard hyperbolic if it satisfies the punctured-disk extension property outside a proper Zariski closed subset, an…

Gauduchon-to-balanced hyperbolicity conjecture. If $X$ admits a balanced metric $\omega$ which is Gauduchon hyperbolic, then $X$ is balanced hyperbolic.

Irreducibility conjecture. For every $cepsilon>0$, there exists a nonzero negatively twisted invariant logarithmic $2$-jet differential $comega$ in this space with

Hyperbolicity conjecture. The linear system $|K_X+mL|$ is hyperbolic if $m\geq 3n+1$.

Let $X$ be a smooth quasi-projective variety. Define the special loci … as follows: $\mathrm{Sp}_{\mathrm{sab}}(X)$ is the Zariski closure of the union of images of non-constant ra…

Kähler–sG hyperbolicity conjecture.

Let $k$ be a number field and let $k^{\mathrm{alg}}$ be an algebraic closure of $k$. Let $X$ be a smooth projective geometrically irreducible variety over $k$. The properties of…

Let $Y$ be a projective variety defined over a number field $k$. Write $Y_\mathbb{C}$ for its base change to $\mathbb{C}$, and let $L$ be a number field extending $k$. A complex pr…

Let $f$ be a rational map of degree $d\geq 2$ on the Riemann sphere, and let $\mathcal{M}_f$ be its moduli space of quasiconformal conjugacy classes. A rational map is flexible Lat…

Extended Lang–Vojta conjecture. The following conditions are equivalent:

Viehweg hyperbolicity conjecture for KSB-stable families. If $f$ has maximal variation, then $V$ is of log general type.

Let $X$ be a smooth projective variety over $L$. Consider the four properties: algebraic hyperbolicity over $L$, boundedness over $L$, grouplessness over $L$, and hyperbolicity ove…

Let $X$ be a projective complex algebraic variety. Define $\,\operatorname{Sp}(X)$ as the union of the positive-dimensional integral closed subvarieties of $X$ that are not of gene…

Let $X$ be a smooth projective variety of general type. Consider the irreducible positive-dimensional subvarieties of $X$ that are not of general type. Geometric Lang's conjecture.…

Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $X/K$ be a projective variety. Say that $X$ is groupless over $K$ when it…