87 problems

Let $X$ be a complex algebraic variety of log general type. Green-Griffiths-Lang conjecture. There exists a proper closed subset $Z\subsetneq X$ such that, for every non-constant h…

Debarre's conjecture. The cotangent bundle $\Omega_X^1$ is ample.

Lang–Vojta's pseudo-algebraic-hyperbolicity conjecture. A projective integral variety $X$ over $k$ is of general type if and only if $X$ is pseudo-algebraically hyperbolic over $k$…

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…

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…

The paper considers varieties in the setting where boundedness and grouplessness are properties of a variety, with grouplessness defined as a weaker property than boundedness. Grou…

Let $X$ be a compact Kähler manifold, let $K_X$ denote its canonical bundle, and suppose that $X$ is Kobayashi hyperbolic. Kobayashi–Lang conjecture. The canonical bundle $K_X$ is…

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

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…

Orbifold Green–Griffiths–Lang conjecture. If $\mathcal{X}$ is of general type, then there exists a proper sub-orbifold $\mathcal{Z}\subset\mathcal{X}$ containing the image of every…

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…

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 $D\subset \mathbb P^2$ be a sufficiently general plane curve of degree $d\ge 5$, where sufficiently general means that $D$ lies in…

Let ${\cal C}(d_1,\ldots,d_k)$ be the space of $k$-tuples of hypersurfaces $C=(C_1,\ldots,C_k)$ in $\mathbb{P}^n$, where $\operatorname{deg}(C_i)=d_i$. Put … For $d\geq 2n+1$, let…

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 $\varrho:\pi_1(X)\twoheadrightarrow G$…

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

Let $H_1,\dots,H_c\subset\mathbb{P}^N$ be a generic choice of high-degree hypersurfaces, with $c\geqslant N/2$, and set … The cotangent bundle $T_X^*$ should be ample. Equivalently…

In the complex projective plane $bbP^2$, let $cmathcal{C} csubset cmathbb{P}^2$ be a generic algebraic curve of degree $d cgeqslant 5$. Kobayashi's logarithmic hyperbolicity conjec…

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…