43 problems

Let $\mathcal{F}$ be a two-dimensional holomorphic foliation on $\mathbb{P}^{3}$. Brunella's conjecture. The foliation $\mathcal{F}$ either admits an invariant algebraic surface or…

Let $\mathcal{F}$ be a generic holomorphic foliation on the complex projective plane $\mathbb{CP}^2$. A leaf of $\mathcal{F}$ is a Riemann surface, and a topological cylinder is a…

Let $\mathcal F$ be a holomorphic foliation of $\mathbb{C}\mathbb{P}^2$, and let $L_1$ and $L_2$ be non-invariant algebraic lines. For a holonomy germ … write $S$ for the countable…

Rationality conjecture. If the polynomials $\varphi_i$ have rational coefficients, then

Topological invariance conjecture. The foliation $\mathcal{F}_1$ admits two holomorphic first integrals if and only if $\mathcal{F}_2$ does.

Let $X$ be a connected compact Kähler manifold, and let $Y$ be a nonsingular hypersurface of $X$ whose normal bundle $N_{Y/X}=[Y]|_Y$ is unitary flat. Here $[Y]$ denotes the holomo…

Generalized Brunella's conjecture. If $n\geq 2r+1$, then every leaf of ${\mathscr{F}}$ accumulates to the singular set ${\rm Sing}({\mathscr{F}})$.

Let $\mathcal{F}$ be a germ of a foliation in $(\mathbb{C}^{2},0)$ with a finite number of separatrices, meaning a finite number of analytic invariant curves through the origin. Th…

A holomorphic foliation by Riemann surfaces in $\mathbb P^k$ is called generic in the sense of belonging to the generic class under consideration. Ilyashenko's conjecture. A generi…

Bounded-height conjecture. There exists a bound for $h(\mathcal{F})$ depending only on the degree of $\mathcal{F}$.

McQuillan's classification conjecture. If $\operatorname{kod}(\mathcal{F})<0$, then $\mathcal{F}$ is a rational fibration or a Hilbert modular foliation.

Let $f,g\in\mathcal O_{\mathbb C^2,0}$ be non-trivial, coprime germs at the origin, and define the Milnor number of the pair by … For the pencil $\alpha f+\beta g$, let … be its Mi…

Touzet's fibration conjecture. The foliation $\mathcal{F}$ is induced by a fibration $X\rightarrow Y$ onto a projective manifold.

Let $\mathcal{F}$ be a foliation on $\mathbb{P}^2_{\mathbb{C}}$ that is logarithmic, non-dicritical, and quasihomogeneous. Suppose that $\mathcal{F}$ has a unique singular point…

A foliation in the stability component of the Jouanolou foliation of degree $d$ on $\mathbb P^2$ is a holomorphic foliation with hyperbolic singularities and no foliated cycle; its…

Low-dimensional foliation singular-locus conjecture. The stated dimension bound should hold for every such foliation.

Druel's conjecture. The stated dimension bound holds for the singular locus of every such foliation.

Let $\mathscr{F}$ be a convex prefolliations of degree $d\geq 3$ on $\mathbb{P}^{2}_{\mathbb{C}}$. Its Legendre web is the $d$-web denoted by $\mathrm{Leg}\mathscr{F}$. Flatness co…

Let $\mathcal{F}$ be a codimension-one foliation on a projective manifold of dimension $n\geq 3$. Its normal bundle is the quotient of the tangent bundle by the tangent bundle of t…

Product-cover conjecture. There exist possibly noncompact Kähler manifolds $\mathsf{Y}$ and $\mathsf{Z}$, with $\omega_\mathsf{Y}$ trivial, and a covering map

Foliation splitting conjecture. There exists a foliation $\mathcal{G}$ on $\mathsf{X}$ such that

Cohomological Kähler foliation conjecture. Every regular foliation on a compact Kähler manifold is cohomologically Kähler.

Cerveau–Lins Neto conjecture. If the transverse action of $f\in\operatorname{End}(X,\mathcal F)$ is infinite, then $\mathcal F$ is virtually transversely additive.

Let $X$ be a compact Kähler manifold whose canonical bundle $K_X$ is pseudo-effective, and let $\mathcal{F}\subset T_X$ be a holomorphic foliation with $c_1(\mathcal{F})=0$. A foli…

Let $\mathcal{H}$ be a homogeneous convex foliation of degree $d\geq2$ on $\mathbb{P}^{2}_{\mathbb{C}}$, and let $s\in\ell_\infty$ be a non-radial singularity of $\mathcal{H}$. For…