20 problems

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

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.

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

Let $\mathcal{F}$ and $\mathcal{G}$ be holomorphic foliations by curves at $(\mathbb{C}^{n},0)$, with algebraic multiplicities denoted by $\nu(\mathcal{F},0)$ and…

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 $\mathcal{H}$ 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…

Let $d\geq 2$. For a homogeneous convex foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$, let $\mathcal{CS}_d$ denote the set of Camacho–Sad indices at non-ra…

Semipositivity conjecture. The line bundle $[Y]$ admits a $C^\infty$ Hermitian metric with semi-positive curvature if and only if the pair $(Y,X)$ is of type $(\beta)$.

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

Brunella's conjecture. Every leaf of ${\mathscr{F}}$ accumulates to the singular set ${\rm Sing}({\mathscr{F}})$.

Let $X$ be a Stein manifold of dimension $n\ge 2$ equipped with a Riemannian metric. A nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces is a foliat…

Let ${\mathscr{F}}$ be a codimension one foliation on ${\mathbb{P}}^{3}$, with tangent sheaf $T_{\mathscr{F}}$. A coherent sheaf is split if it is a direct sum of line bundles. Sem…

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

Topological classification conjecture. The two foliation germs are topologically equivalent if and only if: (1) the ray configurations of their eigenvalue tuples are equivalent; an…

Let $\mathcal{F}$ be a singular holomorphic foliation in $\mathbb{C}\mathbb{P}^2$. Let $L_1$ and $L_2$ be two non-invariant projective lines, and let…

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be foliations by curves on $(\mathbb{C}^3,0)$. Suppose they are topologically equivalent and do not admit invariant surfaces over which the…

Let $\mathcal{F}$ be a holomorphic foliation of dimension one with non-degenerate singularities in the complex projective space $\mathbb{C}\mathbb{P}^{3}$, admitting a stable graph…

Let $M$ be a complex manifold and let $S \subset M$ be a subset of complex codimension at least two. A holomorphic Engel $2$-plane field is a holomorphic $2$-plane field satisfying…

Let $X$ be a compact connected complex manifold of dimension $n\ge 3$, and let ${\mathcal F}$ be a codimension one holomorphic foliation on $X$. Its normal bundle $N_{\mathcal F}$…