39 problems

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…

Belgun–Moroianu's conjecture. Every closed Weyl structure on a compact manifold is flat, exact, or has irreducible holonomy.

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…

GIP conjecture. Any compact complex manifold with vanishing first Chern class admits a Hermitian metric and a Hermitian connection with totally skew-symmetric torsion and restricte…

Consider the conformal class $[g_F]$ defined by the metric in equation, arising from the defining function … where $a_i$, $i=0,1,\ldots,6$, and $b$ are real constants. Let $H$ deno…

Soient $P,Q\in\mathbb C[x,y]$ et considérons l'équation différentielle … Supposons que ni $P$ ni $Q$ ne soient identiquement nuls et que $Q$ ne contienne aucun facteur vertical, de…

Soient $P,Q\in\mathbb C[x,y]$ et considérons l'équation différentielle … Supposons que ni $P$ ni $Q$ ne soient identiquement nuls. Considérons deux droites verticales $T_0$ et…

Normal holonomy conjecture. The normal holonomy group is isomorphic to a subgroup of $\operatorname{SO}(n)$.

Let $M$ be a compact complex manifold of real dimension $2n$ with vanishing first Chern class. A hermitian structure on $M$ is a Riemannian metric compatible with its complex struc…

Let $M$ be a manifold, let $\Omega(M,*)$ denote the based loop space, and let $C(M,*)$ be the groupoid of based loops and homotopies between them. Let $N(M,*)$ be the normal monoid…

Full holonomy conjecture. For $d\geq 3$, $n>d$, and initial conditions with $D(P(0))$ not scalar (non-regular graph), the restricted holonomy group of Laplacian dynamics is

Let $\mathrm{SU}(5)$ carry a Joyce hypercomplex structure parametrised by a matrix $\left(\begin{smallmatrix} a_1 & a_3 \\ a_2 & a_4 \end{smallmatrix}\right)$. In the complementary…

Let $(M,g)$ be a compact connected Riemannian manifold with reducible holonomy, and let $D$ be a conformal product structure on $(M,g)$ different from the Levi-Civita connection of…

Lichnerowicz–Salamon conjecture. The only complete quaternionic Kähler manifolds of positive scalar curvature are symmetric spaces of compact type.

Let $\mathcal{I}$ be a surface, and let $\Gamma_{\mathcal{I}}$ be the set of geodesic loops on it, each endowed with a holonomy transformation. Let $T\mathcal{I}$ be the tangent bu…

A Weyl connection on a compact conformal manifold is a torsion-free connection preserving the conformal class; it is closed when it is locally the Levi-Civita connection of a metri…

Let $G/H$ be a reductive Lorentzian homogeneous space of dimension $m\ge 4$, with isotropy group $H$ acting indecomposably but non-irreducibly. A plane wave is a Lorentzian manifol…

$G_2$-Goldberg conjecture. An Einstein closed $G_2$-structure $\varphi$ must be torsion-free; in particular, $g_{\varphi}$ is Ricci flat.

Belgun–Moroianu's conjecture. The Weyl structure $D$ has reducible holonomy if and only if it is flat.

Global injectivity conjecture. The generic assumption of the local injectivity theorem is unnecessary: if $\dim(M) \geq 3$ and $\dim(M)$ is odd, then $\mathcal{T}^\sharp$ is global…

Wolf-space exhaustion conjecture. Wolf spaces exhaust all complete quaternionic Kähler manifolds with positive scalar curvature.

Let $X$ be a normal compact Kähler space with klt singularities and such that $c_1(X)=0$. After replacing $X$ by a finite quasi-étale cover, suppose that its tangent sheaf has a ho…

Let $M^n$ carry the metric … and density potential $\phi=x_1+x_2+\cdots+x_n$. Compute the holonomy Lie algebra at the origin $(0,0,\ldots,0)$. Strictly upper triangular holonomy co…

Holonomy–Teichmüller conjecture. For any Delaunay angle structure $\Theta$ on a closed triangulated surface with genus $g>1$, the holonomy map is an embedding of a manifold homeomo…

Guichard-Wienhard conjecture. There exists a generalized flag variety $X$ and a compact fiber bundle $M\to S$ such that $\overline{\rho}$ is the holonomy of a locally homogeneous…