16 problems

Fujiki-manifold conjecture. If a Fujiki manifold admits a holomorphic Cartan geometry, then it admits a flat holomorphic Cartan geometry with the same model, and the Fujiki manifol…

Let $k$ be a field of positive characteristic $p$, and let $X$ be a sufficiently general smooth proper curve of genus $g$ over $k$. For each $N\in\mathbb{Z}_{>0}$, let…

Flatness conjecture. The manifold $M$ also admits a flat holomorphic Cartan geometry with model $(G,H)$ and with corresponding injective developing map into $G/H$.

Cartan geometry model conjecture. Every compact complex simply connected manifold admitting a holomorphic Cartan geometry with model $(G,H)$ is biholomorphic to the model $G/H$.

Translation-invariance conjecture. Every flat holomorphic Cartan geometry on a complex torus is translation invariant.

Curved-orbit conjecture. Such a holonomy reduction should decompose $M$ into so-called curved orbits depending on certain $K$-orbits in

Let $M$ be a compact complex manifold with finite fundamental group bearing a holomorphic Cartan geometry. Cartan-geometry conjecture. $M$ is biholomorphic to the model of its Cart…

Let $M$ be a Calabi–Yau manifold, meaning a compact Kähler manifold with trivial canonical bundle, and suppose that $M$ admits a holomorphic Cartan geometry. McKay's conjecture. Th…

The paper studies immersed curves in Cartan geometries, using Cartan's method of moving frames to construct differential invariants and solve equivalence problems. Cartan-geometry…

Calabi–Yau Cartan-geometry conjecture. Calabi–Yau manifolds bear no holomorphic Cartan geometry.

Let $G$ be the split real form of a semisimple Lie group with parabolic subgroup $P$, and let $P\to E\to M$ be a $G/P$-geometry. Suppose that the Lie algebra $\mathfrak{g}$ has its…

Let a nondegenerate $2$-plane field be equipped with its associated morphism curvature and with the curvature of its $G_2/P_2$-geometry. Morphism-curvature conjecture. The morphism…

Let $S$ and $S'$ be Riemannian surfaces whose Gauss curvatures differ at every pair of points, and let $M$ be the space of linear isometries between tangent planes of $S$ and $S'$.…

Let an $\mathcal{O}(n)$-geometry be a Cartan geometry modelled on the corresponding model, and call it tame according to the source's definition. Tameness conjecture. An…

Let $M$ be a manifold with a normal projective connection, and suppose that its geodesics are embedded curves. Projective Blaschke conjecture. If all geodesics are closed embedded…

Let $E_0\to M_0$ and $E_1\to M_1$ be Cartan geometries, and suppose that $E_0\to E_1$ is a morphism modelled on a model epimorphism. Assume that $M_0$ is complete and $M_1$ is conn…