9 problems

Let $X$ be a projective contact manifold with nef canonical divisor $K_X$. Since projective contact manifolds have $\kappa(X)=-\infty$, the abundance conjecture predicts that this…

Let $X$ be a Fano contact manifold with $b_2(X)=1$. A Fano contact manifold conjecture asserts that $X$ is a homogeneous variety which is the unique closed orbit in the projectiviz…

Let $Y$ be a Fano complex contact manifold with second Betti number $b_2=1$. For a simple Lie group $G$, let $\mathfrak{g}$ denote its Lie algebra, and let $G$ act on the projectiv…

Rational-homogeneity conjecture. The manifold $X$ should be rational-homogeneous, equivalently, there should exist an embedding

Let $X$ be a projective contact manifold, meaning a projective complex manifold whose tangent bundle fits into an exact sequence … where $L$ is a line bundle and the induced map fr…

A Fano contact manifold of Picard number one is a Fano manifold equipped with a contact structure and whose Picard number is one. An adjoint variety is the unique closed orbit of a…

Let $\Gamma\subset(M,\xi)$ be a contact divisor bounding a Liouville hypersurface in a contact manifold $(M,\xi)$ of dimension at least $5$. Assume that $\Gamma$ is considered in t…

The setting is that of the sub-Riemannian Gauss–Bonnet formula for a surface in a contact manifold, with condition $(\mathrm{Star})$ referring to the hypothesis used in Theorem 3.1…

Let $\mathbb M$ be the contact manifold considered in the paper, with curvature-dimension parameters $\rho_1$, $\rho_2$, $\rho_3$ and $\kappa$. Compactness conjecture. If … then th…