10 problems

Let $G$ be a complex simple Lie group and let $P\subset G$ be a maximal parabolic subgroup. Assume that $G/P$ is not a compact Hermitian symmetric space, or, if $G/P$ is a compact…

A projective connection is a Cartan geometry modelled on complex projective space $\mathbb{P}^{n}=\mathbb{P}\operatorname{SL}\left(n+1,\mathbb{C}\right)/G_0$, where $G_0$ is the st…

Abundance conjecture. It would follow from the abundance conjecture that the typical fiber is a $4$-torus.

Abundance conjecture. It would follow from the abundance conjecture that $M$, after perhaps replacing it by a finite covering space, is an abelian group scheme over a curve of genu…

Abundance conjecture. It would follow from the abundance conjecture that $M^5$ is a Calabi--Yau manifold.

Let $p,q,n$ be as in the two homogeneous pairs $(O(p+2,q),O(p,q))$ and $(O(p+1,q+1),O(p,q))$, let $t$ be the parameter, and let $(z_1,\ldots,z_n,w)$ be coordinates on…

Invariant bilinear differential pairings on parabolic geometries are constructed using tractor bundles and splitting operators, but certain geometric weights must be excluded. Excl…

Let $H$ be a distribution with co-rank $s$ and rank $r$, so that its derived distribution satisfies $[H,H]=T$ and hence $s<r(r-1)/2$. Square-root growth conjecture. The minimal pos…

Negative-first-Chern-class parabolic-geometry conjecture. Either $M$ admits no parabolic geometry, or $M$ admits a parabolic geometry modelled on a compact Hermitian symmetric spac…

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…