7 problems

Let a simple complex manifold mean a complex manifold satisfying the source’s informal notion of simplicity, and likewise let a simple real form mean a real form satisfying the cor…

Let $M(\mathbb C)$ be a smooth projective manifold. A real form of $M(\mathbb C)$ is a real algebraic manifold whose complexification is $M(\mathbb C)$. Finiteness conjecture. The…

Deré's conjecture. If $L$ is isomorphic to its complex conjugate, then $L$ is defined over $\mathbb{R}$; equivalently, there exists a real Lie algebra $L_{\mathbb{R}}$ such that

Let $L$ be a finite-dimensional Lie algebra over $\mathbb{C}$, and let $\overline{L}$ denote its complex conjugate. A complex Lie algebra is isomorphic to its conjugate if…

Let $\mathfrak{g}$ be a complex Lie algebra. A complex Lie algebra is defined over $\mathbb{R}$ if it is obtained by extension of scalars from a real Lie algebra. Direct-sum real-f…

Let $\mathfrak{g}$ be a complex Lie algebra, and let $\overline{\mathfrak{g}}$ denote its complex conjugate Lie algebra. A complex Lie algebra is defined over $\mathbb{R}$ if it is…

Bruhat trajectory conjecture. There exist trajectories connecting $w$ and $w'$ if and only if the corresponding elements are comparable in Bruhat order. In that case, the space of…