14 problems

Mok's generalized Frankel conjecture. The manifold $M$ is biholomorphic to a compact Hermitian symmetric space.

Let $S$ be an irreducible Hermitian symmetric space of the compact type. Kodaira–Spencer's rigidity conjecture. The complex manifold $S$ is rigid under deformation. This conjecture…

Let $G_{\Bbb R}/K_{\Bbb R}$ be an irreducible Hermitian symmetric space of rank $r$, let $X=\frak p^+$, and let $\Lambda$ be the set of partitions of length at most $r$. For…

Let $G_{{\Bbb R}}/K_{{\Bbb R}}$ be an irreducible Hermitian symmetric space of non-compact type, let $\frak g=\frak k\oplus\frak p^+\oplus\frak p^-$, let $K$ be the complexificatio…

Conjecture C. If $(\mathbb{C}^n,J_F^*g_F)$ admits a Kähler dual, then $(F,g_F)$ is biholomorphically isometric to a Hermitian symmetric space of compact type.

Let $M$ be a simply connected non-positively curved Kähler manifold. Suppose that its metric is locally symmetric outside a compact subset of $M$. Seshadri–Verma's conjecture. Then…

The discussion concerns compact submanifolds in an irreducible compact Hermitian symmetric space, where the main gap-rigidity theorem assumes that the submanifold is tangent to sui…

Let $\Gamma<\operatorname{SU}(1,n)$ be a lattice, with $n\geq 2$, and let $\rho:\Gamma\to\operatorname{PO}_{\mathbf R}(2,\infty)$ be a representation. Nonexistence conjecture. Ther…

Let $n\geq 2$, let $\Gamma<\operatorname{SU}(1,n)$ be a lattice, and let $\rho:\Gamma\to\operatorname{PO}_{\mathbf C}(p,\infty)$ be a maximal representation. The associated invaria…

For $G=\operatorname{SO}^*(2n)$, let a highest weight be called good when it satisfies the conditions required for the Ansatz in the preceding discussion. Good-highest-weight conje…

Let $\Omega$ be a flag manifold, let $p\in\Omega$, let $V_p$ be the maximal domain of definition of the diastasis relative to $p$, and let $\operatorname{Cut}_p$ be the cut locus o…

Let $L$ be a compact connected minimal Lagrangian submanifold of an irreducible Hermitian symmetric space of compact type. Hamiltonian non-displaceability conjecture. The submanifo…

Holomorphicity conjecture. A tight map $\rho\colon\mathcal{X}_1\rightarrow\mathcal{X}_2$ must be holomorphic or antiholomorphic.

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…