15 problems

Let $X$ be a compact complex manifold. A balanced metric is a Hermitian metric whose fundamental form $omega$ satisfies $domega^{n-1}=0$, and a pluriclosed metric is a Hermitian me…

Let $M=\Gamma\backslash G$ be a nilmanifold, where $G$ is a nilpotent Lie group, and let $J$ be a left-invariant complex structure on $M$. Suppose that $M$ admits a locally conform…

Let $G$ be a completely solvable Lie group, let $\Gamma$ be a lattice in $G$, and let $M=\Gamma\backslash G$ be a compact solvmanifold of completely solvable type. Suppose that $M$…

Generalized Alekseevskii conjecture. Every expanding homogeneous Ricci soliton is isometric to a simply-connected solvmanifold.

Solvmanifold deformation conjecture. (i) Every left-invariant complex structure on $G$ is Stein, respectively biholomorphic to ${\bf C}^n$ in the nilpotent case. (ii) Every small d…

Standardness conjecture. Solvmanifolds exhaust the class of noncompact homogeneous Einstein manifolds.

Flat tori conjecture. Then $\Gamma\backslash G$ is diffeomorphic to a flat torus.

Let $X$ be a compact complex manifold of complex dimension greater than $2$. Among balanced, pluriclosed, and locally conformally Kähler (LCK) metrics, suppose that $X$ admits two…

Let $M$ be a compact complex manifold. An SKT metric is a Hermitian metric whose Bismut torsion tensor is $d$-closed, equivalently satisfying $\partial\overline{\partial}\omega=0$,…

Hermitian space-form conjecture. If the Chern connection of $M$ has constant holomorphic sectional curvature, then $M$ must be either Kähler, hence a complex space form, or Chern f…

Let $M$ be a compact completely solvable manifold, and let $\omega$ be a symplectic structure on $M$. The Hard Lefschetz invariance conjecture. If $\omega$ satisfies the Hard Lefsc…

Let $M=G/\Gamma$ be a compact pseudo-Riemannian homogeneous Einstein solvmanifold, where $G$ is a connected solvable Lie group and $\Gamma$ is a lattice in $G$. Equivalently, $M$ i…

An expanding homogeneous Ricci soliton is a homogeneous Ricci soliton with soliton constant $\lambda<0$. A solvmanifold is a homogeneous space associated with a solvable Lie group;…

A solvmanifold is a quotient $G/\Gamma$ of a solvable Lie group $G$ by a discrete subgroup $\Gamma$. A Lie group is at most 3-step solvable when its derived series reaches the triv…

Let $M^3$ be the closed $3$-dimensional solvmanifold described in the source, let $\pi=\pi_1M^3$, and let $\mathcal P$ be the bounding class of polynomial functions. Let $E\pi$ be…