80 problems

A closed manifold is infra-nilmanifold if it is homeomorphic to a compact quotient of a nilpotent Lie group by a discrete group of isometries. An Anosov diffeomorphism is a diffeom…

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 $(M,J)$ be a complex manifold. A $p$-Kähler structure on $(M,J)$ is a closed, pointwise transverse $(p,p)$-form. Alessandrini–Bassanelli conjecture. If $(M,J)$ admits a $p$-Käh…

Classification conjecture. Any Anosov diffeomorphism is topologically conjugate to an Anosov automorphism of an infranilmanifold.

Let $M$ be a compact nilmanifold and let $g$ be an Einstein metric on $M$. The Einstein metric conjecture. The metric $g$ is flat, and $M$ is a torus. This concerns the possible Ei…

Let $X=\Gamma\backslash G$ be a nilmanifold with invariant complex structure $J$, and let $\mathfrak{g}_{\mathds{C}}$ be the complexification of its Lie algebra. The Dolbeault coho…

Let $X=(M,J)$ be a complex nilmanifold, where $M$ is a nilmanifold and $J$ is an invariant complex structure. A pseudo-Kähler metric on $X$ is a pseudo-Riemannian metric compatible…

Dolbeault cohomology conjecture. The Dolbeault cohomology of $M$ coincides with its left-invariant Dolbeault cohomology:

Let $N$ be a connected simply connected nilpotent Lie group of step $r$, let $\Gamma\setminus N$ be a compact nilmanifold, and let $X_1,\dots,X_k$ be constant coefficient vector fi…

All-point equidistribution conjecture. The effective equidistribution result may hold for all points on any nilmanifolds.

Invariant-form conjecture. The Dolbeault cohomology of $G/\Gamma$ can be computed in terms of invariant forms on $G$, i.e. in terms of the pair $({\mathfrak g},J)$.

Anosov diffeomorphism classification conjecture. Every Anosov diffeomorphism is topologically conjugate to a hyperbolic infra-nilmanifold automorphism.

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…

Fukaya–Yamaguchi's fibering conjecture. A finite cover of $M$ fibers over a nilmanifold with a fiber that has nonnegative Ricci curvature and finite fundamental group.

Emptiness conjecture. The classes $\mathcal{Z}_1$ and $\mathcal{Z}_4$ are both empty.

Ouyang–Pesce conjecture. All isospectral deformations of nilmanifolds must arise from the Gordon–Wilson method for constructing isospectral nilmanifolds.

Let $W=N/\Lambda$ be a connected nilmanifold, let $Y=\pi(H)$ be a connected subnilmanifold of $W$, where $H$ is a connected closed subgroup of $N$ and $\pi:N\to W$ is the quotient…

Let $(G/\Gamma,\mu_{G/\Gamma},R_\alpha)$ be an ergodic nilsystem, and let $k_1,\ldots,k_\ell\in\mathbb Z$ be nonzero and distinct. For functions…

Let $X$ be a compact complex manifold. A Hermitian metric on $X$ is balanced if $dF^{n-1}=0$, and astheno-Kähler if $partial\bar{\partial}F^{n-2}=0$, where $F$ is its fundamental f…

Nilmanifold formality conjecture. The double complex of differential forms for nilmanifolds should be quasi-isomorphic to the double complex of left-invariant differential forms.

Brown–Graham–Landman conjecture. If $S$ is not a set of $d$-nilBohr recurrence for some $d$, then $S$ is not $2$-large.

Leibman's conjecture. The orbit closure

Consider partially hyperbolic systems on nilmanifolds, with one system algebraic and satisfying a suitable irreducibility hypothesis, and suppose there is a $C^1$ conjugacy between…

Let $(X=G/\Gamma,\mu_X,T)$ be a connected nilsystem, with $T$ induced by left translation by $\tau\in G$. Write $G=G_0\Gamma$, where $G_0$ is the identity component, and write…

A nilmanifold is a compact quotient of a nilpotent Lie group by a discrete subgroup, and an invariant complex structure is a complex structure induced by left-invariant tensor fiel…