9 problems

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…

Let $A_\theta$ be the noncommutative two-torus, and let the loci $\Theta_\tau$ and ${\cal H}_{r,\tau}$ be as defined above, with $r_0$ the unique number of the form $\pm\theta+n$ f…

Yau's conjecture. For $n\geq 3$, any closed $2n$-manifold admitting an almost complex structure will also admit a complex structure.

Let $(M,J_0)$ be a Fano manifold admitting a Kähler-Ricci soliton $\omega_{KS}\in 2\pi c_1(M,J_0)$. Let $H_1$ be the operator whose negative eigenspaces define the linear subspace…

Let ${\mathfrak g}$ be a $2n$-dimensional nilpotent Lie algebra, and let $J$ be a complex structure on ${\mathfrak g}$. The ascending $J$-compatible series is defined by … for…

Let $\mathfrak{g}$ be of type $D_n$, and let $\hat{\mathcal{T}}_{\mathfrak{g}}$ be the moduli space of compatible $\mathfrak{g}$-complex structures modulo higher diffeomorphisms of…

Let $\mathfrak{g}$ be a Lie algebra admitting an embedding $\rho:\mathfrak{g}\hookrightarrow\mathfrak{sl}_m$, and construct the associated moduli space using this representation. R…

Let $M$ be a smooth manifold equipped with a Riemannian metric $g$. Consider the space of complex structures on $M$ that are orthogonal to $g$, together with its dimension and numb…

Let $J_\lambda$ be local almost complex structures on a Riemannian manifold with a $Spin(7)$-structure, determined by … where $P$ is the $Spin(7)$ cross product and the indices ran…