This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem...

Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties, we prove convergence results for sequences of densities $|{\varphi }_n{|}^2=|s_N{|}^2/\left|\right|s_N{\left|\right|}_{L^2}^2$ for eigensections $s_N\in \Gamma (X,L^N)$ approaching a semiclassical ray. Here $X$ is a normal compact toric variety and $L$ is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus. Our work was motivated by and extends that of Shiffman, Tate and Zelditch....

On utilise les variétés LV-M pour construire des compactifications équivariantes $M$ d’un groupe ${\left({ℂ}^{*}\right)}^m$ avec une variété d’Albanèse nulle mais telles que l’espace des formes holomorphes fermées de degré 1 soit non nul et de dimension inférieure à ${\dim }_{ℂ}{H}^1(M,{𝒪}_M)$.

As a natural extension of bounded complete Reinhardt domains in ${ℂ}^N$ to spaces of continuous functions, continuous Reinhardt domains (CRD) are bounded open connected solid sets in commutative C*-algebras with respect to the natural ordering. We give a complete parametric description for the structure of holomorphic isomorphisms between CRDs and characterize the partial Jordan triple structures which can be associated with some CRDs. On the basis of these results, we test two conjectures concerning...

In this paper we study the degeneration of both the cohomology and the cohomotopy Frölicher spectral sequences in a special class of complex manifolds, namely the class of compact nilmanifolds endowed with a nilpotent complex structure. Whereas the cohomotopy spectral sequence is always degenerate for such a manifold, there exist many nilpotent complex structures on compact nilmanifolds for which the classical Frölicher spectral sequence does not collapse even at the second term.

We prove that the exceptional complex Lie group ${\mathrm{F}}_4$ has a transitive action on the hyperplane section of the complex Cayley plane ${𝕆\mathbb{P}}^2$. Although the result itself is not new, our proof is elementary and constructive. We use an explicit realization of the vector and spin actions of $\mathrm{Spin}(9,ℂ)\le {\mathrm{F}}_4$. Moreover, we identify the stabilizer of the ${\mathrm{F}}_4$-action as a parabolic subgroup ${\mathrm{P}}_4$ (with Levi factor ${\mathrm{B}}_3{\mathrm{T}}_1$) of the complex Lie group ${\mathrm{F}}_4$. In the real case we obtain an analogous realization of ${{\mathrm{F}}_4}^{(-20)}/\mathbb{P}$.

Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989.In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations...