Positive projectively flat manifolds are locally conformally flat-K\"ahler Hopf manifolds

We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-K\"ahler metrics on Hopf manifolds, explicitly characterized by Vaisman. Finally, we review the properties of zero projectively flat metrics. As applications, we refine a list of possible projectively flat metrics by Li, Yau, and Zheng; moreover we prove that projectively flat astheno-K\"ahler metrics are in fact K\"ahler and globally conformally flat.Comment: 11 pages; v2: section 2 modifie

The Calabi's metric for the space of Kaehler metrics

Given any closed Kaehler manifold we define, following an idea by Eugenio Calabi, a Riemannian metric on the space of Kaehler metrics regarded as an infinite dimensional manifold. We prove several geometrical features of the resulting space, some of which we think were already known to Calabi. In particular, the space is a portion of an infinite dimensional sphere and admits explicit unique smooth solutions for the Cauchy and the Dirichlet problems for the geodesic equation.Comment: 30 page

Moduli space of families of positive (n−1)(n-1)-weights

We show the geometrical structure of the moduli space of positive-weighted trees with $n$ labels $1,\ldots , n$ which realize the same family of positive $(n-1)$-weights and we characterize them as a family of positive multi-weights.Comment: 10 pages; misprints correcte

On Calabi extremal Kaehler-Ricci solitons

In this note we give a characterization of Kaehler metrics which are both Calabi extremal and Kaehler-Ricci solitons in terms of complex Hessians and the Riemann curvature tensor. We apply it to prove that, under the assumption of positivity of the holomorphic sectional curvature, these metrics are Einstein.Comment: 8 pages, comments and feedbacks are welcom

Toric extremal Kaehler-Ricci solitons are Kaehler-Einstein

In this short note, we prove that a Calabi extremal Kaehler-Ricci soliton on a compact toric Kaehler manifold is Einstein. This solves for the class of toric manifolds a general problem stated by the authors that they solved only under some curvature assumptions.Comment: 4 pages. Some changes in the references, modified introduction and misprints corrected. Comments are welcom

Treelike quintet systems

Let $X$ be a finite set. We give criterion to say if a system of trees ${\cal P}=\{T_i\}_i$ with leaf sets $L(T_i) \in {X \choose 5}$ can be amalgamated into a supertree, that is, if there exists a tree $T$ with $L(T)=X$ such that $T$ restricted to $L(T_i)$ is equal to $T_i$.Comment: 12 page

Cohomologies of generalized complex manifolds and nilmanifolds

We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.Comment: The previous version has been completely rewritten in joint work with the further author. Comments are welcome

On Chern-Yamabe problem

We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar curvature. In this note, we set the problem and we provide a positive answer when the expected constant Chern scalar curvature is non-positive. In particular, this includes the case when the Kodaira dimension of the manifold is non-negative. Finally, we give some remarks on the positive curvature case, showing existence in some special cases and the failure, in general, of uniqueness of the solution

Remarks on Chern-Einstein Hermitian metrics

We study some basic properties and examples of Hermitian metrics on complex manifolds whose traces of the curvature of the Chern connection are proportional to the metric itself.Comment: minor changes, to appear in Math.

On cohomological decomposition of generalized-complex structures

We study properties concerning decomposition in cohomology by means of generalized-complex structures. This notion includes the $\mathcal{C}^\infty$-pure-and-fullness introduced by Li and Zhang in the complex case and the Hard Lefschetz Condition in the symplectic case. Explicit examples on the moduli space of the Iwasawa manifold are investigated