The aim of the article is to give a conceptual understanding of Kontsevich’s construction of the universal element of the cohomology of the coarse moduli space of smooth algebraic curves with given genus and punctures. In a first step the author presents a toy model of tree graphs coloured by an operad $𝒫$ for which the graph complex and the universal cycle will be constructed. The universal cycle has coefficients in the operad for $\Omega \left({𝒫}^{*}\right)$-algebras with trivial differential over the (dual) cobar construction...

Generalized planar curves (A-curves) are more general analogues of F-planar curves and geodesics. In particular, several well known geometries are described by more than one affinor. The best known example is the almost quaternionic geometry. A new approach to this topic (A-structures) was started in our earlier papers. In this paper we expand the concept of A-structures to projective A-structures.

The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for $\left|1\right|$-graded parabolic geometries and for almost Grassmannian structures, in particular. As an application of two general constructions with parabolic geometries, we present an example of non-flat Grassmannian symmetric space. Next we observe there is a distinguished torsion-free...

If N is a simply connected real nilpotent Lie group with a Γ-rational complex structure, where Γ is a lattice in N, then [...] for each s, t.We study relations between invariant complex structures and Hodge numbers of compact nilmanifolds from a viewpoint of Lie algberas.

Starting by the famous paper by Kirillov, local Lie algebras of functions over smooth manifolds were studied very intensively by mathematicians and physicists. In the present paper we study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.

The postulates of macroscopic thermodynamics give us the possibility to endow the set of thermodynamic states with the structure of a riemannian manifold. Two alternatives are available: the first one is to introduce on the set of thermodynamic equilibrium states a metric induced by an embedding metric space (extrinsic approach), the second one is to introduce the stability metric (intrinsic approach). Between the two choices the second one looks more promising on the basis of its capability of...

The notion of special symplectic connections is closely related to parabolic contact geometries due to the work of M. Cahen and L. Schwachhöfer. We remind their characterization and reinterpret the result in terms of generalized Weyl connections. The aim of this paper is to provide an alternative and more explicit construction of special symplectic connections of three types from the list. This is done by pulling back an ambient linear connection from the total space of a natural scale bundle over...

We consider symmetries on filtered manifolds and we study the $\left|1\right|$-graded parabolic geometries in more details. We discuss the existence of symmetries on the homogeneous models and we conclude some simple observations on the general curved geometries.

In this paper we study the balanced metrics on some Hartogs triangles of exponent $\gamma \in {\mathbb{Z}}^{+}$, i.e., $${\Omega }_n\left(\gamma \right)=\{z=(z_1,\cdots ,z_n)\in {ℂ}^n:|z_1{|}^{1/\gamma }<|z_2|<\cdots <|z_n|<1\}$$ equipped with a natural Kähler form ${\omega }_{g\left(\mu \right)}:=\frac{1}{2}(i/\pi )\partial \overline{\partial }{\Phi }_n$ with $${\Phi }_n\left(z\right)=-{\mu }_{1}ln\left(\right|z_2{|}^{2\gamma }-\left|z_1{|}^2\right)-\sum _{i=2}^{n-1}{\mu }_{i}ln\left(\right|z_{i+1}{|}^2-\left|z_i{|}^2\right)-{\mu }_{n}ln(1-|z_n{|}^2),$$ where $\mu =({\mu }_1,\cdots ,{\mu }_n)$, ${\mu }_i>0$, depending on $n$ parameters. The purpose of this paper is threefold. First, we compute the explicit expression for the weighted Bergman kernel function for $({\Omega }_n\left(\gamma \right),g\left(\mu \right))$ and we prove that $g\left(\mu \right)$ is balanced if and only if ${\mu }_1>1$ and $\gamma {\mu }_1$ is an integer, ${\mu }_i$ are integers such that ${\mu }_i\ge 2$ for all $i=2,...,n-1$, and ${\mu }_n>1$. Second, we prove that $g\left(\mu \right)$ is Kähler-Einstein if and only if ${\mu }_1={\mu }_2=\cdots ={\mu }_n=2\lambda$, where $\lambda$ is a nonzero...