Invariant and equivariant cohomology classes on the space of Kähler forms are defined. Relations to the obstructions to the existence of Kähler-Einstein metrics and Kähler metrics of harmonic Chern forms are discussed.

We will prove the existence of closed quasigeodesics in compact Alexandrov spaces which can be approximated by Riemannian manifolds in the Lipschitz sense. By applying it, we will prove that every convex hypersurface in Euclidean spaces has a closed quasigeodesic.

We construct a generic polynomial for $\mathrm{Mod}_{2^{n+2}}$, the modular 2-group of order $2^{n+2}$, with two parameters over the $2^n$-th cyclotomic field $k$. Our construction is based on an explicit answer for linear Noether's problem. This polynomial, which has a remarkably simple expression, gives every $\mathrm{Mod}_{2^{n+2}}$-extension $L/K$ with $K \supset k$, $\sharp K = \infty$ by specialization of the parameters. Moreover, we derive a new generic polynomial for the cyclic group of order $2^{n+1}$ from our construction.

In the present and the next notes of this series, we shall try to illuminate a geometric structure behind the interactions that have recently been observed between mean values of zeta-functions and automorphic representations. Our discussion is hoped to be a precursor of a unified theory of mean values of automorphic $L$-functions that we are going to forge. In this note we shall deal with the spectral structure over the modular group. In the next note the Picard group will be treated, as a typical case in the complex situation. We stress that we have been inspired by the work [2] due to Cogdell and Pyatetskii-Shapiro.