We prove a result of Chern–Weil type for canonically metrized line bundles on one-parameter families of smooth complex curves. Our result generalizes a result due to J. I. Burgos Gil, J. Kramer, and U. Kühn that deals with a line bundle of Jacobi forms on the universal elliptic curve over the modular curve with full level structure, equipped with the Petersson metric. Our main tool, as in the work by Burgos Gil, Kramer, and Kühn, is the notion of a b-divisor.

Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semidistance $d_{\mathrm{CK}}$, which he introduced for analytic spaces defined over a non-Archimedean metrized field $k$. We prove various characterizations of smooth projective varieties for which $d_{\mathrm{CK}}$ is an actual distance.

Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve $X$ defined over $k$. We prove a non-Archimedean analogue of the equivalence between having a negative Euler characteristic and the normality of certain families of analytic maps taking values in $X$.

We investigate singular hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^2$ estimates and vanishing theorems usually require an assumption that the vector bundles are Nakano positive. However, there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^2$ estimates and vanishing theorems by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.

For every Banach space $X$ with a Schauder basis, consider the Banach algebra $\mathbb{R}I\oplus {\mathcal{K}}_{\mathrm{diag}}(X)$ of all diagonal operators that are of the form $\lambda I+K$. We prove that $\mathbb{R}I\oplus {\mathcal{K}}_{\mathrm{diag}}(X)$ is a Calkin algebra, that is, there exists a Banach space ${\mathcal{Y}}_X$ such that the Calkin algebra of ${\mathcal{Y}}_X$ is isomorphic as a Banach algebra to $\mathbb{R}I\oplus {\mathcal{K}}_{\mathrm{diag}}(X)$. Among other applications of this theorem, we obtain that certain hereditarily indecomposable spaces and the James spaces $J_p$ and their duals endowed with natural multiplications are Calkin algebras; that all nonreflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras; and that sums of reflexive spaces with unconditional bases with certain James–Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.

This paper presents a systematic study for classical aspects of functions with absolutely convergent Fourier series over homogeneous spaces of compact groups. Let $G$ be a compact group, $H$ be a closed subgroup of $G$, and $\mu$ be the normalized $G$-invariant measure over the left coset space $G/H$ associated with Weil’s formula with respect to the probability measures of $G$ and $H$. We introduce the abstract notion of functions with absolutely convergent Fourier series in the Banach function space $L^1(G/H,\mu )$. We then present some analytic characterizations for the linear space consisting of functions with absolutely convergent Fourier series over the compact homogeneous space $G/H$.