In this article, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete–Szegő problem for normalized concave functions with a pole in the unit disk.

We extend the Faltings modular heights of Abelian varieties to general arithmetic varieties, show direct relations with the Kähler–Einstein geometry, the minimal model program, and Bost–Zhang’s heights and give some applications. Along the way, we propose the “arithmetic Yau–Tian–Donaldson conjecture” (the equivalence of a purely arithmetic property of a variety and its metrical property) and partially confirm it.

We complete the construction of the fundamental diagram of various partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. The diagram includes the space of nilpotent orbits, the space of $\mathrm{SL}\left(2\right)$-orbits, and the space of Borel–Serre orbits. We give amplifications of this fundamental diagram and amplify the relations of these spaces. We describe how this work is useful in understanding asymptotic behaviors of Beilinson regulators and of local height pairings in degeneration. We discuss mild degenerations in which regulators converge.

We develop relative trace formulas of unitary hyperbolic spaces for split rank $1$ unitary groups over totally real number fields.