It is conjectured that the lifted Futaki invariant of an $n$-dimensional compact complex manifold vanishes if it admits an Einstein-Kähler metric. If the conjecture holds for $n = 1$, the lifted Futaki invariants for Riemann surfaces must vanish because Riemann surfaces always admit Einstein-Kähler metrics. In this paper, we prove the vanishing of the lifted Futaki invariants for Riemann surfaces under a certain assumption. Our main result is Theorem 1.3.

In this paper we consider the support property and the short time aymptotic behavior of the transition density for (possibly degenerate) processes of jump type whose Lévy measure is singular.

Let $S$ be a non-empty set of prime numbers; $1 \leq |S| \leq \infty$. Let $\mathbf{Q}^S$ denote the abelian extension of the rational field $\mathbf{Q}$ whose Galois group over $\mathbf{Q}$ is topologically isomorphic to the direct product of the additive groups of $l$-adic integers for all $l \in S$. In this note, we shall give simple examples of $S$ such that, for some $l \in S$, the Hilbert $l$-class field over $\mathbf{Q}^S$ is a nontrivial extension of $\mathbf{Q}^S$. Our results imply that, if $S$ contains 2, 3, 31, and 73, then there exists an unramified cyclic extension of degree $2263 = 31 \cdot 73$ over $\mathbf{Q}^S$.

In this paper, we discuss modification of Helgason's support theorem for the Radon transform. It is essential in this theorem to assume that the function decays rapidly towards infinity. We restrict this condition to an open cone.

Kawamoto [5, 6] proved that for any prime number $p$ and any $a \in \mathbf{Q}^{\times}$, the cyclic extenstion $\mathbf{Q}(\zeta_p, a^{1/p}) / \mathbf{Q}(\zeta_p)$ has a normal integral basis (NIB) if it is tame. We show that this property is peculier to the rationals $\mathbf{Q}$. Namely, we show that for a number field $K$ with $K \neq \mathbf{Q}$, there exist infinitely many pairs $(p, a)$ of a prime number $p$ and $a \in K^{\times}$ for which $K(\zeta_p, a^{1/p}) / K(\zeta_p)$ is tame but has no NIB. Our result is an analogue of the theorem of Greither et al. [3] on Hilbert-Speiser number fields.

A group $G$ is said to enjoy ``Hasse principle'' if every local coboundary of $G$ is a global coboundary. It is proved that every group of order $p^4$ and every metacyclic group enjoy ``Hasse principle''.