ON QUASI-ABELIAN VARIETIES OF KIND k

Abstract

Abstract. Gherardelli and Andreotti defined a quasi-abelian variety of kind k. However, this definition is somewhat vague and we do not know the real meaning of the 'kind'. We give an example of a quasi-abelian variety which is of kind k > 0 but not of kind 0, in the sense of Gherardelli and Andreotti. We prove that if a quasiabelian variety X = C n / has an ample Riemann form of kind k, then it has an ample Riemann form of kind k for any k with 2k 2k n − m, where rank = n + m. Next we consider the pair (X, L) of a quasi-abelian variety X and a positive line bundle L on it. We characterize an extendable line bundle L to a compactification X of X.

Meromorphic function fields closed by partial derivatives

“…. If H is an ample Riemann form for a quasi-abelian variety C n /Γ, then rankA Γ = 2(m + k) with 0 ≦ 2k ≦ n − m. In this case we say that the ample Riemann form H is of kind k. The kind of a quasi-abelian variety C n /Γ is defined by the smallest kind of ample Riemann forms for C n /Γ ( [6]). A quasi-abelian variety C n /Γ with rank Γ = n + m is of kind 0 if and only if it is a principal (C * ) n−mbundle ρ : C n /Γ −→ A over an m-dimensional abelian variety A.…”

Analytic study of singular curves

“…If a quasi-abelian variety G has an ample Riemann form of kind k, then it also has an ample Riemann form of kind k ′ for any k ′ with 2k ≤ 2k ′ ≤ n − m ( [6]). Then we defined the kind of a quasi-abelian variety as follows in [6].…”