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Diophantine approximations and Kodaira’s lemma. (English) Zbl 1054.32026

The the authors obtain results concerning the validity of the \(\partial \overline\partial\)-lemma, under arithmetical conditions. More precisely, if \(X\) is a complex manifold one says that the \(\partial \overline\partial\)-lemma holds if for any given, exact \((1,1)\) form \(\varphi\) on \(X\), \(\varphi\in C^\infty\), there is a \(C^\infty\) function \(\psi\) on \(X\) such that \(\varphi = \partial \overline\partial \psi\). The situation considered by the authors is the following: Let \(\mathbb{T} \) be the complex torus \(\mathbb{C}/\mathbb{Z}\{1, 0\}\) (of dimension 1) where \(\mathbb{Z}\{1, w\}\) is a discrete lattice, \(\text{Pic}^0(\mathbb{T})\) the Picard variety of \(\mathbb{T}\), which can be identified with \(\mathbb{T}\) itself by a canonical isomorphism \(V\). Then, the main result of the paper is: suppose \(E \in \text{Pic}^0(\mathbb{T})\) satisfies \(V(E) = [t]\) with \(t\) irrational and \(0 < t < 1\). Then the \(\partial \overline\partial\)-lemma holds on \(E\) iff (*) \(\sup \{ \frac{\log q_{n + 1}}{q_n}\), \(n \in \mathbb{N}\} < \infty\) where \(q_n\) is the denominator of the \(n\)-th convergent of the continued fraction of \(t\) (\(t = 1/c_1 + 1/c_2 + \cdots + 1/c_n \dots\) and \(\frac{p_n}{q_n} = 1/c_1 + \cdots+ 1/c_n\)).
The the authors show that the condition (*) is weaker than the necessary and sufficient condition for the linearization of analytic functions.

MSC:

32W99 Differential operators in several variables
11J70 Continued fractions and generalizations
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