Logarithmic transformations on elliptic fiber spaces. II: Elliptic bundle case
[For part I see J. Math. Kyoto Univ. 28, No.1, 91-110 (1988; Zbl 0654.14021).] Kodaira introduced the notion of logarithmic transformations and showed that a non-Kähler elliptic surface and a Kähler one can be transformed to each other via logarithmic transformations. The main purpose of this paper is to prove the counterpart of Kodaira's theorem for elliptic bundles. Here by an elliptic bundle f: \(Y\to X\) over X, we mean that Y is a principal fiber bundle over a complex manifold X whose typical fiber and structure groups are non-singular elliptic curves. For an elliptic bundle f: \(Y\to X\) over X, we shall define generalized logarithmic transformations along an arbitrary \(Cartier\quad divisor\quad D\) on X, which is not necessarily assumed to be reduced, irreducible or effective. We state our main theorem: Theorem (A). Let f: \(Y\to X\) be an elliptic bundle over a projective manifold X, which satisfies either of the following conditions: (1) Y is Kähler; (2) \(h^{2,0}(X):=\dim (H^ 0(X,\Omega^ 2_ X))=0.\) Then Y can be obtained from the trivial elliptic bundle over X by a succession of generalized logarithmic transformations. However, theorem (A) does not necessarily hold if we drop the assumption that Y is Kähler. We can give the characterization of elliptic bundles which are obtained from the trivial bundle by generalized logarithmic transformations.
scientific article; zbMATH DE number 4132324
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