A criterion for algebraicity of certain analytic elliptic surfaces (Q1906501)

Let \(f : S \to C\) be a compact complex analytic elliptic surface obtained by logarithmic transformations from a trivial elliptic fibration over a nonsingular algebraic curve \(C\). In this paper, the author proves that the algebraicity of \(S\) is equivalent to the existence of a meromorphic 1-form on \(C\) which satisfies some conditions. Furthermore, the author obtains the result that when the genus of \(C\) is greater than or equal to 1, the algebraicity of \(S\) is closely related to the relation between points over which the elliptic fibration has multiple fibres.