A geometric algebraicity property for moduli spaces of compact Kähler manifolds with \(h^{2,0}=1\) (Q909074)

The authors prove the following Theorem: Let f: \(X\to S\) be a surjective holomorphic map with connected fibres, where X is a compact connected Kähler manifold. Assume that some fibre \(X_ 0\) of f is non-singular, projective, with \(h^{2,0}=1\). Then either f is locally trivial with fibre \(X_ 0\) over some non-empty Zariski-open subset of S, or the algebraic dimension a(S) of S is positive. A corollary says that if furthermore \(Aut^ 0(X_ 0)=0\) and \(a(S)=0\), then there exists a finite surjective map g: S\({}'\to S\) and a bimeromorphic map b: \(X\times_ SS'\to X_ 0\times S'\) over \(S'\).