Pluri-canonical divisors on Kähler manifolds. II (Q1084560)

The paper proves the following theorem. Let \(p: X\to D\) be a holomorphic family of connected compact manifolds over the unit disc \(D\) in \({\mathbb{C}}\), and suppose \(X_0=p_{-1}(0)\) is dominated by a compact Kähler manifold. Fix a positive integer \(m\), and suppose that the divisor \(p(s)\) (l.c.) of a general section of \(K(X_0)_m\) is reduced and the corresponding \(m\)-sheeted covering of \(X_0\) ramified over \((s)\) has only ``mild'' singularities (e.g. only quotient singularities). Then the \(m\)-th pluri-genus of the fibre of \(p\) is constant in a neighbourhood of \(0\). In part I of this paper [Invent. Math. 74, 293-303 (1983; Zbl 0647.53051)] the author has proved the same result assuming the smoothness of \(p(s)\).