Deformation and degeneration of complex hyperbolic pseudometrics and pseudovolumes (Q1066318)

Let \(f: M\to \Delta\) be a holomorphic family of complex manifolds over a unit disc \(\Delta\) nondegenerate over \(\Delta^*=\Delta \setminus \{0\}\) (i.e. df has maximal rank over \(\Delta^*)\). The paper contains announcements of some semicontinuity theorems for the hyperbolic pseudometrics and pseudovolumes on the fibers \(M_ z=f^{-1}(z)\) when \(z\to 0\), where \(M_ 0\) is replaced by its regular part \(M'_ 0=M_ 0\setminus \sin g M_ 0.\) Theorem 1 which contains the local infinitesimal versions was recently strengthened by the author who showed that upper semicontinuity holds in any cases (it partly strengthens also corresponding results of M. Wright and M. Kalka for nondegenerate case). Uniform estimates for integral pseudometrics (pseudovolumes) are valid under additional hypothesis of the continuity of corresponding forms on the fiber \(M'_ 0\) (theorem 2). The uniform continuity in z of these forms is stated under some conditions which guarantee that \(M_ 0\) has normal crossings and there exists an hyperbolic embedded neighborhood \(U=f^{- 1}(\Delta_{\epsilon})\) for some \(\epsilon >0\) (theorems 3 and 4). In the case when dim M\(=2\) as a consequence we get an inequality \(\chi (M_ z)\leq \Sigma k_ i\chi (M'_{0,i})\) for Euler characteristic of general fiber, where \(\{M'_{0,i}\}\) are the irreducible components of the curve \(M'_ 0\) and \(\{k_ i\}\) are their multiplicities. For the proofs see the author's paper in Mat. Sb., Nov. Ser. 127(169), No.1(5), 55-71 (1985; see the following review).