Quasi-hodge Metrics and Canonical Singularities

Throughout this paper we work over the base field C. The basic problem in algebraic geometry treated here is the filling-in problem (or degeneration problem). Given a smooth projective family X× → ∆×, we would like to know when we can fill in a reasonably nice special fiber X0 to form a projective family X → ∆, perhaps up to a finite base change on ∆×. For example, when can X0 be smooth? When can it be irreducible? Or when can it be irreducible with at most certain type of mild singularities? We would like to search for conditions depending only on the punctured family. In this paper, for any smooth projective family X → S over a smooth base S such that Xs has semi-ample canonical bundle, we shall define for each large m ∈ N a Kähler metric gm on S, called the m-th quasiHodge metric. When S = ∆×, we propose that the incompleteness of gm near 0 for suitable m’s provides a necessary and sufficient condition for the existence of X0 to be irreducible and with at most canonical singularities (c.f. Remark 2.5). Notice that the metric incompleteness condition is insensitive to base changes. More precisely, for a smooth projective family π : X → S with pg(Xs) 6= 0, the (possibly degenerate) quasi-Hodge metric gH = g1 on S is given by the semi-positive first Chern form of the rank pg Hodge bundle F n = π∗KX/S. When S = ∆ ×, let T be the monodromy operator acting on H(Xs,C) where n = dimXs, s 6= 0. Theorem 1.1. For any smooth projective family π : X → ∆×,