Families of canonically polarized manifolds over log Fano varieties (Q2845017)

If \(f: Y\to X\) is a smooth projective family of canonically polarized manifolds, where \(Y\) is a quasi-projective manifold, Viehweg's conjecture states that if \(f\) has maximal variation, then \(Y\) is necessarily of log general type. A refined version of the conjecture was proved in [\textit{S. Kebekus} and \textit{S. J. Kovács}, Duke Math. J. 155, No. 1, 1--33 (2010; Zbl 1208.14027)] when \(\dim X\leq3\). The conjecture was recently proved in any dimension in [\textit{F.\ Campana} and \textit{M.\ Păun}, \url{arXiv:1303.3169}].NEWLINENEWLINEThe paper under review studies what happens when there is an additional assumption on the base manifold \(X\), namely that it is a dlt log Fano variety. Here dlt stands for a particular class of singularities of pairs which appears naturally in the context of the Minimal Model Program. The main result of the paper is the following.NEWLINENEWLINETheorem. Let \((X,\Delta)\) be a projective dlt pair, where \(\Delta\) is an effective \(\mathbb R\)-divisor and \({-}(K_X+\Delta)\) is \(\mathbb R\)-ample. Let \(T\subseteq X\) be a subvariety of codimension at least \(2\) such that \(X^0=X\setminus (T\cup \mathrm{Supp}\lfloor\Delta\rfloor)\) is smooth. Then any smooth family of canonically polarized varieties over \(X^0\) is isotrivial. \vskip 2mm The proof uses recent techniques of Kebekus and Kovács [op.\ cit.], and also establishes a structure theorem for the cone of movable curves on a dlt projective pair \((X,\Delta)\), i.e.\ the cone dual to the pseudo-effective cone; this is Theorem 1.4 in the paper under review.