Compactifications of smooth families and of moduli spaces of polarized manifolds

Let Mh be the moduli scheme of canonically polarized manifolds with Hilbert polynomial h. We construct for ν≥2 with h(ν)>0 a projective compactification M¯¯¯¯h of the reduced moduli scheme (Mh)red such that the ample invertible sheaf λν , corresponding to det(f∗ωνX0/Y0) on the moduli stack, has a natural extension λ¯ν∈Pic(M¯¯¯¯h)Q . A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero. In both cases "natural" means that the pullback of λ¯ν to a curve φ:C→M¯¯¯¯h, induced by a family f0:X0→C0=φ−1(Mh) , is isomorphic to det(f∗ωνX/C) whenever f0 extends to  a semistable model f:X→C .

Besides of the weak semistable reduction of Abramovich-Karu and the extension theorem of Gabber there are new tools, hopefully of interest by themselves. In particular we will need a theorem on the flattening of multiplier sheaves in families, on their compatibility with pullbacks and on base change for their direct images, twisted by certain semiample sheaves.