# Compactifications of smooth families and of moduli spaces of polarized manifolds | Annals of Mathematics

Abstract 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.

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### 数学年刊（Annals of Mathematics）

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