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Compactifications of smooth families and of moduli spaces of polarized manifolds

Eckart Viehweg

Number 29
Author Eckart Viehweg
Project C11
Year 2008

Let Mh be the moduli scheme of canonically polarized manifolds with Hilbert polynomial h. We construct for ν2 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(fωνX0/Y0) on the moduli stack, has a natural extension λ¯νPic(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:CM¯¯¯¯h, induced by a family f0:X0C0=φ1(Mh) , is isomorphic to det(fωνX/C) whenever f0 extends to  a semistable model f:XC .

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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