Compactifications of smooth families and of moduli spaces of polarized manifolds


Let $M_h$ be the moduli scheme of canonically polarized manifolds with Hilbert polynomial $h$. We construct for $\nu\geq 2$ with $h(\nu)>0$ a projective compactification $\overline{M}_h$ of the reduced moduli scheme $(M_h)_{\rm red}$ such that the ample invertible sheaf $\lambda_\nu$, corresponding to ${\rm det}(f_*\omega_{X_0/Y_0}^\nu)$ on the moduli stack, has a natural extension $\overline{\lambda}_\nu\in {\rm Pic}(\overline{M}_h)_\Bbb{Q}$. A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero. In both cases “natural” means that the pullback of $\overline{\lambda}_\nu$ to a curve $\varphi:C\to \overline{M}_h$, induced by a family $f_0:X_0\to C_0=\varphi^{-1}(M_h)$, is isomorphic to ${\rm det}(f_*\omega_{X/C}^\nu)$ whenever $f_0$ extends to a semistable model $f:X\to C$.

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.


Eckart Viehweg

Correspondence should be addressed to: Hélène Esnault
Universität Duisburg-Essen
45117 Essen