Compactifications of smooth families and of moduli spaces of polarized manifolds

Abstract

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.