Compactifications of smooth families and of moduli spaces of polarized manifolds

TL;DR

This paper constructs projective compactifications of moduli spaces of polarized manifolds, extending natural line bundles in a way compatible with families, using new tools like flattening of multiplier sheaves and semi-stable reduction.

Contribution

It introduces a method to extend natural line bundles over compactified moduli spaces of polarized manifolds, employing novel flattening techniques for multiplier sheaves.

Findings

Constructed projective compactifications of moduli spaces.

Extended natural line bundles over these compactifications.

Developed new flattening theorems for multiplier sheaves.

Abstract

Let M be the moduli scheme of canonically polarized manifolds with Hilbert polynomial h. We construct for a given finite set I of natural numbers m>1 with h(m)>0 a projective compactification M' of the reduced scheme underlying M such that the ample invertible sheaf L corresponding to the determinant of the direct image of the m-th power of the relative dualizing sheaf on the moduli stack, has a natural extension L' to M'. A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero. In both cases "natural" means that the pullback of L' to a curve C --> M', induced by a family f:X --> C is isomorphic to the determinant of the direct image of the m-th power of the relative dualizing sheaf whenever f is birational to a semi-stable family. Besides of the weak semistable reduction of Abramovich-Karu and the extension theorem of Gabber there are new tools, hopefully of interest by itself. 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. Following suggestions of a referee, we reorganized the article, we added several comments explaining the main line of the proof, and we changed notations a little bit.