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Reflexive pull-backs and base extension. (English) Zbl 1081.14017

This paper deals with the problem to find a natural compactification of the moduli space of smooth canonically polarized varieties of dimension \(d\). Despite recent progress by Alexeev, Kollár, Shepherd-Barron, Viehweg and others, if \(d>1\) it is not entirely clear what the correct definition of the moduli functor should be in order to obtain a compact moduli space. The main problem is to decide about the right notion of admissible family for such a functor. The problems are caused by the fact that forming the reflexive hull of a coherent sheaf does not in general commute with base change. In the previous decade, Kollár and Viehweg introduced definitions of moduli functors which seem to be suitable in this context. The main result of the paper under review shows that Viehweg’s moduli functor is locally closed in the surface case. This was the missing piece in order to establish the existence of a compactified moduli space of canonically polarized surfaces in characteristic zero. The proof carefully pays attention to the interplay between base change and forming the reflexive hull of a coherent sheaf. A generalisation of this result to varieties of dimension \(d>2\) fails due to the lack of a proof of the existence of minimal models in dimension \(d+1\).

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14J10 Families, moduli, classification: algebraic theory
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