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Positivity of relative canonical bundles and applications. (English) Zbl 1258.32005
This paper in algebraic geometry, dedicated to the memory of Eckart Vieweg, uses methods of differential geometry (e.g., Kähler-Einstein metrics) and geometric analysis (e.g., potential theory as in Hodge theory, complex Monge-Ampère equations, and deformation theory) in the manner of the schools of Siu and Demailly to analyze the moduli space \({\mathcal M}_{\mathrm{can}}\) of canonically polarized varieties and to show that it is quasi-projective. It has eleven sections and features many theorems, including the following Main Theorem.
Main Theorem. Let \({\mathcal X}\to S\) be a holomorphic family of canonically polarized, compact, complex manifolds, which is nowhere infinitesimally trivial. Then the curvature of the hermitian metric on the relative canonical bundle \({\mathcal K}_{{\mathcal X}/S}\) that is induced by the Kähler-Einstein metrics on the fibers is strictly positive.
A large part of this long paper is devoted to differential geometric tensorial calculations to compute explicitly the curvature tensor of an appropriate direct image.

MSC:
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14D20 Algebraic moduli problems, moduli of vector bundles
32Q20 Kähler-Einstein manifolds
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