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Harmonic metrics on Higgs sheaves and uniformization of varieties of general type. (English) Zbl 1453.32029

The structure of the article is split into two parts.
In the first part the authors begin with a review of Mochizuki’s theory of tame and purely imaginary harmonic bundles on quasi-projective varieties including the particular case of the smooth locus of a Kawamata log terminal (klt for short) variety. Also they review the notion of a Higgs sheaf on singular spaces, and discuss the stability of Higgs bundles defined on the smooth locus of a klt variety. Then they show a central existence result for harmonic structures.
In the second part, the authors show how existence of harmonic structures leads to nonabelian Hodge correspondences pertaining to local systems on the smooth locus of a klt space. Hereafter, they prove their main result on quasi-étale uniformisation (Theorem 1.5).

MSC:

32Q30 Uniformization of complex manifolds
14E20 Coverings in algebraic geometry
14E30 Minimal model program (Mori theory, extremal rays)
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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