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Moduli of Kähler manifolds equipped with Hermite-Einstein vector bundles. (English) Zbl 0813.32019
It is well known that irreducible Hermite-Einstein (or stable) holomorphic vector bundles over a compact Kähler manifold admit analytic moduli spaces which possess natural Kähler metrics. It is also well known that moduli spaces exist for polarised non-uniruled compact Kähler manifolds, and that for Kähler-Einstein manifolds the moduli spaces possess natural Kähler metrics (generalised Petersson-Weil metrics). The object of this paper is to combine these two results and construct coarse moduli spaces for pairs \((X,E)\) consisting of a polarised non-uniruled compact Kähler manifold \(X\) and a stable holomorphic bundle \(E\) on \(X\). In the case where \(X\) is Kähler-Einstein and \(E\) is projectively flat (with an extra condition when \(c_ 1(E)\neq 0)\), the authors construct a natural Kähler metric on the regular part of the moduli space.

32G13 Complex-analytic moduli problems
32Q20 Kähler-Einstein manifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
14J15 Moduli, classification: analytic theory; relations with modular forms
32J27 Compact Kähler manifolds: generalizations, classification