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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.

##### MSC:
 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