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Singular Hermitian metrics on holomorphic vector bundles. (English) Zbl 1345.32021
The article introduces and studies the notions of singular hermitian metrics on holomorphic vector bundles. Let $$E \rightarrow X$$ be a holomorphic vector bundle over a complex manifold $$X$$. A possible singular hermitian metric $$h$$ on $$E$$ is by definition, a measurable map from $$X$$ to the space of non-negative hermitian forms on the fibres [B. Berndtsson and M. Păun, Duke Math. J. 145, No. 2, 341–378 (2008; Zbl 1181.32025)]. The main results of the article is a study of the connections and curvatures of $$(E,h)$$ when $$(E,h)$$ is negatively (or positively) curved in the sense of Griffiths.
The author defines first the notion of a positively (or negatively) curved metric as follows. Let $$E$$ be a holomorphic vector bundle with a possibly singular hermitian metric $$h$$. We say that $$(E,h)$$ is negatively curved in the sense of Griffiths if $$\|u\|_h ^2$$ is plurisubharmonic for any local holomorphic section $$u$$. We say that $$(E ,h)$$ is positively curved, if the dual metric is negatively curved. The author shows that this definition is equivalent to the definition of [loc. cit.]. Note that, when $$h$$ is a smooth hermitian metric, then $$(E ,h)$$ is positively curved if and only if $$\Theta_h (E)$$ is Griffiths semipositive.
The main part of the article is devoted to studying the connection matrix and the curvatures of $$h$$ when $$(E,h)$$ is negatively curved. One of the powerful results of the article reads as follows: Let $$E$$ be a holomorphic vector bundle with a possible singular metric $$h$$. If $$(E,h)$$ is negatively curved and $$\det h > \epsilon$$ for some $$\epsilon >0$$, then the curvature matrix $$\Theta_h (E)$$ is well defined (with measure coefficients).
The article is well written and has many nice applications in complex geometry (cf. for example [M. Păun and S. Takayama, “Positivity of twisted relative pluricanonical bundles and their direct images”, Preprint, arXiv:1409.5504]).
Reviewer: Junyan Cao (Paris)

##### MSC:
 32L05 Holomorphic bundles and generalizations 32U99 Pluripotential theory
Zbl 1181.32025
Full Text:
##### References:
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