Singular Hermitian metrics on holomorphic vector bundles.

*(English)*Zbl 1345.32021The 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]).

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)

##### Keywords:

holomorphic vector bundles; singular hermitian metrics; positively curved bundles; negatively curved bundles##### Citations:

Zbl 1181.32025**OpenURL**

##### References:

[1] | Berndtsson, B., Curvature of vector bundles associated to holomorphic fibrations, Ann. of Math., 169, 531-560, (2009) · Zbl 1195.32012 |

[2] | Berndtsson, B.; Păun, M., Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J., 145, 341-378, (2008) · Zbl 1181.32025 |

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[4] | Demailly, J.-P., Estimations \(L\)\^{}{2} pour l’opérateur [InlineEquation not available: see fulltext.] d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Éc. Norm. Supér.15 (1982), 457-511. · Zbl 0507.32021 |

[5] | Demailly, J.-P., Singular Hermitian metrics on positive line bundles, Bayreuth, 1990, Berlin · Zbl 0784.32024 |

[6] | Nadel, A. M., Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. of Math., 132, 549-596, (1990) · Zbl 0731.53063 |

[7] | Raufi, H., The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for vector bundles, Preprint, Göteborg, 2012. arXiv:1212.4417. · Zbl 0902.32012 |

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