In this note, we obtain various formulas for the curvature of the metric on the direct image of the relative canonical bundle twisted by a holomorphic line bundle endowed with a positively curved metric with analytic singularities, generalizing some of Berndtsson’s seminal results in the smooth case. When the twist is assumed to be relatively big, we further provide a very explicit lower bound for the curvature of the metric.
Dans cette note, nous obtenons diverses formules pour la courbure de la métriques sur l’image directe du fibré canonique relatif tordu par un fibré en droites holomorphe muni d’une métrique à courbure positive avec singularités analytiques, généralisant certains des résultats fondateurs de Berndtsson dans le cas lisse. Quand le fibré par lequel on tord est gros, nous pouvons de plus donner une borne inférieure explicite de la courbure de la métrique .
Junyan Cao 1; Henri Guenancia 2; Mihai Păun 3
CC-BY 4.0
@article{AFST_2022_6_31_3_861_0,
author = {Junyan Cao and Henri Guenancia and Mihai P\u{a}un},
title = {Curvature formula for direct images of twisted relative canonical bundles endowed with a singular metric},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {861--905},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 31},
number = {3},
year = {2022},
doi = {10.5802/afst.1707},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1707/}
}
TY - JOUR AU - Junyan Cao AU - Henri Guenancia AU - Mihai Păun TI - Curvature formula for direct images of twisted relative canonical bundles endowed with a singular metric JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2022 SP - 861 EP - 905 VL - 31 IS - 3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1707/ DO - 10.5802/afst.1707 LA - en ID - AFST_2022_6_31_3_861_0 ER -
%0 Journal Article %A Junyan Cao %A Henri Guenancia %A Mihai Păun %T Curvature formula for direct images of twisted relative canonical bundles endowed with a singular metric %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2022 %P 861-905 %V 31 %N 3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1707/ %R 10.5802/afst.1707 %G en %F AFST_2022_6_31_3_861_0
Junyan Cao; Henri Guenancia; Mihai Păun. Curvature formula for direct images of twisted relative canonical bundles endowed with a singular metric. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 3, pp. 861-905. doi : 10.5802/afst.1707. https://afst.centre-mersenne.org/articles/10.5802/afst.1707/
[1] The space of Poincaré type Kähler metrics on the complement of a divisor, J. Reine Angew. Math., Volume 722 (2017), pp. 1-64 | DOI | MR | Zbl
[2] Curvature of vector bundles associated to holomorphic fibrations, Ann. Math., Volume 169 (2009), pp. 531-560 | DOI | MR | Zbl
[3] Strict and nonstrict positivity of direct image bundles, Math. Z., Volume 269 (2011) no. 3-4, pp. 1201-1218 | DOI | MR | Zbl
[4] Introduction to Hodge theory. Transl. from the French by James Lewis and Chris Peters, SMF/AMS Texts and Monographs, 8, American Mathematical Society, 2002, ix+232 pages | Zbl
[5] On the volume of a line bundle., Int. J. Math., Volume 13 (2002) no. 10, pp. 1043-1063 | DOI | MR | Zbl
[6] Ohsawa-Takegoshi extension theorem for compact Kähler manifolds and applications, Complex and symplectic geometry (Springer INdAM Series), Volume 21, Springer, 2017, pp. 19-38 | MR | Zbl
[7] On extension of pluricanonical forms defined on the central fiber of a Kähler family (2020) | arXiv
[8] Differential equations on Riemannian manifolds and their geometric applications, Commun. Pure Appl. Math., Volume 28 (1975) no. 3, pp. 333-354 | DOI | MR | Zbl
[9] Complex Analytic and Differential Geometry, 2012 (OpenContent Book, freely available from the author’s web site, http://www-fourier.ujf-grenoble.fr/~demailly/books.html)
[10] Positivity of holomorphic vector bundles in terms of -conditions of (2020) | arXiv
[11] Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor, Ann. Inst. Fourier, Volume 64 (2014) no. 6, pp. 1291-1330 | DOI | Numdam | MR | Zbl
[12] Kähler–Einstein metrics on an open algebraic manifolds, Osaka J. Math., Volume 21 (1984), pp. 399-418
[13] Complex manifolds and deformation of complex structures. Transl. from the Japanese by Kazuo Akao, Grundlehren der Mathematischen Wissenschaften, 283, Springer, 1986 | DOI | Zbl
[14] Injectivity theorems with multiplier ideal sheaves for higher direct images under Kähler morphisms (2016) (to appear in Algebr. Geom.) | arXiv
[15] Positivity of direct images with a Poincaré type twist (2020) | arXiv
[16] Singular Hermitian metrics and positivity of direct images of pluricanonical bundles, Algebraic geometry: Salt Lake City 2015 (Proceedings of Symposia in Pure Mathematics), Volume 97, American Mathematical Society, 2018, pp. 519-553 | MR | Zbl
[17] Positivity of twisted relative pluricanonical bundles and their direct images, J. Algebr. Geom., Volume 27 (2018) no. 2, pp. 211-272 | DOI | MR | Zbl
[18] Positivity of relative canonical bundles and applications, Invent. Math., Volume 190 (2012) no. 1, pp. 1-56 | DOI | MR | Zbl
[19] Curvature of the Weil–Petersson metric in the moduli space of compact Kähler–Einstein manifolds of negative first Chern class, Contributions to several complex variables (Aspects of Mathematics), Volume E9, Vieweg & Sohn, 1986, pp. 261-298 | MR | Zbl
[20] Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (San Diego, Calif., 1986) (Advanced Series in Mathematical Physics), Volume 1, World Scientific, 1987, pp. 574-628 | DOI
Cited by Sources: