L 2 -extension indices, sharper estimates and curvature positivity
[L 2 -extension indices, sharper estimates and curvature positivity]
Inayama, Takahiro1
1 Department of Mathematics Faculty of Science and Technology Tokyo University of Science 2641 Yamazaki, Noda Chiba, 278-8510 (Japan)
Annales de l'Institut Fourier, Online first, 29 p.

In this paper, we introduce a new concept of L 2 -extension indices. This index is a function that gives the minimum constant with respect to the L 2 -estimate of an Ohsawa–Takegoshi-type extension at each point. By using this notion, we propose a new way to study the positivity of curvature. We prove that there is an equivalence between how sharp the L 2 -extension is and how positive the curvature is. New examples of sharper L 2 -extensions are also systematically given. As applications, we use the L 2 -extension index to study Prékopa-type theorems and to study the positivity of a certain direct image sheaf. We also provide new characterizations of pluriharmonicity and curvature flatness.

Dans cet article, nous introduisons un nouveau concept d’indices d’extension L 2 . Cet indice est une fonction qui donne la constante minimale par rapport à l’estimation L 2 d’une extension de type Ohsawa–Takegoshi en chaque point. En utilisant cette notion, nous proposons une nouvelle façon d’étudier la positivité de la courbure. Nous prouvons qu’il existe une équivalence entre le degré de netteté de l’extension L 2 et le degré de positivité de la courbure. De nouveaux exemples d’extensions L 2 plus nettes sont également systématiquement donnés. Comme applications, nous utilisons l’indice d’extension en L 2 pour étudier des théorèmes de type Prékopa et pour étudier la positivité d’un certain faisceau d’image directe. Nous fournissons également de nouvelles caractérisations de la pluriharmonicité et de la planéité de la courbure.

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Révisé le :
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DOI : 10.5802/aif.3738
Classification : 32A36, 32U05
Keywords: Ohsawa–Takegoshi extension theorem, $L^2$-extension, Plurisubharmonic function, Griffiths positivity, $L^2$-extension index
Mots-clés : théorème d’extension d’Ohsawa–Takegoshi, extension $L^2$, fonction plurisousharmonique, positivité de Griffiths, indice d’extension $L^2$

Inayama, Takahiro 1

1 Department of Mathematics Faculty of Science and Technology Tokyo University of Science 2641 Yamazaki, Noda Chiba, 278-8510 (Japan) 
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Inayama, Takahiro. $L^2$-extension indices, sharper estimates and curvature positivity. Annales de l'Institut Fourier, Online first, 29 p.

[1] Berndtsson, Bo Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions, Math. Ann., Volume 312 (1998) no. 4, pp. 785-792 | DOI | Zbl | MR

[2] Berndtsson, Bo Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier, Volume 56 (2006) no. 6, pp. 1633-1662 | DOI | MR | Zbl | Numdam

[3] Berndtsson, Bo Curvature of vector bundles associated to holomorphic fibrations, Ann. Math. (2), Volume 169 (2009) no. 2, pp. 531-560 | DOI | Zbl

[4] Błocki, Zbigniew Suita conjecture and the Ohsawa–Takegoshi extension theorem, Invent. Math., Volume 193 (2013) no. 1, pp. 149-158 | Zbl | MR | DOI

[5] Demailly, Jean-Pierre Complex analytic and differential geometry, http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf, 2012

[6] Deng, Fusheng; Ning, Jiafu; Wang, Zhiwei Characterizations of plurisubharmonic functions, Sci. China, Math., Volume 64 (2021) no. 9, pp. 1959-1970 | DOI | Zbl | MR

[7] Deng, Fusheng; Ning, Jiafu; Wang, Zhiwei; Zhou, Xiangyu Positivity of holomorphic vector bundles in terms of L p -estimates for ∂ ¯, Math. Ann., Volume 385 (2023) no. 1-2, pp. 575-607 | DOI | Zbl | MR

[8] Deng, Fusheng; Wang, Zhiwei; Zhang, Liyou; Zhou, Xiangyu New characterizations of plurisubharmonic functions and positivity of direct image sheaves (2018) | arXiv

[9] Deng, Fusheng; Zhang, Huiping; Zhou, Xiangyu Positivity of direct images of positively curved volume forms, Math. Z., Volume 278 (2014) no. 1-2, pp. 347-362 | DOI | Zbl | MR

[10] Guan, Qi’an; Zhou, Xiangyu A solution of an L 2 extension problem with an optimal estimate and applications, Ann. Math. (2), Volume 181 (2015) no. 3, pp. 1139-1208 | DOI | Zbl

[11] Hacon, Christopher; Popa, Mihnea; Schnell, Christian Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and Păun, Local and global methods in algebraic geometry (Contemporary Mathematics), Volume 712, American Mathematical Society, 2018, pp. 143-195 | DOI | Zbl

[12] Hosono, Genki On sharper estimates of Ohsawa–Takegoshi L 2 -extension theorem, J. Math. Soc. Japan, Volume 71 (2019) no. 3, pp. 909-914 | DOI | Zbl | MR

[13] Hosono, Genki; Inayama, Takahiro A converse of Hörmander’s L 2 -estimate and new positivity notions for vector bundles, Sci. China, Math., Volume 64 (2021) no. 8, pp. 1745-1756 | DOI | Zbl | MR

[14] Inayama, Takahiro Optimal L 2 -Extensions on Tube Domains and a Simple Proof of Prékopa’s Theorem, J. Geom. Anal., Volume 32 (2021) no. 1, 32, 10 pages | DOI | Zbl | MR

[15] Inayama, Takahiro A note on characterizing pluriharmonic functions via the Ohsawa–Takegoshi extension theorem, J. Math. Sci., Tokyo, Volume 30 (2023) no. 3, pp. 365-369 | MR | Zbl

[16] Khare, Apoorva; Pingali, Vamsi Pritham On an asymptotic characterisation of Griffiths semipositivity, Bull. Sci. Math., Volume 167 (2021), 102956, 8 pages | DOI | Zbl | MR

[17] Kikuchi, Shota On sharper estimates of Ohsawa–Takegoshi L 2 -extension theorem in higher dimensional case, Manuscr. Math., Volume 170 (2023) no. 3, pp. 453-469 | DOI | Zbl | MR

[18] Kiselman, Christer O. The partial Legendre transformation for plurisubharmonic functions, Invent. Math., Volume 49 (1978) no. 2, pp. 137-148 | DOI | Zbl | MR

[19] Liu, Zhuo; Xu, Wang Characterizations of Griffiths Positivity, Pluriharmonicity and Flatness (2022) | arXiv

[20] Ohsawa, Takeo; Takegoshi, Kensho On the extension of L 2 holomorphic functions, Math. Z., Volume 195 (1987) no. 2, pp. 197-204 | DOI | Zbl | MR

[21] Siu, Yum-Tong Invariance of plurigenera, Invent. Math., Volume 134 (1998) no. 3, pp. 661-673 | DOI | Zbl | MR

[22] Xu, Wang A Quantitative Characterization of Griffiths Positivity (2022) | arXiv

[23] Xu, Wang; Zhou, Xiangyu Optimal L 2 Extensions of Openness Type (2022) | arXiv

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