On the positivity of direct image bundles

Zhi Li ; Xiangyu Zhou

Izvestiya. Mathematics , Tome 87 (2023) no. 5, pp. 987-1010.

Résumé

In the present paper, we obtain an equivalent relation between the log-plurisubharmonicity of the relative Bergman kernel, the Griffiths and Nakano positivity for the direct image with the natural $L^2$ metric, by finding a converse of Berndtsson's theorem on the direct image. A converse of Berndtsson's generalization of Kiselman minimal principle is also obtained.

Détail

Keywords: $L^2$-methods, plurisubharmonic functions, positive hermitian holomorphic vector bundles, minimal principles, relative Bergman kernel.
Mots-clés : direct images

Bibliographie

[1] B. Berndtsson, “Curvature of vector bundles associated to holomorphic fibrations”, Ann. of Math. (2), 169:2 (2009), 531–560 | DOI | MR | Zbl

[2] B. Berndtsson, “Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains”, Ann. Inst. Fourier (Grenoble), 56:6 (2006), 1633–1662 | DOI | MR | Zbl

[3] B. Berndtsson and M. Păun, “Bergman kernels and the pseudoeffectivity of relative canonical bundles”, Duke Math. J., 145:2 (2008), 341–378 | DOI | MR | Zbl

[4] Fusheng Deng, Zhiwei Wang, Liyou Zhang, and Xiangyu Zhou, New characterizations of plurisubharmonic functions and positivity of direct image sheaves, arXiv: 1809.10371

[5] Fusheng Deng, Jiafu Ning, Zhiwei Wang, and Xiangyu Zhou, “Positivity of holomorphic vector bundles in terms of $L^p$-conditions for $\overline{\partial}$”, Math. Ann., 385:1-2 (2023), 575–607 ; arXiv: 2001.01762 | DOI | MR | Zbl

[6] T. Inayama, “Nakano positivity of singular Hermitian metrics and vanishing theorems of Demailly–Nadel–Nakano type”, Algebr. Geom., 9:1 (2022), 69–92 | DOI | MR | Zbl

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

[8] Qi'an Guan and Xiangyu Zhou, “A solution of an $L^2$ extension problem with an optimal estimate and applications”, Ann. of Math. (2), 181:3 (2015), 1139–1208 | DOI | MR | Zbl

[9] Qi'an Guan and XiangYu Zhou, “Strong openness of multiplier ideal sheaves and optimal $L^2$ extension”, Sci. China Math., 60:6 (2017), 967–976 | DOI | MR | Zbl

[10] B. Berndtsson and L. Lempert, “A proof of the Ohsawa–Takegoshi theorem with sharp estimates”, J. Math. Soc. Japan, 68:4 (2016), 1461–1472 | DOI | MR | Zbl

[11] A. Prékopa, “On logarithmic concave measures and functions”, Acta Sci. Math. (Szeged), 34 (1973), 335–343 | MR | Zbl

[12] C. O. Kiselman, “The partial Legendre transformation for plurisubharmonic functions”, Invent. Math., 49:2 (1978), 137–148 | DOI | MR | Zbl

[13] B. Berndtsson, “Prekopa's theorem and Kiselman's minimum principle for plurisubharmonic functions”, Math. Ann., 312:4 (1998), 785–792 | DOI | MR | Zbl

[14] K. Ball, F. Barthe, and A. Naor, “Entropy jumps in the presence of a spectral gap”, Duke Math. J., 119:1 (2003), 41–63 | DOI | MR | Zbl

[15] D. Cordero-Erausquin, “On Berndtsson's generalization of Prékopa's theorem”, Math. Z., 249:2 (2005), 401–410 | DOI | MR | Zbl

[16] L. Hörmander, “$L^2$ estimates and existence theorems for the $\overline{\partial}$ operator”, Acta. Math., 113:1 (1965), 89–152 | DOI | MR | Zbl

[17] So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Stud. Adv. Math., 19, Amer. Math. Soc., Providence, RI; International Press, Boston, MA, 2001 | DOI | MR | Zbl

[18] J.-P. Demailly, “Estimations $L^2$ pour l'opérateur $\overline{\partial}$ d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété Kählerienne complète”, Ann. Sci. École Norm. Sup. (4), 15:3 (1982), 457–511 | DOI | MR | Zbl

[19] J.-P. Demailly, Analytic methods in algebraic geometry, Surv. Mod. Math., 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2010/2012 | MR | Zbl

[20] T. Ohsawa, $L^2$ approaches in several complex variables. Development of Oka–Cartan theory by $L^2$ estimates for the $\overline\partial$ operator, Springer Monogr. Math., Springer, Tokyo, 2015 | DOI | MR | Zbl

[21] Xiangyu Zhou, “A survey on $L^2$ extension problem”, Complex geometry and dynamics, Abel Symp., 10, Springer, Cham, 2015, 291–309 | DOI | MR | Zbl

[22] T. Ohsawa and K. Takegoshi, “On the extension of $L^2$ holomorphic functions”, Math. Z., 195:2 (1987), 197–204 | DOI | MR | Zbl

[23] W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987 | MR | Zbl

[24] V. Guedj and A. Zeriahi, Degenerate complex Monge–Ampère equations, EMS Tracts Math., 26, Eur. Math. Soc. (EMS), Zürich, 2017 | DOI | MR | Zbl

[25] L. Hörmander, Notions of convexity, Progr. Math., 127, Birkhäuser Boston, Inc., Boston, MA, 1994 | DOI | MR | Zbl

[26] L. Lempert, “Modules of square integrable holomorphic germs”, Analysis meets geometry, Trends Math., Birkhäuser/Springer, Cham, 2017, 311–333 | DOI | MR | Zbl

[27] Zhuo Liu, Hui Yang, and Xiangyu Zhou, “New properties of multiplier submodule sheaves”, C. R. Math. Acad. Sci. Paris, 360 (2022), 1205–1212 | DOI | MR | Zbl

[28] Zhuo Liu, Hui Yang, and Xiangyu Zhou, On the multiplier submodule sheaves associated to singular Nakano semi-positive metrics, arXiv: 2111.13452

[29] H. Raufi, Log concavity for matrix-valued functions and a matrix-valued Prékopa theorem, arXiv: 1311.7343

[30] F. Maitani and H. Yamaguchi, “Variation of Bergman metrics on Riemann surfaces”, Math. Ann., 330:3 (2004), 477–489 | DOI | MR | Zbl