×

A solution of an \(L^{2}\) extension problem with an optimal estimate and applications. (English) Zbl 1348.32008

The authors give the proof of the generalization of the Ohsawa-Takegoshi extension theorem announced in [C. R. Math. Acad. Sci. Paris, 350 (2012), 753-756]. A slightly less general version, sufficient for several applications, is given below. Let \(c_A (t)\) denote a smooth function defined on \((-A, +\infty )\) and such that \(c_A (t) e^{-t}\) is decreasing. Let \(M\) be either a Stein manifold or a complex projective manifold, and \(S\) a closed complex subvariety. Let \(S'\) be a set containing \(S\) and negligible for \(L^2\) functions on \(M\). Further, suppose \(\Psi\) is almost plurisubharmonic on \(M\), smooth on \(M\setminus S'\), with \(S\) contained in the \(-\infty\) set of \(\Psi\), with singularities prescribed in the following way: if \(S\) is \(k\)-dimensional around a given regular point and in a local chart \(S\) is given by \(z_{k+1} = ... = z_n =0\), then in this chart \(\Psi\) has the same singularities as \((n-k)\log \sum _{k+1}^n |z_j |^2 \). Consider also a holomorphic vector bundle \(E\) on \(M\) of rank \(r\), and \(h\) a smooth metric on \(E\) with the property that \(h e^{-\Psi}\) is semipositive in the sense of Nakano on \(M\setminus S'\). Then, for a holomorphic section \(f\) of \(K_M\) tensor \(E\) restricted to \(S\), there exists an extension \(F\) to the whole tensor product on \(M\) with the estimate
\[ \int _M c_A (\psi )|F|^2 _h dV_M \leq \int _A ^{\infty }c_A (t)e^{-t} dt \sum _{k=1}^{n} \frac{\pi ^k}{k!} \int _{S_{n-k}}|f|_h ^2 dV_M [\Psi ],\tag{1} \] provided the right hand side is finite. Here \(dV_M [\Psi ]\) is defined as a minimal element of the set of positive measures \(\mu\) satisfying \[ \int _{S_k} fd\mu \geq \limsup _{t\to \infty }\frac{2(n-k)}{\sigma _{2n-2k-1} }\int _M fe^{-\Psi }\chi _{\{ -1-t<\Psi <-t \} } dV_M , \] for any continuous, compactly supported \(f\), where \(\chi\) denotes the characteristic function and \(\sigma _k \) the area of the unit \(k\)-dimensional sphere. The important point is that the estimate (1) is no longer true if the right hand side is multiplied by a constant smaller than 1.
The authors give several applications of this result. One of them is the characterisation of the sets for which the equality holds in the Suita conjecture [N. Suita, Arch. Ration. Mech. Anal. 46, 212–217 (1972; Zbl 0245.30014)]. Others include a related conjecture of A. Yamada, a question of T. Ohsawa on boundedness of the extension operators [Contemp. Math. 332, 235–239 (2003; Zbl 1049.32010)], the log-plurisubharmonicity of the Bergman kernel, and the optimal constant in various versions of the \(L^2\) extension theorem.

MSC:

32L05 Holomorphic bundles and generalizations
32S05 Local complex singularities
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton, N.J.: Princeton Univ. Press, 1960, vol. 26. · Zbl 0196.33801
[2] B. Berndtsson, ”The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 46, iss. 4, pp. 1083-1094, 1996. · Zbl 0853.32024 · doi:10.5802/aif.1541
[3] B. Berndtsson, ”Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions,” Math. Ann., vol. 312, iss. 4, pp. 785-792, 1998. · Zbl 0938.32021 · doi:10.1007/s002080050246
[4] B. Berndtsson, ”Integral formulas and the Ohsawa-Takegoshi extension theorem,” Sci. China Ser. A, vol. 48, iss. suppl., pp. 61-73, 2005. · Zbl 1126.32002 · doi:10.1007/BF02884696
[5] B. Berndtsson, ”Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 56, iss. 6, pp. 1633-1662, 2006. · Zbl 1120.32021 · doi:10.5802/aif.2223
[6] B. Berndtsson, ”Curvature of vector bundles associated to holomorphic fibrations,” Ann. of Math., vol. 169, iss. 2, pp. 531-560, 2009. · Zbl 1195.32012 · doi:10.4007/annals.2009.169.531
[7] B. Berndtsson and M. Puaun, A Bergman kernel proof of the Kawamata subadjunction theorem.
[8] B. Berndtsson and M. Puaun, ”Bergman kernels and the pseudoeffectivity of relative canonical bundles,” Duke Math. J., vol. 145, iss. 2, pp. 341-378, 2008. · Zbl 1181.32025 · doi:10.1215/00127094-2008-054
[9] B. Berndtsson and M. Puaun, Bergman kernels and subadjunction.
[10] Z. Błocki, ”On the Ohsawa-Takegoshi extension theorem,” Univ. Iagel. Acta Math., vol. 50, pp. 53-61, 2012. · Zbl 1295.32008 · doi:10.4467/20843828AM.12.004.1122
[11] Z. Błocki, ”Suita conjecture and the Ohsawa-Takegoshi extension theorem,” Invent. Math., vol. 193, iss. 1, pp. 149-158, 2013. · Zbl 1282.32014 · doi:10.1007/s00222-012-0423-2
[12] K. Diederich and G. Herbort, ”Extension of holomorphic \(L^2\)-functions with weighted growth conditions,” Nagoya Math. J., vol. 126, pp. 141-157, 1992. · Zbl 0759.32002
[13] J. Demailly, ”On the Ohsawa-Takegoshi-Manivel \(L^2\) extension theorem,” in Complex Analysis and Geometry, Birkhäuser, Basel, 2000, vol. 188, pp. 47-82. · Zbl 0959.32019
[14] J. Demailly, Complex Analytic and Differential Geometry. · Zbl 1102.14300
[15] J. Demailly, Analytic Methods in Algebraic Geometry, Somerville, MA; Higher Education Press, Beijing: International Press, 2012, vol. 1. · Zbl 1271.14001
[16] J. Demailly, C. D. Hacon, and M. Puaun, ”Extension theorems, non-vanishing and the existence of good minimal models,” Acta Math., vol. 210, iss. 2, pp. 203-259, 2013. · Zbl 1278.14022 · doi:10.1007/s11511-013-0094-x
[17] J. E. Fornaess and R. Narasimhan, ”The Levi problem on complex spaces with singularities,” Math. Ann., vol. 248, iss. 1, pp. 47-72, 1980. · Zbl 0411.32011 · doi:10.1007/BF01349254
[18] H. M. Farkas and I. Kra, Riemann Surfaces, New York: Springer-Verlag, 1980, vol. 71. · Zbl 0475.30001
[19] H. Grauert, ”Charakterisierung der Holomorphiegebiete durch die vollständige Kählersche Metrik,” Math. Ann., vol. 131, pp. 38-75, 1956. · Zbl 0073.30203 · doi:10.1007/BF01354665
[20] H. Grauert and R. Remmert, Coherent Analytic Sheaves, New York: Springer-Verlag, 1984, vol. 265. · Zbl 0537.32001 · doi:10.1007/978-3-642-69582-7
[21] P. Griffiths and J. Harris, Principles of Algebraic Geometry, New York: Wiley-Interscience [John Wiley & Sons], 1978. · Zbl 0408.14001
[22] Q. Guan, X. Zhou, and L. Zhu, ”On the Ohsawa-Takegoshi \(L^2\) extension theorem and the twisted Bochner-Kodaira identity,” C. R. Math. Acad. Sci. Paris, vol. 349, iss. 13-14, pp. 797-800, 2011. · Zbl 1227.32014 · doi:10.1016/j.crma.2011.06.001
[23] Q. Guan and X. Zhou, ”Optimal constant problem in the \(L^2\) extension theorem,” C. R. Math. Acad. Sci. Paris, vol. 350, iss. 15-16, pp. 753-756, 2012. · Zbl 1256.32009 · doi:10.1016/j.crma.2012.08.007
[24] Q. Guan and X. Zhou, ”Generalized \(L^2\) extension theorem and a conjecture of Ohsawa,” C. R. Math. Acad. Sci. Paris, vol. 351, iss. 3-4, pp. 111-114, 2013. · Zbl 1272.32011 · doi:10.1016/j.crma.2013.01.012
[25] Q. Guan and X. Zhou, Optimal constant in an \(L^2\) extension problem and a proof of a conjecture of Ohsawa. · Zbl 1484.32015 · doi:10.1007/s11425-014-4946-4
[26] Q. Guan and X. Zhou, ”An \(L^2\) extension theorem with optimal estimate,” C. R. Math. Acad. Sci. Paris, vol. 352, iss. 2, pp. 137-141, 2014. · Zbl 1288.32015 · doi:10.1016/j.crma.2013.12.007
[27] L. Manivel, ”Un théorème de prolongement \(L^2\) de sections holomorphes d’un fibré hermitien,” Math. Z., vol. 212, iss. 1, pp. 107-122, 1993. · Zbl 0789.32015 · doi:10.1007/BF02571643
[28] J. D. McNeal and D. Varolin, ”Analytic inversion of adjunction: \(L^2\) extension theorems with gain,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 57, iss. 3, pp. 703-718, 2007. · Zbl 1208.32011 · doi:10.5802/aif.2273
[29] T. Ohsawa, ”On the extension of \(L^2\) holomorphic functions. II,” Publ. Res. Inst. Math. Sci., vol. 24, iss. 2, pp. 265-275, 1988. · Zbl 0653.32012 · doi:10.2977/prims/1195175200
[30] T. Ohsawa, ”On the extension of \(L^2\) holomorphic functions. III. Negligible weights,” Math. Z., vol. 219, iss. 2, pp. 215-225, 1995. · Zbl 0823.32006 · doi:10.1007/BF02572360
[31] T. Ohsawa, ”On the extension of \(L^2\) holomorphic functions. IV. A new density concept,” in Geometry and analysis on complex manifolds, World Sci. Publ., River Edge, NJ, 1994, pp. 157-170. · Zbl 0884.32012
[32] T. Ohsawa, ”On the extension of \(L^2\) holomorphic functions. V. Effects of generalization,” Nagoya Math. J., vol. 161, pp. 1-21, 2001. · Zbl 0986.32002
[33] T. Ohsawa, ”Erratum to \citeohsawa5a,” Nagoya Math. J., vol. 163, p. 229, 2001.
[34] T. Ohsawa, ”On the extension of \(L^2\) holomorphic functions. VI. A limiting case,” in Explorations in Complex and Riemannian Geometry, Providence, RI: Amer. Math. Soc., 2003, vol. 332, pp. 235-239. · Zbl 1049.32010 · doi:10.1090/conm/332/05939
[35] T. Ohsawa and K. Takegoshi, ”On the extension of \(L^2\) holomorphic functions,” Math. Z., vol. 195, iss. 2, pp. 197-204, 1987. · Zbl 0625.32011 · doi:10.1007/BF01166457
[36] J. Ortega-Cerdà, A. Schuster, and D. Varolin, ”Interpolation and sampling hypersurfaces for the Bargmann-Fock space in higher dimensions,” Math. Ann., vol. 335, iss. 1, pp. 79-107, 2006. · Zbl 1092.32006 · doi:10.1007/s00208-005-0726-3
[37] M. Puaun, ”Siu’s invariance of plurigenera: a one-tower proof,” J. Differential Geom., vol. 76, iss. 3, pp. 485-493, 2007. · Zbl 1122.32014
[38] L. Sario and K. Oikawa, Capacity Functions, New York: Springer-Verlag, 1969, vol. 149. · Zbl 0184.10503
[39] M. Schiffer and D. C. Spencer, Functionals of Finite Riemann Surfaces, Princeton, NJ: Princeton Univ. Press, 1954. · Zbl 0059.06901
[40] Y. Siu, ”The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi,” in Geometric Complex Analysis, River Edge, NJ: World Sci. Publ., 1996, pp. 577-592. · Zbl 0941.32021
[41] Y. Siu, ”Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type,” in Complex Geometry, Berlin: Springer-Verlag, 2002, pp. 223-277. · Zbl 1007.32010
[42] N. Suita, ”Capacities and kernels on Riemann surfaces,” Arch. Rational Mech. Anal., vol. 46, pp. 212-217, 1972. · Zbl 0245.30014 · doi:10.1007/BF00252460
[43] N. Suita and A. Yamada, ”On the Lu Qi-keng conjecture,” Proc. Amer. Math. Soc., vol. 59, iss. 2, pp. 222-224, 1976. · Zbl 0319.30013 · doi:10.2307/2041472
[44] M. Tsuji, Potential Theory in Modern Function Theory, Tokyo: Maruzen Co., Ltd., 1959. · Zbl 0087.28401
[45] A. Yamada, ”Topics related to reproducing kernels, theta functions and the Suita conjecture,” S\Burikaisekikenky\Busho K\Boky\Buroku, iss. 1067, pp. 39-47, 1998. · Zbl 0938.30509
[46] L. Zhu, Q. Guan, and X. Zhou, ”On the Ohsawa-Takegoshi \(L^2\) extension theorem and the Bochner-Kodaira identity with non-smooth twist factor,” J. Math. Pures Appl., vol. 97, iss. 6, pp. 579-601, 2012. · Zbl 1244.32005 · doi:10.1016/j.matpur.2011.09.010
[47] Y. Siu, ”Invariance of plurigenera,” Invent. Math., vol. 134, iss. 3, pp. 661-673, 1998. · Zbl 0955.32017 · doi:10.1007/s002220050276
[48] T. Ohsawa, ”Addendum to \citeohsawa95a,” Nagoya Math. J., vol. 137, pp. 145-148, 1995. · Zbl 0817.32013
[49] T. Ohsawa, ”On the Bergman kernel of hyperconvex domains,” Nagoya Math. J., vol. 129, pp. 43-52, 1993. · Zbl 0774.32016
[50] S. Chen and M. Shaw, Partial Differential Equations in Several Complex Variables, Amer. Math. Soc., 2001, vol. 19. · Zbl 0963.32001
[51] J. Demailly, ”Estimations \(L^2\) pour l’opérateur \(\bar \partial \) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète,” Ann. Sci. École Norm. Sup., vol. 15, iss. 3, pp. 457-511, 1982. · Zbl 0507.32021
[52] J. E. Fornaess and N. Sibony, ”Some open problems in higher dimensional complex analysis and complex dynamics,” Publ. Mat., vol. 45, iss. 2, pp. 529-547, 2001. · Zbl 0993.32001 · doi:10.5565/PUBLMAT_45201_11
[53] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Englewood Cliffs, N.J.: Prentice-Hall, 1965. · Zbl 0141.08601
[54] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Third ed., North-Holland Publishing Co., Amsterdam, 1990. · Zbl 0685.32001
[55] C. O. Kiselman, ”Plurisubharmonic functions and potential theory in several complex variables,” in Development of Mathematics 1950-2000, Boston: Birkhäuser, 2000, pp. 655-714. · Zbl 0962.31001
[56] Y. Siu, Function Theory of Several Complex Variables, 2013.
[57] E. J. Straube, Lectures on the \(\mathcalL^2\)-Sobolev Theory of the \(\overline{\partial}\)-Neumann Problem, Zürich: European Mathematical Society (EMS), 2010. · Zbl 1247.32003 · doi:10.4171/076
[58] X. Zhou, ”Some results related to group actions in several complex variables,” in Proceedings of the International Congress of Mathematicians, Vol. II, Beijing, 2002, pp. 743-753. · Zbl 1004.32004
[59] X. Zhou, ”Invariant holomorphic extension in several complex variables,” Sci. China Ser. A, vol. 49, iss. 11, pp. 1593-1598, 2006. · Zbl 1111.32006 · doi:10.1007/s11425-006-2072-7
[60] X. Zhou and L. F. Zhu, ”\(L^2\)-extension theorem: revisited,” in Fifth International Congress of Chinese Mathematicians. Part 1, 2, Providence, RI: Amer. Math. Soc., 2012, vol. 51. · Zbl 1250.32008
[61] H. Grauert and R. Remmert, Theory of Stein Spaces, New York: Springer-Verlag, 1979, vol. 236. · Zbl 0433.32007 · doi:10.1007/978-1-4757-4357-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.