Abstract
In this paper, we prove an $L^2$ extension theorem with an optimal estimate in a precise way, which implies optimal estimate versions of various well-known $L^2$ extension theorems. As applications, we give proofs of a conjecture of Suita on the equality condition in Suita’s conjecture, the so-called $L$-conjecture, and the extended Suita conjecture. As other applications, we give affirmative answer to a question by Ohsawa about limiting case for the extension operators between the weighted Bergman spaces, and we present a relation of our result to Berndtsson’s important result on log-plurisubharmonicity of the Bergman kernel.
-
[ahlfors] L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton, N.J.: Princeton Univ. Press, 1960, vol. 26.
@book{ahlfors, mrkey = {0114911},
author = {Ahlfors, Lars V. and Sario, Leo},
title = {Riemann Surfaces},
series = {Princeton Math. Ser.},
volume = {26},
publisher = {Princeton Univ. Press},
address = {Princeton, N.J.},
year = {1960},
pages = {xi+382},
mrclass = {30.00},
mrnumber = {0114911},
mrreviewer = {M. H. Heins},
zblnumber = {0196.33801},
} -
[berndtsson]
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.@article{berndtsson, mrkey = {1415958},
author = {Berndtsson, Bo},
title = {The extension theorem of {O}hsawa-{T}akegoshi and the theorem of {D}onnelly-{F}efferman},
journal = {Ann. Inst. Fourier $($Grenoble$)$},
fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
volume = {46},
year = {1996},
number = {4},
pages = {1083--1094},
issn = {0373-0956},
coden = {AIFUA7},
mrclass = {32D15 (32F15)},
mrnumber = {1415958},
mrreviewer = {Laurent Manivel},
doi = {10.5802/aif.1541},
zblnumber = {0853.32024},
} -
[berndtsson98] B. Berndtsson, "Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions," Math. Ann., vol. 312, iss. 4, pp. 785-792, 1998.
@article{berndtsson98, mrkey = {1660227},
author = {Berndtsson, Bo},
title = {Prekopa's theorem and {K}iselman's minimum principle for plurisubharmonic functions},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {312},
year = {1998},
number = {4},
pages = {785--792},
issn = {0025-5831},
coden = {MAANA},
mrclass = {32U05 (32A07 32W05)},
mrnumber = {1660227},
mrreviewer = {Mihnea Coltoiu},
doi = {10.1007/s002080050246},
zblnumber = {0938.32021},
} -
[berndtsson05] B. Berndtsson, "Integral formulas and the Ohsawa-Takegoshi extension theorem," Sci. China Ser. A, vol. 48, iss. suppl., pp. 61-73, 2005.
@article{berndtsson05, mrkey = {2156491},
author = {Berndtsson, Bo},
title = {Integral formulas and the {O}hsawa-{T}akegoshi extension theorem},
journal = {Sci. China Ser. A},
fjournal = {Science in China. Series A. Mathematics},
volume = {48},
year = {2005},
number = {suppl.},
pages = {61--73},
issn = {1006-9283},
mrclass = {32A26 (32W05)},
mrnumber = {2156491},
mrreviewer = {Dror Varolin},
doi = {10.1007/BF02884696},
zblnumber = {1126.32002},
} -
[berndtsson06] 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.
@article{berndtsson06, mrkey = {2282671},
author = {Berndtsson, Bo},
title = {Subharmonicity properties of the {B}ergman kernel and some other functions associated to pseudoconvex domains},
journal = {Ann. Inst. Fourier $($Grenoble$)$},
fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
volume = {56},
year = {2006},
number = {6},
pages = {1633--1662},
issn = {0373-0956},
coden = {AIFUA7},
mrclass = {32U05 (32A25 32A36 32U25)},
mrnumber = {2282671},
mrreviewer = {Gregor Herbort},
doi = {10.5802/aif.2223},
zblnumber = {1120.32021},
} -
[berndtsson09] B. Berndtsson, "Curvature of vector bundles associated to holomorphic fibrations," Ann. of Math., vol. 169, iss. 2, pp. 531-560, 2009.
@article{berndtsson09, mrkey = {2480611},
author = {Berndtsson, Bo},
title = {Curvature of vector bundles associated to holomorphic \hbox{fibrations}},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {169},
year = {2009},
number = {2},
pages = {531--560},
issn = {0003-486X},
coden = {ANMAAH},
mrclass = {32L05 (32A36 32L15)},
mrnumber = {2480611},
mrreviewer = {Bianca Santoro},
doi = {10.4007/annals.2009.169.531},
zblnumber = {1195.32012},
} -
[B-P08b] B. Berndtsson and M. Puaun, A Bergman kernel proof of the Kawamata subadjunction theorem.
@misc{B-P08b,
author = {Berndtsson, Bo and P{\u{a}}un, Mihai},
title = {A {B}ergman kernel proof of the {K}awamata subadjunction theorem},
arxiv = {0804.3884v2},
sortyear = {2020},
} -
[B-P08] 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.
@article{B-P08, mrkey = {2449950},
author = {Berndtsson, Bo and P{\u{a}}un, Mihai},
title = {Bergman kernels and the pseudoeffectivity of relative canonical bundles},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {145},
year = {2008},
number = {2},
pages = {341--378},
issn = {0012-7094},
coden = {DUMJAO},
mrclass = {32L15 (14J40)},
mrnumber = {2449950},
mrreviewer = {James McKernan},
doi = {10.1215/00127094-2008-054},
zblnumber = {1181.32025},
} -
[B-P10] B. Berndtsson and M. Puaun, Bergman kernels and subadjunction.
@misc{B-P10,
author = {Berndtsson, Bo and P{\u{a}}un, Mihai},
title = {Bergman kernels and subadjunction},
arxiv = {1002.4145v1},
sortyear = {2021},
} -
[blocki12] Z. Błocki, "On the Ohsawa-Takegoshi extension theorem," Univ. Iagel. Acta Math., vol. 50, pp. 53-61, 2012.
@article{blocki12, mrkey = {3235002},
author = {B{\l}ocki, Zbigniew},
title = {On the {O}hsawa-{T}akegoshi extension theorem},
journal = {Univ. Iagel. Acta Math.},
fjournal = {Universitatis Iagellonicae. Acta Mathematica},
volume = {50},
year = {2012},
pages = {53--61},
issn = {0083-4386},
isbn = {978-83-233-3648-8},
mrclass = {32W05 (32A25)},
mrnumber = {3235002},
mrreviewer = {Norman Levenberg},
zblnumber = {06303076},
doi = {10.4467/20843828AM.12.004.1122},
} -
[blocki12a] Z. Błocki, "Suita conjecture and the Ohsawa-Takegoshi extension theorem," Invent. Math., vol. 193, iss. 1, pp. 149-158, 2013.
@article{blocki12a, mrkey = {3069114},
author = {B{\l}ocki, Zbigniew},
title = {Suita conjecture and the {O}hsawa-{T}akegoshi extension theorem},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {193},
year = {2013},
number = {1},
pages = {149--158},
issn = {0020-9910},
mrclass = {32W05 (30C35 30C40 30C85 32D15)},
mrnumber = {3069114},
mrreviewer = {Norman Levenberg},
doi = {10.1007/s00222-012-0423-2},
zblnumber = {1282.32014},
} -
[D-H92] K. Diederich and G. Herbort, "Extension of holomorphic $L^2$-functions with weighted growth conditions," Nagoya Math. J., vol. 126, pp. 141-157, 1992.
@article{D-H92, mrkey = {1171597},
author = {Diederich, Klas and Herbort, Gregor},
title = {Extension of holomorphic {$L\sp 2$}-functions with weighted growth conditions},
journal = {Nagoya Math. J.},
fjournal = {Nagoya Mathematical Journal},
volume = {126},
year = {1992},
pages = {141--157},
issn = {0027-7630},
coden = {NGMJA2},
mrclass = {32D15 (32F15)},
mrnumber = {1171597},
mrreviewer = {Kensho Takegoshi},
url = {http://projecteuclid.org/euclid.nmj/1118783179},
zblnumber = {0759.32002},
} -
[demailly99] J. Demailly, "On the Ohsawa-Takegoshi-Manivel $L^2$ extension theorem," in Complex Analysis and Geometry, Birkhäuser, Basel, 2000, vol. 188, pp. 47-82.
@incollection{demailly99, mrkey = {1782659},
author = {Demailly, Jean-Pierre},
title = {On the {O}hsawa-{T}akegoshi-{M}anivel {$L\sp 2$} extension theorem},
booktitle = {Complex Analysis and Geometry},
venue = {{P}aris, 1997},
series = {Progr. Math.},
volume = {188},
pages = {47--82},
publisher = {Birkhäuser, Basel},
year = {2000},
mrclass = {32L10 (32D15 32J25 32U05)},
mrnumber = {1782659},
mrreviewer = {Thierry Bouche},
zblnumber = {0959.32019},
} -
[demailly-book] J. Demailly, Complex Analytic and Differential Geometry.
@misc{demailly-book,
author = {Demailly, Jean-Pierre},
title = {{\it Complex Analytic and Differential Geometry}},
url = {http://www-fourier.ujf-grenoble.fr/~demailly/books.html},
} -
[demailly2010] J. Demailly, Analytic Methods in Algebraic Geometry, Somerville, MA; Higher Education Press, Beijing: International Press, 2012, vol. 1.
@book{demailly2010, mrkey = {2978333},
author = {Demailly, Jean-Pierre},
title = {Analytic Methods in Algebraic Geometry},
series = {Surv. Modern Math.},
volume = {1},
publisher = {International Press},
address = {Somerville, MA; Higher Education Press, Beijing},
year = {2012},
pages = {viii+231},
isbn = {978-1-57146-234-3},
mrclass = {32-02 (14C30 14F18 32J25 32Q15 32U40)},
mrnumber = {2978333},
mrreviewer = {Valentino Tosatti},
zblnumber = {1271.14001},
} -
[D-P] 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.
@article{D-P, mrkey = {3070567},
author = {Demailly, Jean-Pierre and Hacon, Christopher D. and P{\u{a}}un, Mihai},
title = {Extension theorems, non-vanishing and the existence of good minimal models},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {210},
year = {2013},
number = {2},
pages = {203--259},
issn = {0001-5962},
mrclass = {14E30},
mrnumber = {3070567},
mrreviewer = {Vladimir Lazi{ć}},
doi = {10.1007/s11511-013-0094-x},
zblnumber = {1278.14022},
} -
[FN1980] J. E. Fornaess and R. Narasimhan, "The Levi problem on complex spaces with singularities," Math. Ann., vol. 248, iss. 1, pp. 47-72, 1980.
@article{FN1980, mrkey = {0569410},
author = {Forn{æ}ss, John Erik and Narasimhan, Raghavan},
title = {The {L}evi problem on complex spaces with singularities},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {248},
year = {1980},
number = {1},
pages = {47--72},
issn = {0025-5831},
coden = {MAANA3},
mrclass = {32E10},
mrnumber = {0569410},
mrreviewer = {Peter Pflug},
doi = {10.1007/BF01349254},
zblnumber = {0411.32011},
} -
[farkas] H. M. Farkas and I. Kra, Riemann Surfaces, New York: Springer-Verlag, 1980, vol. 71.
@book{farkas, mrkey = {0583745},
author = {Farkas, Hershel M. and Kra, Irwin},
title = {Riemann Surfaces},
series = {Grad. Texts in Math.},
volume = {71},
publisher = {Springer-Verlag},
year = {1980},
pages = {xi+337},
isbn = {0-387-90465-4},
mrclass = {30Fxx (14-01)},
mrnumber = {0583745},
mrreviewer = {M. Herv{é}},
address = {New York},
zblnumber = {0475.30001},
} -
[grauert] H. Grauert, "Charakterisierung der Holomorphiegebiete durch die vollständige Kählersche Metrik," Math. Ann., vol. 131, pp. 38-75, 1956.
@article{grauert, mrkey = {0077651},
author = {Grauert, Hans},
title = {Charakterisierung der {H}olomorphiegebiete durch die vollständige {K}ählersche {M}etrik},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {131},
year = {1956},
pages = {38--75},
issn = {0025-5831},
mrclass = {32.0X},
mrnumber = {0077651},
mrreviewer = {P. Lelong},
doi = {10.1007/BF01354665},
zblnumber = {0073.30203},
} -
[grauert-remmert] H. Grauert and R. Remmert, Coherent Analytic Sheaves, New York: Springer-Verlag, 1984, vol. 265.
@book{grauert-remmert, mrkey = {0755331},
author = {Grauert, Hans and Remmert, Reinhold},
title = {Coherent Analytic Sheaves},
series = {Grundl. Math. Wissen.},
volume = {265},
publisher = {Springer-Verlag},
year = {1984},
pages = {xviii+249},
isbn = {3-540-13178-7},
mrclass = {32-02 (32Bxx 32C30)},
mrnumber = {0755331},
mrreviewer = {Daniel Barlet},
doi = {10.1007/978-3-642-69582-7},
address = {New York},
zblnumber = {0537.32001},
} -
[G-H] P. Griffiths and J. Harris, Principles of Algebraic Geometry, New York: Wiley-Interscience [John Wiley & Sons], 1978.
@book{G-H, mrkey = {0507725},
author = {Griffiths, Phillip and Harris, Joseph},
title = {Principles of Algebraic Geometry},
series = {Pure Appl. Math.},
publisher = {Wiley-Interscience [John Wiley \& Sons]},
address = {New York},
year = {1978},
pages = {xii+813},
isbn = {0-471-32792-1},
mrclass = {14-01},
mrnumber = {0507725},
mrreviewer = {Gerhard Pfister},
zblnumber = {0408.14001},
} -
[guan-zhou-zhu11] 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.
@article{guan-zhou-zhu11, mrkey = {2825944},
author = {Guan, Qi'an and Zhou, Xiangyu and Zhu, Langfeng},
title = {On the {O}hsawa-{T}akegoshi {$L\sp 2$} extension theorem and the twisted {B}ochner-{K}odaira identity},
journal = {C.~R. Math. Acad. Sci. Paris},
fjournal = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
volume = {349},
year = {2011},
number = {13-14},
pages = {797--800},
issn = {1631-073X},
mrclass = {32D15 (32Q15 32Q28 32U10)},
mrnumber = {2825944},
mrreviewer = {Sergey Ivashkovich},
doi = {10.1016/j.crma.2011.06.001},
zblnumber = {1227.32014},
} -
[guan-zhou12] 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.
@article{guan-zhou12, mrkey = {2981347},
author = {Guan, Qi'an and Zhou, Xiangyu},
title = {Optimal constant problem in the {${L}\sp 2$} extension theorem},
journal = {C.~R. Math. Acad. Sci. Paris},
fjournal = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
volume = {350},
year = {2012},
number = {15-16},
pages = {753--756},
issn = {1631-073X},
mrclass = {32Q28 (32D15 32U05)},
mrnumber = {2981347},
mrreviewer = {Rasul Shafikov},
doi = {10.1016/j.crma.2012.08.007},
zblnumber = {1256.32009},
} -
[guan-zhou12a] 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.
@article{guan-zhou12a, mrkey = {3037997},
author = {Guan, Qi'an and Zhou, Xiangyu},
title = {Generalized {$L\sp 2$} extension theorem and a conjecture of {O}hsawa},
journal = {C.~R. Math. Acad. Sci. Paris},
fjournal = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
volume = {351},
year = {2013},
number = {3-4},
pages = {111--114},
issn = {1631-073X},
mrclass = {32W05},
mrnumber = {3037997},
mrreviewer = {{\.Z}ywomir Dinew},
doi = {10.1016/j.crma.2013.01.012},
zblnumber = {1272.32011},
} -
[guan-zhou12p] Q. Guan and X. Zhou, Optimal constant in an ${L}^{2}$ extension problem and a proof of a conjecture of Ohsawa.
@misc{guan-zhou12p,
author = {Guan, Qi'an and Zhou, Xiangyu},
title = {Optimal constant in an ${L}^{2}$ extension problem and a proof of a conjecture of {O}hsawa},
note = {details of \cite{guan-zhou12a}; \emph{Sci. China Math.},
published online, 2014},
sortyear = {2014},
doi = {10.1007/s11425-014-4946-4},
} -
[guan-zhou13a] 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.
@article{guan-zhou13a, mrkey = {3151882},
author = {Guan, Qi'an and Zhou, Xiangyu},
title = {An {$L\sp 2$} extension theorem with optimal estimate},
journal = {C.~R. Math. Acad. Sci. Paris},
fjournal = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
volume = {352},
year = {2014},
number = {2},
pages = {137--141},
issn = {1631-073X},
mrclass = {32L05 (32D15 32U15)},
mrnumber = {3151882},
mrreviewer = {{\.Z}ywomir Dinew},
doi = {10.1016/j.crma.2013.12.007},
zblnumber = {1288.32015},
} -
[manivel93] 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.
@article{manivel93, mrkey = {1200166},
author = {Manivel, Laurent},
title = {Un théorème de prolongement {$L\sp 2$} de sections holomorphes d'un fibré hermitien},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {212},
year = {1993},
number = {1},
pages = {107--122},
issn = {0025-5874},
coden = {MAZEAX},
mrclass = {32L10},
mrnumber = {1200166},
mrreviewer = {Antonella Nannicini},
doi = {10.1007/BF02571643},
zblnumber = {0789.32015},
} -
[mcneal-varolin] 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.
@article{mcneal-varolin, mrkey = {2336826},
author = {McNeal, Jeffery D. and Varolin, Dror},
title = {Analytic inversion of adjunction: {$L\sp 2$} extension theorems with gain},
journal = {Ann. Inst. Fourier $($Grenoble$)$},
fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
volume = {57},
year = {2007},
number = {3},
pages = {703--718},
issn = {0373-0956},
coden = {AIFUA7},
mrclass = {32D15 (32Q15)},
mrnumber = {2336826},
mrreviewer = {Shanyu Ji},
doi = {10.5802/aif.2273},
zblnumber = {1208.32011},
} -
[ohsawa2] T. Ohsawa, "On the extension of $L^2$ holomorphic functions. II," Publ. Res. Inst. Math. Sci., vol. 24, iss. 2, pp. 265-275, 1988.
@article{ohsawa2, mrkey = {0944862},
author = {Ohsawa, Takeo},
title = {On the extension of {$L\sp 2$} holomorphic functions. {II}},
journal = {Publ. Res. Inst. Math. Sci.},
fjournal = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
volume = {24},
year = {1988},
number = {2},
pages = {265--275},
issn = {0034-5318},
coden = {KRMPBV},
mrclass = {32D15 (32A10)},
mrnumber = {0944862},
mrreviewer = {Harold P. Boas},
doi = {10.2977/prims/1195175200},
zblnumber = {0653.32012},
} -
[ohsawa3] T. Ohsawa, "On the extension of $L^2$ holomorphic functions. III. Negligible weights," Math. Z., vol. 219, iss. 2, pp. 215-225, 1995.
@article{ohsawa3, mrkey = {1337216},
author = {Ohsawa, Takeo},
title = {On the extension of {$L\sp 2$} holomorphic functions. {III}. {N}egligible weights},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {219},
year = {1995},
number = {2},
pages = {215--225},
issn = {0025-5874},
coden = {MAZEAX},
mrclass = {32D15 (32A10)},
mrnumber = {1337216},
mrreviewer = {Harold P. Boas},
doi = {10.1007/BF02572360},
zblnumber = {0823.32006},
} -
[ohsawa4] 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.
@incollection{ohsawa4, mrkey = {1463969},
author = {Ohsawa, Takeo},
title = {On the extension of {$L\sp 2$} holomorphic functions. {IV}. {A} new density concept},
booktitle = {Geometry and analysis on complex manifolds},
pages = {157--170},
publisher = {World Sci. Publ., River Edge, NJ},
year = {1994},
mrclass = {32D15 (32A10)},
mrnumber = {1463969},
mrreviewer = {Harold P. Boas},
zblnumber = {0884.32012},
} -
[ohsawa5a] T. Ohsawa, "On the extension of $L^2$ holomorphic functions. V. Effects of generalization," Nagoya Math. J., vol. 161, pp. 1-21, 2001.
@article{ohsawa5a, mrkey = {1820210},
author = {Ohsawa, Takeo},
title = {On the extension of {$L\sp 2$} holomorphic functions. {V}. {E}ffects of generalization},
journal = {Nagoya Math. J.},
fjournal = {Nagoya Mathematical Journal},
volume = {161},
year = {2001},
pages = {1--21},
issn = {0027-7630},
coden = {NGMJA2},
mrclass = {32A36 (32D15)},
mrnumber = {1820210},
mrreviewer = {Harold P. Boas},
url = {http://projecteuclid.org/euclid.nmj/1114631550},
zblnumber = {0986.32002},
note={erratum: {\it Nagoya Math. J.} {\bf 163} (2001), 229.},
} -
[ohsawa5] T. Ohsawa, "Erratum to \citeohsawa5a," Nagoya Math. J., vol. 163, p. 229, 2001.
@article{ohsawa5, mrkey = {1855197},
author = {Ohsawa, Takeo},
title = {Erratum to \cite{ohsawa5a}},
journal = {Nagoya Math. J.},
fjournal = {Nagoya Mathematical Journal},
volume = {163},
year = {2001},
SORTYEAR={2002},
pages = {229},
MRNUMBER = {1855197},
URL = {http://projecteuclid.org/euclid.nmj/1114631628},
} -
[ohsawa6] 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.
@incollection{ohsawa6, mrkey = {2018342},
author = {Ohsawa, Takeo},
title = {On the extension of {$L\sp 2$} holomorphic functions. {VI}. {A} limiting case},
booktitle = {Explorations in Complex and {R}iemannian Geometry},
series = {Contemp. Math.},
volume = {332},
pages = {235--239},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2003},
mrclass = {32D15 (32A36)},
mrnumber = {2018342},
mrreviewer = {Harold P. Boas},
doi = {10.1090/conm/332/05939},
zblnumber = {1049.32010},
} -
[ohsawa-takegoshi] T. Ohsawa and K. Takegoshi, "On the extension of $L^2$ holomorphic functions," Math. Z., vol. 195, iss. 2, pp. 197-204, 1987.
@article{ohsawa-takegoshi, mrkey = {0892051},
author = {Ohsawa, Takeo and Takegoshi, Kensh{ō}},
title = {On the extension of {$L\sp 2$} holomorphic functions},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {195},
year = {1987},
number = {2},
pages = {197--204},
issn = {0025-5874},
coden = {MAZEAX},
mrclass = {32D15 (32A10)},
mrnumber = {0892051},
mrreviewer = {Harold P. Boas},
doi = {10.1007/BF01166457},
zblnumber = {0625.32011},
} -
[ort06] 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.
@article{ort06, mrkey = {2217685},
author = {Ortega-Cerd{à},
Joaquim and Schuster, Alexander and Varolin, Dror},
title = {Interpolation and sampling hypersurfaces for the {B}argmann-{F}ock space in higher dimensions},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {335},
year = {2006},
number = {1},
pages = {79--107},
issn = {0025-5831},
coden = {MAANA},
mrclass = {32A20 (32A15 32A37)},
mrnumber = {2217685},
mrreviewer = {Kristian Seip},
doi = {10.1007/s00208-005-0726-3},
zblnumber = {1092.32006}
} -
[paun07] M. Puaun, "Siu’s invariance of plurigenera: a one-tower proof," J. Differential Geom., vol. 76, iss. 3, pp. 485-493, 2007.
@article{paun07, mrkey = {2331528},
author = {P{\u{a}}un, Mihai},
title = {Siu's invariance of plurigenera: a one-tower proof},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {76},
year = {2007},
number = {3},
pages = {485--493},
issn = {0022-040X},
coden = {JDGEAS},
mrclass = {32L10 (32J25 32W05)},
mrnumber = {2331528},
mrreviewer = {Julien Keller},
url = {http://projecteuclid.org/euclid.jdg/1180135695},
zblnumber = {1122.32014},
} -
[sario] L. Sario and K. Oikawa, Capacity Functions, New York: Springer-Verlag, 1969, vol. 149.
@book{sario, mrkey = {0254232},
author = {Sario, L. and Oikawa, K.},
title = {Capacity Functions},
series={Grundl. Math. Wissen.},
volume = {149},
publisher = {Springer-Verlag},
year = {1969},
pages = {xvii+361},
mrclass = {30.45},
mrnumber = {0254232},
mrreviewer = {Alexander Dinghas},
address = {New York},
zblnumber = {0184.10503},
} -
[schiffer] M. Schiffer and D. C. Spencer, Functionals of Finite Riemann Surfaces, Princeton, NJ: Princeton Univ. Press, 1954.
@book{schiffer, mrkey = {0065652},
author = {Schiffer, Menahem and Spencer, Donald C.},
title = {Functionals of Finite {R}iemann Surfaces},
publisher = {Princeton Univ. Press},
address = {Princeton, NJ},
year = {1954},
pages = {x+451},
mrclass = {30.0X},
mrnumber = {0065652},
mrreviewer = {M. Heins},
zblnumber = {0059.06901},
} -
[siu96] 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.
@incollection{siu96, mrkey = {1453639},
author = {Siu, Yum-Tong},
title = {The {F}ujita conjecture and the extension theorem of {O}hsawa-{T}akegoshi},
booktitle = {Geometric Complex Analysis},
venue = {{H}ayama, 1995},
pages = {577--592},
publisher = {World Sci. Publ.},
address = {River Edge, NJ},
year = {1996},
mrclass = {32L10 (14C20 32J25)},
mrnumber = {1453639},
mrreviewer = {Thierry Bouche},
zblnumber = {0941.32021},
} -
[siu00] 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.
@incollection{siu00, mrkey = {1922108},
author = {Siu, Yum-Tong},
title = {Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type},
booktitle = {Complex Geometry},
venue = {{G}öttingen, 2000},
pages = {223--277},
publisher = {Springer-Verlag},
year = {2002},
mrclass = {32L10 (14J15 32G13 32J18)},
mrnumber = {1922108},
mrreviewer = {Laurent Manivel},
address = {Berlin},
url = {http://projecteuclid.org/euclid.nmj/1114631736},
zblnumber = {1007.32010},
} -
[suita] N. Suita, "Capacities and kernels on Riemann surfaces," Arch. Rational Mech. Anal., vol. 46, pp. 212-217, 1972.
@article{suita, mrkey = {0367181},
author = {Suita, Nobuyuki},
title = {Capacities and kernels on {R}iemann surfaces},
journal = {Arch. Rational Mech. Anal.},
fjournal = {Archive for Rational Mechanics and Analysis},
volume = {46},
year = {1972},
pages = {212--217},
issn = {0003-9527},
mrclass = {30A44},
mrnumber = {0367181},
mrreviewer = {M. Ohtsuka},
doi = {10.1007/BF00252460},
zblnumber = {0245.30014},
} -
[suita76] N. Suita and A. Yamada, "On the Lu Qi-keng conjecture," Proc. Amer. Math. Soc., vol. 59, iss. 2, pp. 222-224, 1976.
@article{suita76, mrkey = {0425185},
author = {Suita, Nobuyuki and Yamada, Akira},
title = {On the {L}u {Q}i-keng conjecture},
journal = {Proc. Amer. Math. Soc.},
fjournal = {Proceedings of the American Mathematical Society},
volume = {59},
year = {1976},
number = {2},
pages = {222--224},
issn = {0002-9939},
mrclass = {32H10 (30A31)},
mrnumber = {0425185},
mrreviewer = {Peter Russev},
doi = {10.2307/2041472},
zblnumber = {0319.30013},
} -
[tsuji] M. Tsuji, Potential Theory in Modern Function Theory, Tokyo: Maruzen Co., Ltd., 1959.
@book{tsuji, mrkey = {0114894},
author = {Tsuji, M.},
title = {Potential Theory in Modern Function Theory},
publisher = {Maruzen Co., Ltd.},
address = {Tokyo},
year = {1959},
pages = {590},
mrclass = {30.00},
mrnumber = {0114894},
mrreviewer = {A. J. Lohwater},
zblnumber = {0087.28401},
} -
[yamada98] A. Yamada, "Topics related to reproducing kernels, theta functions and the Suita conjecture," Sūrikaisekikenkyūsho Kōkyūroku, iss. 1067, pp. 39-47, 1998.
@article{yamada98, mrkey = {1703023},
author = {Yamada, Akira},
title = {Topics related to reproducing kernels, theta functions and the {S}uita conjecture},
note = {The theory of reproducing kernels and their applications (Japanese) (Kyoto, 1998)},
journal = {Sūrikaisekikenkyūsho Kōkyūroku},
fjournal = {Sūrikaisekikenkyūsho Kōkyūroku},
number = {1067},
year = {1998},
pages = {39--47},
mrclass = {30C40},
mrnumber = {1703023},
zblnumber = {0938.30509},
} -
[guan-zhou-zhu10] 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.
@article{guan-zhou-zhu10, mrkey = {2921602},
author = {Zhu, Langfeng and Guan, Qi'an and Zhou, Xiangyu},
title = {On the {O}hsawa-{T}akegoshi {$L\sp 2$} extension theorem and the {B}ochner-{K}odaira identity with non-smooth twist factor},
journal = {J. Math. Pures Appl.},
fjournal = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
volume = {97},
year = {2012},
number = {6},
pages = {579--601},
issn = {0021-7824},
mrclass = {32F32 (32D15 32Q28 32U05 32W05)},
mrnumber = {2921602},
mrreviewer = {Debraj Chakrabarti},
doi = {10.1016/j.matpur.2011.09.010},
zblnumber = {1244.32005},
} -
[siu98] Y. Siu, "Invariance of plurigenera," Invent. Math., vol. 134, iss. 3, pp. 661-673, 1998.
@article {siu98, MRKEY = {1660941},
AUTHOR = {Siu, Yum-Tong},
TITLE = {Invariance of plurigenera},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {134},
YEAR = {1998},
NUMBER = {3},
PAGES = {661--673},
ISSN = {0020-9910},
MRCLASS = {32L10 (32G13 32J18)},
MRNUMBER = {1660941},
MRREVIEWER = {J{á}nos Koll{á}r},
DOI = {10.1007/s002220050276},
} -
[ohsawa95] T. Ohsawa, "Addendum to \citeohsawa95a," Nagoya Math. J., vol. 137, pp. 145-148, 1995.
@article {ohsawa95, MRKEY = {1324546},
AUTHOR = {Ohsawa, Takeo},
TITLE = {Addendum to \cite{ohsawa95a}},
JOURNAL = {Nagoya Math. J.},
FJOURNAL = {Nagoya Mathematical Journal},
VOLUME = {137},
YEAR = {1995},
PAGES = {145--148},
ISSN = {0027-7630},
CODEN = {NGMJA2},
MRCLASS = {32H10},
MRNUMBER = {1324546},
MRREVIEWER = {So-Chin Chen},
ZBLNUMBER = {0817.32013},
URL = {http://projecteuclid.org/euclid.nmj/1118775538},
} -
[ohsawa95a] T. Ohsawa, "On the Bergman kernel of hyperconvex domains," Nagoya Math. J., vol. 129, pp. 43-52, 1993.
@article {ohsawa95a, MRKEY = {1210002},
AUTHOR = {Ohsawa, Takeo},
TITLE = {{O}n the {B}ergman kernel of hyperconvex domains},
JOURNAL = {Nagoya Math. J.},
FJOURNAL = {Nagoya Mathematical Journal},
VOLUME = {129},
YEAR = {1993},
PAGES = {43--52},
MRNUMBER = {1210002},
ZBLNUMBER = {0774.32016},
URL = {http://projecteuclid.org/euclid.nmj/1118783357},
} -
[chen-shaw] S. Chen and M. Shaw, Partial Differential Equations in Several Complex Variables, Amer. Math. Soc., 2001, vol. 19.
@book {chen-shaw, MRKEY = {1800297},
AUTHOR = {Chen, So-Chin and Shaw, Mei-Chi},
TITLE = {Partial Differential Equations in Several Complex Variables},
SERIES = {AMS/IP Stud. Adv. Math.},
VOLUME = {19},
PUBLISHER = {Amer. Math. Soc.},
YEAR = {2001},
PAGES = {xii+380},
ISBN = {0-8218-1062-6},
MRCLASS = {32Wxx (32-02 35A20 35N15 35S05 47N20)},
MRNUMBER = {1800297},
MRREVIEWER = {Harold P. Boas},
ZBLNUMBER = {0963.32001},
} -
[demailly82] 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.
@article {demailly82, MRKEY = {0690650},
AUTHOR = {Demailly, Jean-Pierre},
TITLE = {Estimations {$L\sp{2}$} pour l'opérateur {$\bar \partial $} d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {15},
YEAR = {1982},
NUMBER = {3},
PAGES = {457--511},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {32L15 (32L20)},
MRNUMBER = {0690650},
MRREVIEWER = {Autorreferat},
URL = {http://www.numdam.org/item?id=ASENS_1982_4_15_3_457_0},
ZBLNUMBER = {0507.32021},
} -
[FSi] 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.
@article {FSi, MRKEY = {1876919},
AUTHOR = {Forn{æ}ss, John Erik and Sibony, Nessim},
TITLE = {Some open problems in higher dimensional complex analysis and complex dynamics},
JOURNAL = {Publ. Mat.},
FJOURNAL = {Publicacions Matemàtiques},
VOLUME = {45},
YEAR = {2001},
NUMBER = {2},
PAGES = {529--547},
ISSN = {0214-1493},
MRCLASS = {32H50 (32E40 32W05 37Fxx)},
MRNUMBER = {1876919},
MRREVIEWER = {Marco Abate},
ZBLNUMBER = {0993.32001},
DOI = {10.5565/PUBLMAT_45201_11},
} -
[gunning-rossi] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Englewood Cliffs, N.J.: Prentice-Hall, 1965.
@book {gunning-rossi, MRKEY = {0180696},
AUTHOR = {Gunning, Robert C. and Rossi, Hugo},
TITLE = {Analytic Functions of Several Complex Variables},
PUBLISHER = {Prentice-Hall},
ADDRESS={Englewood Cliffs, N.J.},
YEAR = {1965},
PAGES = {xiv+317},
MRCLASS = {32.00},
MRNUMBER = {0180696},
MRREVIEWER = {R. P. Gilbert},
ZBLNUMBER = {0141.08601},
} -
[hormander] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Third ed., North-Holland Publishing Co., Amsterdam, 1990.
@book {hormander, MRKEY = {1045639},
AUTHOR = {H{ö}rmander, Lars},
TITLE = {An {I}ntroduction to {C}omplex {A}nalysis in {S}everal {V}ariables},
SERIES = {North-Holland Mathematical Library},
NUMBER = {7},
EDITION = {Third},
PUBLISHER = {North-Holland Publishing Co., Amsterdam},
YEAR = {1990},
PAGES = {xii+254},
ISBN = {0-444-88446-7},
MRCLASS = {32-01 (35N15)},
MRNUMBER = {1045639},
ZBLNUMBER = {0685.32001},
} -
[ki2000] C. O. Kiselman, "Plurisubharmonic functions and potential theory in several complex variables," in Development of Mathematics 1950–2000, Boston: Birkhäuser, 2000, pp. 655-714.
@incollection {ki2000, MRKEY = {1796855},
AUTHOR = {Kiselman, Christer O.},
TITLE = {Plurisubharmonic functions and potential theory in several complex variables},
BOOKTITLE = {Development of Mathematics 1950--2000},
PAGES = {655--714},
PUBLISHER = {Birkhäuser},
ADDRESS={Boston},
YEAR = {2000},
MRCLASS = {32Uxx (01A60 31C10 32-03 32W20)},
MRNUMBER = {1796855},
MRREVIEWER = {Norman Levenberg},
ZBLNUMBER = {0962.31001},
} -
[siu13] Y. Siu, Function Theory of Several Complex Variables, 2013.
@misc{siu13,
author = {Siu, Yum-Tong},
TITLE = {{\it Function Theory of Several Complex Variables}},
NOTE={in Chinese; notes by Chen Zhihua and Zhong Jiaqing, Higher Educational Press},
YEAR={2013},
} -
[Straube] E. J. Straube, Lectures on the $\mathcal{L}^2$-Sobolev Theory of the $\overline{\partial}$-Neumann Problem, Zürich: European Mathematical Society (EMS), 2010.
@book {Straube, MRKEY = {2603659},
AUTHOR = {Straube, Emil J.},
TITLE = {Lectures on the {$\mathcal{L}\sp 2$}-{S}obolev Theory of the {$\overline{\partial}$}-{N}eumann Problem},
SERIES = {ESI Lect. Math. Phys.},
PUBLISHER = {European Mathematical Society (EMS)},
ADDRESS={Zürich},
YEAR = {2010},
PAGES = {viii+206},
ISBN = {978-3-03719-076-0},
MRCLASS = {35-02 (32W05 35N15)},
MRNUMBER = {2603659},
MRREVIEWER = {Gerald B. Folland},
DOI = {10.4171/076},
ZBLNUMBER = {1247.32003},
} -
[zhou2002] 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.
@inproceedings {zhou2002, MRKEY = {1957081},
AUTHOR = {Zhou, Xiangyu},
TITLE = {Some results related to group actions in several complex variables},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {II}},
VENUE={{B}eijing, 2002},
PAGES = {743--753},
PUBLISHER = {Higher Ed. Press},
ADDRESS={Beijing},
YEAR = {2002},
MRCLASS = {32T05 (32D05 32M05 32T35)},
MRNUMBER = {1957081},
MRREVIEWER = {Marek Jarnicki},
ZBLNUMBER = {1004.32004},
} -
[zhou] X. Zhou, "Invariant holomorphic extension in several complex variables," Sci. China Ser. A, vol. 49, iss. 11, pp. 1593-1598, 2006.
@article {zhou, MRKEY = {2288216},
AUTHOR = {Zhou, Xiangyu},
TITLE = {Invariant holomorphic extension in several complex variables},
JOURNAL = {Sci. China Ser. A},
FJOURNAL = {Science in China. Series A. Mathematics},
VOLUME = {49},
YEAR = {2006},
NUMBER = {11},
PAGES = {1593--1598},
ISSN = {1006-9283},
MRCLASS = {32D15 (32C35 32E10 32U05)},
MRNUMBER = {2288216},
MRREVIEWER = {Alberto Saracco},
DOI = {10.1007/s11425-006-2072-7},
ZBLNUMBER = {1111.32006},
} -
[zhou-zhu2012] 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.
@incollection{zhou-zhu2012,
author = {Zhou, Xiangyu and Zhu, L. F.},
TITLE = {${L}^2$-extension theorem: revisited},
BOOKTITLE={Fifth International Congress of Chinese Mathematicians. Part 1, 2},
SERIES={AMS/IP Stud. Adv. Math.},
VOLUME={51},
PUBLISHER={Amer. Math. Soc.},
ADDRESS={Providence, RI},
YEAR={2012},
MRNUMBER = {2908088},
ZBLNUMBER = {1250.32008},
} -
[grauert-remmert1] H. Grauert and R. Remmert, Theory of Stein Spaces, New York: Springer-Verlag, 1979, vol. 236.
@book {grauert-remmert1, MRKEY = {0580152},
AUTHOR = {Grauert, Hans and Remmert, Reinhold},
TITLE = {Theory of {S}tein Spaces},
SERIES = {Grundl. Math. Wissen.},
VOLUME = {236},
NOTE = {translated from the German by Alan Huckleberry},
PUBLISHER = {Springer-Verlag},
YEAR = {1979},
PAGES = {xxi+249},
ISBN = {3-540-90388-7},
MRCLASS = {32-01},
MRNUMBER = {0580152},
ADDRESS = {New York},
ZBLNUMBER = {0433.32007},
DOI = {10.1007/978-1-4757-4357-9},
}
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