Siu’s lemma: generalizations and applications. (English) Zbl 1528.32016

Zhou, Xiangyu; Zhu, Langfeng

Hujdurović, Ademir (ed.) et al., European congress of mathematics. Proceedings of the 8th congress, 8ECM, Portorož, Slovenia, June 20–26, 2021. Berlin: European Mathematical Society (EMS). 947-967 (2023).

Summary: In this survey paper, we present some generalizations of Siu’s lemma related to multiplier ideal sheaves and discuss their applications in some problems related to optimal \(L^2\) extension, comparison between singular metrics on exceptional fibers of twisted relative pluricanonical bundles, and subadditivity of Kodaira-Iitaka dimensions with multiplier ideal sheaves. We also discuss some ideas in the proofs.
For the entire collection see [Zbl 1519.00033].

MSC:

32D15	Continuation of analytic objects in several complex variables
14F18	Multiplier ideals
32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results
32W05	\(\overline\partial\) and \(\overline\partial\)-Neumann operators
32U05	Plurisubharmonic functions and generalizations

Keywords:

optimal \(L^2\) extension; multiplier ideal sheaf; singularities of plurisubharmonic functions; strong openness

References:

[1]	U. Angehrn and Y. T. Siu, Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, 291-308 Zbl 0847.32035 MR 1358978 · Zbl 0847.32035
[2]	B. Berndtsson, The openness conjecture and complex Brunn-Minkowski inequalities. In Complex Geometry and Dynamics, pp. 29-44, Abel Symp. 10, Springer, Cham, 2015 Zbl 1337.32001 MR 3587460 · Zbl 1337.32001
[3]	B. Berndtsson and M. Pȃun, Bergman kernels and the pseudoeffectivity of relative canon-ical bundles. Duke Math. J. 145 (2008), no. 2, 341-378 Zbl 1181.32025 MR 2449950 · Zbl 1181.32025
[4]	B. Berndtsson and M. Pȃun, A Bergman kernel proof of the Kawamata subadjunction theorem. 2008, arXiv:0804.3884
[5]	B. Berndtsson and M. Pȃun, Bergman kernels and subadjunction. 2010, arXiv:1002.4145
[6]	J. Cao, Ohsawa-Takegoshi extension theorem for compact Kähler manifolds and appli-cations. In Complex and Symplectic Geometry, pp. 19-38, Springer INdAM Ser. 21, Springer, Cham, 2017 Zbl 1405.32029 MR 3645303 · Zbl 1405.32029
[7]	J.-P. Demailly, Singular Hermitian metrics on positive line bundles. In Complex Algebraic Varieties (Bayreuth, 1990), pp. 87-104, Lecture Notes in Math. 1507, Springer, Berlin, 1992 Zbl 0784.32024 MR 1178721 · Zbl 0784.32024
[8]	J.-P. Demailly, Analytic Methods in Algebraic Geometry. Surv. Mod. Math. 1, Interna-tional Press, Somerville, MA; Higher Education Press, Beijing, 2012 Zbl 1271.14001 MR 2978333
[9]	J.-P. Demailly, Extension of holomorphic functions defined on non reduced analytic sub-varieties. In The Legacy of Bernhard Riemann After One Hundred and Fifty Years. Vol. I, pp. 191-222, Adv. Lect. Math. (ALM) 35, Int. Press, Somerville, MA, 2016 Zbl 1360.14025 MR 3525916 · Zbl 1360.14025
[10]	J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 4, 525-556 Zbl 0994.32021 MR 1852009 · Zbl 0994.32021
[11]	J. E. Fornaess, Several complex variables. 2015, arXiv:1507.00562
[12]	Q. Guan and X. Zhou, Effectiveness of Demailly’s strong openness conjecture and related problems. Invent. Math. 202 (2015), no. 2, 635-676 Zbl 1333.32014 MR 3418242 · Zbl 1333.32014
[13]	Q. Guan and X. Zhou, A proof of Demailly’s strong openness conjecture. Ann. of Math. (2) 182 (2015), no. 2, 605-616 Zbl 1329.32016 MR 3418526 · Zbl 1329.32016
[14]	Q. Guan and X. Zhou, A solution of an L 2 extension problem with an optimal estimate and applications. Ann. of Math. (2) 181 (2015), no. 3, 1139-1208 Zbl 1348.32008 MR 3296822 · Zbl 1348.32008
[15]	Q. A. Guan, Z. Q. Li, and X. Y. Zhou, Estimation of weighted L 2 norm related to Demailly’s Strong Openness Conjecture. 2016, arXiv:1603.05733
[16]	C. Hacon, M. Popa, and C. Schnell, Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Pȃun. In Local and Global Methods in Algebraic Geometry, pp. 143-195, Contemp. Math. 712, Amer. Math. Soc., Providence, RI, 2018 Zbl 1398.14018 MR 3832403 · Zbl 1398.14018
[17]	Y. Kawamata, Characterization of abelian varieties. Compositio Math. 43 (1981), no. 2, 253-276 Zbl 0471.14022 MR 622451 · Zbl 0471.14022
[18]	A. M. Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar cur-vature. Ann. of Math. (2) 132 (1990), no. 3, 549-596 Zbl 0731.53063 MR 1078269 · Zbl 0731.53063
[19]	T. Ohsawa, On the extension of L 2 holomorphic functions. V. Effects of generalization. Nagoya Math. J. 161 (2001), 1-21 Zbl 0986.32002 MR 1820210 · Zbl 0986.32002
[20]	T. Ohsawa, L 2 Approaches in Several Complex Variables, Towards the Oka-Cartan The-ory with Precise Bounds. Springer Monogr. Math., Springer, Tokyo, 2018 Zbl 1439.32003 MR 3887636
[21]	T. Ohsawa and K. Takegoshi, On the extension of L 2 holomorphic functions. Math. Z. 195 (1987), no. 2, 197-204 Zbl 0625.32011 MR 892051 · Zbl 0625.32011
[22]	D. H. Phong and J. Sturm, Algebraic estimates, stability of local zeta functions, and uni-form estimates for distribution functions. Ann. of Math. (2) 152 (2000), no. 1, 277-329 Zbl 0995.11065 MR 1792297 · Zbl 0995.11065
[23]	M. Pȃun and S. Takayama, Positivity of twisted relative pluricanonical bundles and their direct images. J. Algebraic Geom. 27 (2018), no. 2, 211-272 Zbl 1430.14017 MR 3764276 · Zbl 1430.14017
[24]	Siu’s lemma: Generalizations and applications 967
[25]	Y.-T. Siu, The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi. In Geo-metric Complex Analysis (Hayama, 1995), pp. 577-592, World Sci. Publ., River Edge, NJ, 1996 Zbl 0941.32021 MR 1453639
[26]	Y.-T. Siu, Invariance of plurigenera and torsion-freeness of direct image sheaves of pluri-canonical bundles. In Finite or Infinite Dimensional Complex Analysis and Applications, pp. 45-83, Adv. Complex Anal. Appl. 2, Kluwer Acad. Publ., Dordrecht, 2004 Zbl 1044.32016 MR 2058399 · Zbl 1044.32016
[27]	E. Viehweg, Die Additivität der Kodaira Dimension für projektive Faserräume über Vari-etäten des allgemeinen Typs. J. Reine Angew. Math. 330 (1982), 132-142 · Zbl 0466.14009
[28]	J. Wang, On the Iitaka conjecture C n;m for Kähler fibre spaces. Ann. Fac. Sci. Toulouse Math. (6) 30 (2021), no. 4, 813-897 Zbl 07469482 MR 4350100 · Zbl 1535.32022
[29]	L. Yi, An Ohsawa-Takegoshi theorem on compact Kähler manifolds. Sci. China Math. 57 (2014), no. 1, 9-30 Zbl 1302.32035 MR 3146512 · Zbl 1302.32035
[30]	X. Zhou and L. Zhu, A generalized Siu’s lemma. Math. Res. Lett. 24 (2017), no. 6, 1897-1913 Zbl 1394.32027 MR 3762700 · Zbl 1394.32027
[31]	X. Zhou and L. Zhu, An optimal L 2 extension theorem on weakly pseudoconvex Kähler manifolds. J. Differential Geom. 110 (2018), no. 1, 135-186 Zbl 1426.53082 MR 3851746 · Zbl 1426.53082
[32]	X. Zhou and L. Zhu, Optimal L 2 extension of sections from subvarieties in weakly pseu-doconvex manifolds. Pacific J. Math. 309 (2020), no. 2, 475-510 Zbl 1458.32013 MR 4202022 · Zbl 1458.32013
[33]	X. Zhou and L. Zhu, Siu’s lemma, optimal L 2 extension and applications to twisted pluri-canonical sheaves. Math. Ann. 377 (2020), no. 1-2, 675-722 Zbl 1452.32014 MR 4099619 · Zbl 1452.32014
[34]	X. Zhou and L. Zhu, Subadditivity of generalized Kodaira dimensions and extension the-orems. Internat. J. Math. 31 (2020), no. 12, 2050098, 36 Zbl 1457.32059 MR 4184430 · Zbl 1457.32059
[35]	X. Zhou and L. Zhu, Extension of cohomology classes and holomorphic sections defined on subvarieties. J. Algebraic Geom. 31 (2022), no. 1, 137-179 Zbl 07459642 MR 4372411 · Zbl 1487.32108
[36]	X. Y. Zhou and L. F. Zhu, Optimal L 2 extension and Siu’s lemma. Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 8, 1289-1296 Zbl 1402.32013 MR 3843436 · Zbl 1402.32013
[37]	Xiangyu Zhou Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, P. R. China;
[38]	Langfeng Zhu School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China;