## Recent results in several complex variables and complex geometry.(English. Russian original)Zbl 1458.32012

Proc. Steklov Inst. Math. 311, 245-260 (2020); translation from Tr. Mat. Inst. Steklova 311, 264-281 (2020).
Summary: We first recall the background and contents of our recent solutions of the optimal $$L^2$$ extension problem and Demailly’s strong openness conjecture on multiplier ideal sheaves and related results, and then present some new related results in several complex variables and complex geometry.

### MSC:

 32D15 Continuation of analytic objects in several complex variables 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 32Q15 Kähler manifolds 32U05 Plurisubharmonic functions and generalizations 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators
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### References:

 [1] Berndtsson, B., The openness conjecture and complex Brunn-Minkowski inequalities, Complex Geometry and Dynamics: The Abel Symposium 2013, 10, 29-44 (2015) · Zbl 1337.32001 [2] Berndtsson, B.; Păun, M., Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J., 145, 2, 341-378 (2008) · Zbl 1181.32025 [3] B. Berndtsson and M. Păun, “Bergman kernels and subadjunction,” arXiv: 1002.4145v1 [math.AG]. · Zbl 1181.32025 [4] Blel, M.; Mimouni, S. K., Singularité et intégrabilité des fonctions plurisousharmoniques, Ann. Inst. Fourier, 55, 2, 319-351 (2005) · Zbl 1075.31007 [5] Błocki, Z., On the Ohsawa-Takegoshi extension theorem, Univ. Iagel. Acta Math., 50, 53-61 (2012) · Zbl 1295.32008 [6] Błocki, Z., Suita conjecture and the Ohsawa-Takegoshi extension theorem, Invent. Math., 193, 1, 149-158 (2013) · Zbl 1282.32014 [7] Boucksom, S.; Favre, C.; Jonsson, M., Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci., 44, 2, 449-494 (2008) · Zbl 1146.32017 [8] Cao, J.; Demailly, J.-P.; Matsumura, S., A general extension theorem for cohomology classes on non reduced analytic subspaces, Sci. China, Math., 60, 6, 949-962 (2017) · Zbl 1379.32017 [9] Demailly, J.-P., On the Ohsawa-Takegoshi-Manivel $$L^2$$ extension theorem, Complex Analysis and Geometry: Proc. Int. Conf. in Honour of Pierre Lelong on the Occasion of His 85th Birthday, Paris, 1997, 188, 47-82 (2000) · Zbl 0959.32019 [10] Demailly, J.-P., Multiplier ideal sheaves and analytic methods in algebraic geometry, School on Vanishing Theorems and Effective Results in Algebraic Geometry, Trieste, 2000, 6, 1-148 (2001) · Zbl 1102.14300 [11] Demailly, J.-P., Kähler manifolds and transcendental techniques in algebraic geometry, Proc. Int. Congr. Math. 2006,, 0, 153-186 (2007) · Zbl 1141.14007 [12] Demailly, J.-P., Analytic Methods in Algebraic Geometry (2012), Somerville, MA: Int. Press, Somerville, MA · Zbl 1271.14001 [13] Demailly, J.-P., Complex Analytic and Differential Geometry (2012), Grenoble: Univ. Grenoble I, Inst. Fourier, Grenoble [14] Demailly, J.-P., Extension of holomorphic functions defined on non reduced analytic subvarieties, The Legacy of Bernhard Riemann after One Hundred and Fifty Years, 35.1, 191-222 (2016) · Zbl 1360.14025 [15] Demailly, J.-P.; Ein, L.; Lazarsfeld, R., A subadditivity property of multiplier ideals, Mich. Math. J., 48, 137-156 (2000) · Zbl 1077.14516 [16] Demailly, J. P.; Hacon, C. D.; Păun, M., Extension theorems, non-vanishing and the existence of good minimal models, Acta Math., 210, 2, 203-259 (2013) · Zbl 1278.14022 [17] Demailly, J.-P.; Kollár, J., Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér., Sér. 4,, 34, 4, 525-556 (2001) · Zbl 0994.32021 [18] Deng, F.; Wang, Z.; Zhang, L.; Zhou, X., Linear invariants of complex manifolds and their plurisubharmonic variations, J. Funct. Anal., 279, 1 (2020) · Zbl 1436.53049 [19] Deng, F.; Zhang, H.; Zhou, X., Positivity of direct images of positively curved volume forms, Math. Z., 278, 1-2, 347-362 (2014) · Zbl 1304.32022 [20] Deng, F.; Zhang, H.; Zhou, X., Positivity of character subbundles and minimum principle for noncompact group actions, Math. Z., 286, 1-2, 431-442 (2017) · Zbl 1432.32021 [21] Deng, F.; Zhou, X. Y., Rigidity of automorphism groups of invariant domains in homogeneous Stein spaces, Izv. Math., 78, 1, 34-58 (2014) · Zbl 1293.14010 [22] Enoki, I., Kawamata-Viehweg vanishing theorem for compact Kähler manifolds, Einstein Metrics and Yang-Mills Connections: Proc. 27th Taniguchi Int. Symp., Sanda, 1990, 145, 59-68 (1993) · Zbl 0797.53052 [23] Favre, C.; Jonsson, M., Valuative analysis of planar plurisubharmonic functions, Invent. Math., 162, 2, 271-311 (2005) · Zbl 1089.32032 [24] Favre, C.; Jonsson, M., Valuations and multiplier ideals, J. Am. Math. Soc., 18, 3, 655-684 (2005) · Zbl 1075.14001 [25] Forstnerič, F., Stein Manifolds and Holomorphic Mappings. The Homotopy Principle in Complex Analysis (2017), Cham: Springer, Cham · Zbl 1382.32001 [26] Fujino, O., A transcendental approach to Kollár’s injectivity theorem. II, J. Reine Angew. Math., 681, 149-174 (2013) · Zbl 1285.32009 [27] O. Fujino and S. Matsumura, “Injectivity theorem for pseudo-effective line bundles and its applications,” arXiv: 1605.02284 [math.CV]. [28] Gongyo, Y.; Matsumura, S., Versions of injectivity and extension theorems, Ann. Sci. Éc. Norm. Supér., Sér. 4,, 50, 2, 479-502 (2017) · Zbl 1401.14083 [29] Grauert, H., On Levi’s problem and the imbedding of real-analytic manifolds, Ann. Math., Ser. 2,, 68, 460-472 (1958) · Zbl 0108.07804 [30] Grauert, H., Selected Papers (1994), Berlin: Springer, Berlin · Zbl 0820.01028 [31] Grauert, H.; Remmert, R., Theory of Stein Spaces (1979), Berlin: Springer, Berlin · Zbl 0433.32007 [32] Grauert, H.; Remmert, R., Coherent Analytic Sheaves (1984), Berlin: Springer, Berlin · Zbl 0537.32001 [33] Guan, Q.; Zhou, X., Optimal constant problem in the $$L^2$$ extension theorem, C. R., Math., Acad. Sci. Paris, 350, 15-16, 753-756 (2012) · Zbl 1256.32009 [34] Guan, Q.; Zhou, X., Optimal constant in an $$L^2$$ extension problem and a proof of a conjecture of Ohsawa, Sci. China, Math., 58, 1, 35-59 (2015) · Zbl 1484.32015 [35] Guan, Q.; Zhou, X., A solution of an $$L^2$$ extension problem with an optimal estimate and applications, Ann. Math., Ser. 2,, 181, 3, 1139-1208 (2015) · Zbl 1348.32008 [36] Guan, Q.; Zhou, X., A proof of Demailly’s strong openness conjecture, Ann. Math., Ser. 2,, 182, 2, 605-616 (2015) · Zbl 1329.32016 [37] Guan, Q.; Zhou, X., Effectiveness of Demailly’s strong openness conjecture and related problems, Invent. Math., 202, 2, 635-676 (2015) · Zbl 1333.32014 [38] Guan, Q.; Zhou, X., Characterization of multiplier ideal sheaves with weights of Lelong number one, Adv. Math., 285, 1688-1705 (2015) · Zbl 1343.32025 [39] Guan, Q. A.; Zhou, X. Y., Strong openness of multiplier ideal sheaves and optimal $$L^2$$ extension, Sci. China, Math., 60, 6, 967-976 (2017) · Zbl 1386.32031 [40] Guan, Q.; Zhou, X., Restriction formula and subadditivity property related to multiplier ideal sheaves, J. Reine Angew. Math., 769, 1-33 (2020) [41] Guan, Q.; Zhou, X.; Zhu, L., On the Ohsawa-Takegoshi $$L^2$$ extension theorem and the twisted Bochner-Kodaira identity, C. R., Math., Acad. Sci. Paris, 349, 13-14, 797-800 (2011) · Zbl 1227.32014 [42] Hacon, C.; Popa, M.; Schnell, C., Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and Păun, Local and Global Methods in Algebraic Geometry, 712, 143-195 (2018) · Zbl 1398.14018 [43] Hörmander, L., An Introduction to Complex Analysis in Several Variables (1990), Amsterdam: North-Holland, Amsterdam · Zbl 0685.32001 [44] Jonsson, M.; Mustaţă, M., Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier, 62, 6, 2145-2209 (2012) · Zbl 1272.14016 [45] Jonsson, M.; Mustaţă, M., An algebraic approach to the openness conjecture of Demailly and Kollár, J. Inst. Math. Jussieu, 13, 1, 119-144 (2014) · Zbl 1314.32047 [46] Kiselman, C. O., Plurisubharmonic functions and potential theory in several complex variables, Development of Mathematics 1950-2000, 0, 655-714 (2000) · Zbl 0962.31001 [47] Kollár, J., Higher direct images of dualizing sheaves. I, Ann. Math., Ser. 2,, 123, 1, 11-42 (1986) · Zbl 0598.14015 [48] Matsumura, S., A Nadel vanishing theorem via injectivity theorems, Math. Ann., 359, 3-4, 785-802 (2014) · Zbl 1327.14092 [49] Matsumura, S., A Nadel vanishing theorem for metrics with minimal singularities on big line bundles, Adv. Math., 280, 188-207 (2015) · Zbl 1345.14025 [50] S. Matsumura, “Injectivity theorems with multiplier ideal sheaves for higher direct images under Kähler morphisms,” arXiv: 1607.05554v2 [math.CV]. [51] Matsumura, S., An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities, J. Algebr. Geom., 27, 2, 305-337 (2018) · Zbl 1422.32024 [52] Meng, X.; Zhou, X., Pseudo-effective line bundles over holomorphically convex manifolds, J. Algebr. Geom., 28, 1, 169-200 (2019) · Zbl 1402.32020 [53] Ning, J.; Zhang, H.; Zhou, X., On $$p$$-Bergman kernel for bounded domains in $$\mathbb C^n$$, Commun. Anal. Geom., 24, 4, 887-900 (2016) · Zbl 1368.32004 [54] Ning, J.; Zhang, H.; Zhou, X., Proper holomorphic mappings between invariant domains in $$\mathbb C^n$$, Trans. Am. Math. Soc., 369, 1, 517-536 (2017) · Zbl 1351.32003 [55] Ohsawa, T., On the extension of $$L^2$$ holomorphic functions. II, Publ. Res. Inst. Math. Sci., 24, 2, 265-275 (1988) · Zbl 0653.32012 [56] Ohsawa, T., On the extension of $$L^2$$ holomorphic functions. III: Negligible weights, Math. Z., 219, 2, 215-225 (1995) · Zbl 0823.32006 [57] Ohsawa, T., On the extension of $$L^2$$ holomorphic functions. V: Effects of generalization, Nagoya Math. J., 161, 1-21 (2001) · Zbl 0986.32002 [58] Ohsawa, T., On the extension of $$L^2$$ holomorphic functions. VI: A limiting case, Explorations in Complex and Riemannian Geometry: A Volume Dedicated to R. E. Greene, 332, 235-239 (2003) · Zbl 1049.32010 [59] Ohsawa, T., On a curvature condition that implies a cohomology injectivity theorem of Kollár-Skoda type, Publ. Res. Inst. Math. Sci., 41, 3, 565-577 (2005) · Zbl 1103.32005 [60] Ohsawa, T., $$L^2$$ Approaches in Several Complex Variables: Development of Oka-Cartan Theory by $$L^2$$ Estimates for the $$\bar \partial$$ Operator (2015), Tokyo: Springer, Tokyo · Zbl 1355.32001 [61] Ohsawa, T., On the extension of $$L^2$$ holomorphic functions. VIII: A remark on a theorem of Guan and Zhou, Int. J. Math., 28, 9 (2017) · Zbl 1380.32015 [62] Ohsawa, T.; Takegoshi, K., On the extension of $$L^2$$ holomorphic functions, Math. Z., 195, 2, 197-204 (1987) · Zbl 0625.32011 [63] Păun, M.; Takayama, S., Positivity of twisted relative pluricanonical bundles and their direct images, J. Algebr. Geom., 27, 2, 211-272 (2018) · Zbl 1430.14017 [64] Pommerenke, C.; Suita, N., Capacities and Bergman kernels for Riemann surfaces and Fuchsian groups, J. Math. Soc. Japan, 36, 4, 637-642 (1984) · Zbl 0534.30035 [65] Rashkovskii, A., A log canonical threshold test, Analysis Meets Geometry: The Mikael Passare Memorial Volume, 0, 361-368 (2017) · Zbl 1405.32055 [66] Sergeev, A. G., On matrix Reinhardt and circled domains, Several Complex Variables: Proc. Mittag-Leffler Inst., Stockholm, 1987-1988, 38, 573-586 (1993) · Zbl 0778.32002 [67] Sergeev, A. G., On invariant domains of holomorphy, Topics in Complex Analysis: Proc. Semester on Complex Analysis, Warsaw, 1992, 31, 349-357 (1995) · Zbl 0827.32029 [68] Sergeev, A. G.; Zhou, X., On invariant domains of holomorphy, Proc. Steklov Inst. Math., 203, 145-155 (1995) [69] Sergeev, A. G.; Zhou, X. Y., Invariant domains of holomorphy: Twenty years later, Proc. Steklov Inst. Math., 285, 241-250 (2014) · Zbl 1302.32011 [70] Siu, Y.-T., The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi, Geometric Complex Analysis: Proc. 3rd Int. Res. Inst., Math. Soc. Japan, Hayama, 1995, 0, 577-592 (1996) · Zbl 0941.32021 [71] Siu, Y.-T., Invariance of plurigenera, Invent. Math., 134, 3, 661-673 (1998) · Zbl 0955.32017 [72] Siu, Y.-T., Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, Complex Geometry: Collection of Papers Dedicated to Hans Grauert, 0, 223-277 (2002) · Zbl 1007.32010 [73] Siu, Y.-T., Some recent transcendental techniques in algebraic and complex geometry, Proc. Int. Congr. Math. 2002,, 0, 439-448 (2002) · Zbl 1028.32012 [74] Siu, Y.-T., Invariance of plurigenera and torsion-freeness of direct image sheaves of pluricanonical bundles, Finite or Infinite Dimensional Complex Analysis and Applications, 2, 45-83 (2004) · Zbl 1044.32016 [75] Siu, Y.-T., Multiplier ideal sheaves in complex and algebraic geometry, Sci. China, Ser. A, 48, Suppl., 1-31 (2005) · Zbl 1131.32010 [76] Takegoshi, K., Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kähler morphisms, Math. Ann., 303, 3, 389-416 (1995) · Zbl 0843.32018 [77] Tankeev, S. G., On $$n$$-dimensional canonically polarized varieties and varieties of fundamental type, Math. USSR, Izv., 5, 1, 29-43 (1971) · Zbl 0248.14005 [78] Vladimirov, V. S., Methods of the Theory of Functions of Many Complex Variables (1964), Moscow: Nauka, Moscow [79] Vladimirov, V. S., Nikolai Nikolaevich Bogolyubov—mathematician by the grace of God, Mathematical Events of the Twentieth Century, 0, 475-499 (2006) · Zbl 1086.01034 [80] Vladimirov, V. S.; Sergeev, A. G., Complex analysis in the future tube, Complex Analysis—Many Variables-2, 8, 191-266 (1985) · Zbl 0614.32001 [81] Volovich, I. V.; Polivanov, M. K., Quantum field theory, Mathematical Encyclopedia, 0, 829-837 (1979) [82] Wang, Z.; Zhou, X., CR eigenvalue estimate and Kohn-Rossi cohomology, J. Diff. Geom., 0, 0 (0000) [83] Yau, S.-T., On the pseudonorm project of birational classification of algebraic varieties, Geometry and Analysis on Manifolds: In Memory of S. Kobayashi, 308, 327-339 (2015) · Zbl 1322.14033 [84] Zhou, X., On matrix Reinhardt domains, Math. Ann., 287, 1, 35-46 (1990) · Zbl 0672.32003 [85] Zhou, X.-Y., On orbital convexity of domains of holomorphy invariant under a linear action of tori, Dokl. Math., 45, 1, 93-98 (1992) [86] Zhou, X.-Y., On orbit connectedness, orbit convexity, and envelopes of holomorphy, Russ. Acad. Sci., Izv. Math., 44, 2, 403-413 (1995) [87] Zhou, X.-Y., A proof of the extended future tube conjecture, Izv. Math., 62, 1, 201-213 (1998) · Zbl 0922.32007 [88] Zhou, X.-Y., An invariant version of Cartan’s lemma and complexification of invariant domains of holomorphy, Dokl. Math., 59, 3, 460-463 (1999) · Zbl 0967.32021 [89] Zhou, X.-Y., Quotients, invariant version of Cartan’s lemma, and the minimum principle, First International Congress of Chinese Mathematicians: Proc. ICCM98, Beijing, 1998, 20, 335-343 (2001) · Zbl 1048.32013 [90] Zhou, X.-Y., Extension theorems for special holomorphic functions, Geometry and Nonlinear Partial Differential Equations: Dedicated to Prof. Buqing Su in Honor of His 100th Birthday. Proc. Conf., Zhejiang Univ., 2001, 29, 235-237 (2002) · Zbl 1019.32022 [91] Zhou, X., Some results related to group actions in several complex variables, Proc. Int. Congr. Math. 2002,, 0, 743-753 (2002) · Zbl 1004.32004 [92] Zhou, X., Invariant holomorphic extension in several complex variables, Sci. China, Ser. A, 49, 11, 1593-1598 (2006) · Zbl 1111.32006 [93] Zhou, X., A survey on $$L^2$$ extension problem, Complex Geometry and Dynamics: The Abel Symposium 2013, 10, 291-309 (2015) · Zbl 1337.32008 [94] Zhou, X., Roles of plurisubharmonic functions, Proc. Steklov Inst. Math., 306, 288-295 (2019) · Zbl 1436.32104 [95] Zhou, X.; Zhu, L., A generalized Siu’s lemma, Math. Res. Lett., 24, 6, 1897-1913 (2017) · Zbl 1394.32027 [96] Zhou, X.; Zhu, L., An optimal $$L^2$$ extension theorem on weakly pseudoconvex Kähler manifolds, J. Diff. Geom., 110, 1, 135-186 (2018) · Zbl 1426.53082 [97] Zhou, X.; Zhu, L., Siu’s lemma, optimal $$L^2$$ extension and applications to twisted pluricanonical sheaves, Math. Ann., 377, 1-2, 675-722 (2020) · Zbl 1452.32014 [98] Zhou, X.; Zhu, L., Extension of cohomology classes and holomorphic sections defined on subvarieties, J. Algebr. Geom., 0, 0 (0000) [99] Zhu, L.; Guan, Q.; Zhou, X., On the Ohsawa-Takegoshi $$L^2$$ extension theorem and the Bochner-Kodaira identity with non-smooth twist factor, J. Math. Pures Appl., Sér. 9,, 97, 6, 579-601 (2012) · Zbl 1244.32005
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