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Recent Results in Several Complex Variables and Complex Geometry

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Abstract

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.

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References

  1. B. Berndtsson, “The openness conjecture and complex Brunn–Minkowski inequalities,” in Complex Geometry and Dynamics: The Abel Symposium 2013 (Springer, Cham, 2015), Abel Symp. 10, pp. 29–44.

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Berndtsson and M. Păun, “Bergman kernels and the pseudoeffectivity of relative canonical bundles,” Duke Math. J. 145 (2), 341–378 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  3. B. Berndtsson and M. Păun, “Bergman kernels and subadjunction,” arXiv: 1002.4145v1 [math.AG].

  4. M. Blel and S. K. Mimouni, “Singularité et intégrabilité des fonctions plurisousharmoniques,” Ann. Inst. Fourier 55 (2), 319–351 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  5. Z. Błocki, “On the Ohsawa–Takegoshi extension theorem,” Univ. Iagel. Acta Math. 50, 53–61 (2012).

    MathSciNet  MATH  Google Scholar 

  6. Z. Błocki, “Suita conjecture and the Ohsawa–Takegoshi extension theorem,” Invent. Math. 193 (1), 149–158 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Boucksom, C. Favre, and M. Jonsson, “Valuations and plurisubharmonic singularities,” Publ. Res. Inst. Math. Sci. 44 (2), 449–494 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Cao, J.-P. Demailly, and S. Matsumura, “A general extension theorem for cohomology classes on non reduced analytic subspaces,” Sci. China, Math. 60 (6), 949–962 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  9. J.-P. Demailly, “On the Ohsawa–Takegoshi–Manivel \(L^2\) extension theorem,” in Complex Analysis and Geometry: Proc. Int. Conf. in Honour of Pierre Lelong on the Occasion of His 85th Birthday, Paris, 1997 (Birkhäuser, Basel, 2000), Prog. Math. 188, pp. 47–82.

    Article  MathSciNet  MATH  Google Scholar 

  10. J.-P. Demailly, “Multiplier ideal sheaves and analytic methods in algebraic geometry,” in School on Vanishing Theorems and Effective Results in Algebraic Geometry, Trieste, 2000 (Abdus Salam Int. Cent. Theor. Phys., Trieste, 2001), ICTP Lect. Notes 6, pp. 1–148.

    MathSciNet  MATH  Google Scholar 

  11. J.-P. Demailly, “Kähler manifolds and transcendental techniques in algebraic geometry,” in Proc. Int. Congr. Math. 2006, Vol. I: Plenary Lectures and Ceremonies (Eur. Math. Soc., Zürich, 2007), pp. 153–186.

    MathSciNet  MATH  Google Scholar 

  12. J.-P. Demailly, Analytic Methods in Algebraic Geometry (Int. Press, Somerville, MA, 2012), Surv. Mod. Math. 1.

    MATH  Google Scholar 

  13. J.-P. Demailly, Complex Analytic and Differential Geometry (Univ. Grenoble I, Inst. Fourier, Grenoble, 2012); https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

    Google Scholar 

  14. J.-P. Demailly, “Extension of holomorphic functions defined on non reduced analytic subvarieties,” in The Legacy of Bernhard Riemann after One Hundred and Fifty Years (Int. Press, Somerville, MA, 2016), Vol. 1, Adv. Lect. Math. 35.1, pp. 191–222.

    MathSciNet  MATH  Google Scholar 

  15. J.-P. Demailly, L. Ein, and R. Lazarsfeld, “A subadditivity property of multiplier ideals,” Mich. Math. J. 48, 137–156 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  16. J. P. Demailly, C. D. Hacon, and M. Păun, “Extension theorems, non-vanishing and the existence of good minimal models,” Acta Math. 210 (2), 203–259 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  17. J.-P. Demailly and J. Kollár, “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).

    MATH  Google Scholar 

  18. F. Deng, Z. Wang, L. Zhang, and X. Zhou, “Linear invariants of complex manifolds and their plurisubharmonic variations,” J. Funct. Anal. 279 (1), 108514 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Deng, H. Zhang, and X. Zhou, “Positivity of direct images of positively curved volume forms,” Math. Z. 278 (1–2), 347–362 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Deng, H. Zhang, and X. Zhou, “Positivity of character subbundles and minimum principle for noncompact group actions,” Math. Z. 286 (1–2), 431–442 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Deng and X. Y. Zhou, “Rigidity of automorphism groups of invariant domains in homogeneous Stein spaces,” Izv. Math. 78 (1), 34–58 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  22. I. Enoki, “Kawamata–Viehweg vanishing theorem for compact Kähler manifolds,” in Einstein Metrics and Yang–Mills Connections: Proc. 27th Taniguchi Int. Symp., Sanda, 1990 (M. Dekker, New York, 1993), Lect. Notes Pure Appl. Math. 145, pp. 59–68.

    MathSciNet  MATH  Google Scholar 

  23. C. Favre and M. Jonsson, “Valuative analysis of planar plurisubharmonic functions,” Invent. Math. 162 (2), 271–311 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  24. C. Favre and M. Jonsson, “Valuations and multiplier ideals,” J. Am. Math. Soc. 18 (3), 655–684 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Forstnerič, Stein Manifolds and Holomorphic Mappings. The Homotopy Principle in Complex Analysis, 2nd ed. (Springer, Cham, 2017), Ergebn. Math. Grenzgeb., 3. Folge 56.

    Book  MATH  Google Scholar 

  26. O. Fujino, “A transcendental approach to Kollár’s injectivity theorem. II,” J. Reine Angew. Math. 681, 149–174 (2013).

    MathSciNet  MATH  Google Scholar 

  27. O. Fujino and S. Matsumura, “Injectivity theorem for pseudo-effective line bundles and its applications,” arXiv: 1605.02284 [math.CV].

  28. Y. Gongyo and S. Matsumura, “Versions of injectivity and extension theorems,” Ann. Sci. Éc. Norm. Supér., Sér. 4, 50 (2), 479–502 (2017).

    MathSciNet  MATH  Google Scholar 

  29. H. Grauert, “On Levi’s problem and the imbedding of real-analytic manifolds,” Ann. Math., Ser. 2, 68, 460–472 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  30. H. Grauert, Selected Papers (Springer, Berlin, 1994), Vols. I, II.

    MATH  Google Scholar 

  31. H. Grauert and R. Remmert, Theory of Stein Spaces (Springer, Berlin, 1979), Grundl. Math. Wiss. 236.

    Book  MATH  Google Scholar 

  32. H. Grauert and R. Remmert, Coherent Analytic Sheaves (Springer, Berlin, 1984), Grundl. Math. Wiss. 265.

    Book  MATH  Google Scholar 

  33. Q. Guan and X. Zhou, “Optimal constant problem in the \(L^2\) extension theorem,” C. R., Math., Acad. Sci. Paris 350 (15–16), 753–756 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  34. Q. Guan and X. Zhou, “Optimal constant in an \(L^2\) extension problem and a proof of a conjecture of Ohsawa,” Sci. China, Math. 58 (1), 35–59 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  35. Q. Guan and X. Zhou, “A solution of an \(L^2\) extension problem with an optimal estimate and applications,” Ann. Math., Ser. 2, 181 (3), 1139–1208 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  36. Q. Guan and X. Zhou, “A proof of Demailly’s strong openness conjecture,” Ann. Math., Ser. 2, 182 (2), 605–616 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  37. Q. Guan and X. Zhou, “Effectiveness of Demailly’s strong openness conjecture and related problems,” Invent. Math. 202 (2), 635–676 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  38. Q. Guan and X. Zhou, “Characterization of multiplier ideal sheaves with weights of Lelong number one,” Adv. Math. 285, 1688–1705 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  39. Q. A. Guan and X. Y. Zhou, “Strong openness of multiplier ideal sheaves and optimal \(L^2\) extension,” Sci. China, Math. 60 (6), 967–976 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  40. Q. Guan and X. Zhou, “Restriction formula and subadditivity property related to multiplier ideal sheaves,” J. Reine Angew. Math. 769, 1–33 (2020).

    Article  MathSciNet  Google Scholar 

  41. 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 349 (13–14), 797–800 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  42. C. Hacon, M. Popa, and C. Schnell, “Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and Păun,” in Local and Global Methods in Algebraic Geometry (Am. Math. Soc., Providence, RI, 2018), Contemp. Math. 712, pp. 143–195.

    MATH  Google Scholar 

  43. L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd ed. (North-Holland, Amsterdam, 1990), North-Holland Math. Libr. 7.

    MATH  Google Scholar 

  44. M. Jonsson and M. Mustaţă, “Valuations and asymptotic invariants for sequences of ideals,” Ann. Inst. Fourier 62 (6), 2145–2209 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Jonsson and M. Mustaţă, “An algebraic approach to the openness conjecture of Demailly and Kollár,” J. Inst. Math. Jussieu 13 (1), 119–144 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  46. C. O. Kiselman, “Plurisubharmonic functions and potential theory in several complex variables,” in Development of Mathematics 1950–2000 (Birkhäuser, Basel, 2000), pp. 655–714.

    Article  MathSciNet  MATH  Google Scholar 

  47. J. Kollár, “Higher direct images of dualizing sheaves. I,” Ann. Math., Ser. 2, 123 (1), 11–42 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  48. S. Matsumura, “A Nadel vanishing theorem via injectivity theorems,” Math. Ann. 359 (3–4), 785–802 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  49. S. Matsumura, “A Nadel vanishing theorem for metrics with minimal singularities on big line bundles,” Adv. Math. 280, 188–207 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  50. S. Matsumura, “Injectivity theorems with multiplier ideal sheaves for higher direct images under Kähler morphisms,” arXiv: 1607.05554v2 [math.CV].

  51. S. Matsumura, “An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities,” J. Algebr. Geom. 27 (2), 305–337 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  52. X. Meng and X. Zhou, “Pseudo-effective line bundles over holomorphically convex manifolds,” J. Algebr. Geom. 28 (1), 169–200 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  53. J. Ning, H. Zhang, and X. Zhou, “On \(p\)-Bergman kernel for bounded domains in \(\mathbb C^n\),” Commun. Anal. Geom. 24 (4), 887–900 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  54. J. Ning, H. Zhang, and X. Zhou, “Proper holomorphic mappings between invariant domains in \(\mathbb C^n\),” Trans. Am. Math. Soc. 369 (1), 517–536 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  55. T. Ohsawa, “On the extension of \(L^2\) holomorphic functions. II,” Publ. Res. Inst. Math. Sci. 24 (2), 265–275 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  56. T. Ohsawa, “On the extension of \(L^2\) holomorphic functions. III: Negligible weights,” Math. Z. 219 (2), 215–225 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  57. T. Ohsawa, “On the extension of \(L^2\) holomorphic functions. V: Effects of generalization,” Nagoya Math. J. 161, 1–21 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  58. T. Ohsawa, “On the extension of \(L^2\) holomorphic functions. VI: A limiting case,” in Explorations in Complex and Riemannian Geometry: A Volume Dedicated to R. E. Greene (Am. Math. Soc., Providence, RI, 2003), Contemp. Math. 332, pp. 235–239.

    Article  MathSciNet  MATH  Google Scholar 

  59. T. Ohsawa, “On a curvature condition that implies a cohomology injectivity theorem of Kollár–Skoda type,” Publ. Res. Inst. Math. Sci. 41 (3), 565–577 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  60. T. Ohsawa, \(L^2\) Approaches in Several Complex Variables: Development of Oka–Cartan Theory by \(L^2\) Estimates for the \(\bar \partial \) Operator (Springer, Tokyo, 2015), Springer Monogr. Math.

    Book  MATH  Google Scholar 

  61. T. Ohsawa, “On the extension of \(L^2\) holomorphic functions. VIII: A remark on a theorem of Guan and Zhou,” Int. J. Math. 28 (9), 1740005 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  62. T. Ohsawa and K. Takegoshi, “On the extension of \(L^2\) holomorphic functions,” Math. Z. 195 (2), 197–204 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  63. M. Păun and S. Takayama, “Positivity of twisted relative pluricanonical bundles and their direct images,” J. Algebr. Geom. 27 (2), 211–272 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  64. C. Pommerenke and N. Suita, “Capacities and Bergman kernels for Riemann surfaces and Fuchsian groups,” J. Math. Soc. Japan 36 (4), 637–642 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  65. A. Rashkovskii, “A log canonical threshold test,” in Analysis Meets Geometry: The Mikael Passare Memorial Volume (Birkhäuser/Springer, Cham, 2017), Trends Math., pp. 361–368; arXiv: 1501.04831 [math.CV].

    Article  MathSciNet  MATH  Google Scholar 

  66. A. G. Sergeev, “On matrix Reinhardt and circled domains,” in Several Complex Variables: Proc. Mittag-Leffler Inst., Stockholm, 1987–1988 (Princeton Univ. Press, Princeton, NJ, 1993), Math. Notes 38, pp. 573–586.

    MathSciNet  MATH  Google Scholar 

  67. A. G. Sergeev, “On invariant domains of holomorphy,” in Topics in Complex Analysis: Proc. Semester on Complex Analysis, Warsaw, 1992 (Pol. Acad. Sci., Inst. Math., Warszawa, 1995), Banach Cent. Publ. 31, pp. 349–357.

    MathSciNet  MATH  Google Scholar 

  68. A. G. Sergeev and X. Zhou, “On invariant domains of holomorphy,” Proc. Steklov Inst. Math. 203, 145–155 (1995) [transl. from Tr. Mat. Inst. Steklova 203, 159–172 (1994)].

    MathSciNet  Google Scholar 

  69. A. G. Sergeev and X. Y. Zhou, “Invariant domains of holomorphy: Twenty years later,” Proc. Steklov Inst. Math. 285, 241–250 (2014) [transl. from Tr. Mat. Inst. Steklova 285, 253–263 (2014)].

    Article  MathSciNet  MATH  Google Scholar 

  70. Y.-T. Siu, “The Fujita conjecture and the extension theorem of Ohsawa–Takegoshi,” in Geometric Complex Analysis: Proc. 3rd Int. Res. Inst., Math. Soc. Japan, Hayama, 1995 (World Scientific, Singapore, 1996), pp. 577–592.

    MathSciNet  MATH  Google Scholar 

  71. Y.-T. Siu, “Invariance of plurigenera,” Invent. Math. 134 (3), 661–673 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  72. Y.-T. 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: Collection of Papers Dedicated to Hans Grauert (Springer, Berlin, 2002), pp. 223–277.

    Article  MathSciNet  MATH  Google Scholar 

  73. Y.-T. Siu, “Some recent transcendental techniques in algebraic and complex geometry,” in Proc. Int. Congr. Math. 2002, Vol. I: Plenary Lectures and Ceremonies (High. Educ. Press, Beijing, 2002), pp. 439–448.

    MathSciNet  MATH  Google Scholar 

  74. Y.-T. Siu, “Invariance of plurigenera and torsion-freeness of direct image sheaves of pluricanonical bundles,” in Finite or Infinite Dimensional Complex Analysis and Applications (Kluwer, Boston, MA, 2004), Adv. Complex Anal. Appl. 2, pp. 45–83.

    Article  MathSciNet  MATH  Google Scholar 

  75. Y.-T. Siu, “Multiplier ideal sheaves in complex and algebraic geometry,” Sci. China, Ser. A 48 (Suppl.), 1–31 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  76. K. Takegoshi, “Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kähler morphisms,” Math. Ann. 303 (3), 389–416 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  77. S. G. Tankeev, “On \(n\)-dimensional canonically polarized varieties and varieties of fundamental type,” Math. USSR, Izv. 5 (1), 29–43 (1971) [transl. from Izv. Akad. Nauk SSSR, Ser. Mat. 35 (1), 31–44 (1971)].

    Article  MATH  Google Scholar 

  78. V. S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables (Nauka, Moscow, 1964) [in Russian].

    Google Scholar 

  79. V. S. Vladimirov, “Nikolai Nikolaevich Bogolyubov—mathematician by the grace of God,” in Mathematical Events of the Twentieth Century (Springer, Berlin, 2006), pp. 475–499.

    Article  MathSciNet  MATH  Google Scholar 

  80. V. S. Vladimirov and A. G. Sergeev, “Complex analysis in the future tube,” in Complex Analysis—Many Variables–2 (VINITI, Moscow, 1985), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 8, pp. 191–266. Engl. transl.: A. G. Sergeev and V. S. Vladimirov, “Complex analysis in the future tube,” in Several Complex Variables. II: Function Theory in Classical Domains. Complex Potential Theory (Springer, Berlin, 1994), Encycl. Math. Sci. 8, pp. 179–253.

    MathSciNet  MATH  Google Scholar 

  81. I. V. Volovich and M. K. Polivanov, “Quantum field theory,” in Mathematical Encyclopedia, Ed. by I. M. Vinogradov (Sov. Entsiklopediya, Moscow, 1979), Vol. 2, pp. 829–837. Engl. transl. in Encyclopaedia of Mathematics, Ed. by M. Hazewinkel (Kluwer, Dordrecht, 1991), Vol. 7, pp. 403–408.

    Google Scholar 

  82. Z. Wang and X. Zhou, “CR eigenvalue estimate and Kohn–Rossi cohomology,” J. Diff. Geom. (in press); arXiv: 1905.03474 [math.CV].

    Google Scholar 

  83. S.-T. Yau, “On the pseudonorm project of birational classification of algebraic varieties,” in Geometry and Analysis on Manifolds: In Memory of S. Kobayashi (Birkhäuser/Springer, Cham, 2015), Prog. Math. 308, pp. 327–339.

    Article  MathSciNet  MATH  Google Scholar 

  84. X. Zhou, “On matrix Reinhardt domains,” Math. Ann. 287 (1), 35–46 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  85. X.-Y. Zhou, “On orbital convexity of domains of holomorphy invariant under a linear action of tori,” Dokl. Math. 45 (1), 93–98 (1992) [transl. from Dokl. Akad. Nauk 322 (2), 262–267 (1992)].

    MathSciNet  Google Scholar 

  86. X.-Y. Zhou, “On orbit connectedness, orbit convexity, and envelopes of holomorphy,” Russ. Acad. Sci., Izv. Math. 44 (2), 403–413 (1995) [repr. from Izv. Ross. Akad. Nauk, Ser. Mat. 58 (2), 196–205 (1994)].

    MathSciNet  Google Scholar 

  87. X.-Y. Zhou, “A proof of the extended future tube conjecture,” Izv. Math. 62 (1), 201–213 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  88. X.-Y. Zhou, “An invariant version of Cartan’s lemma and complexification of invariant domains of holomorphy,” Dokl. Math. 59 (3), 460–463 (1999).

    MATH  Google Scholar 

  89. X.-Y. Zhou, “Quotients, invariant version of Cartan’s lemma, and the minimum principle,” in First International Congress of Chinese Mathematicians: Proc. ICCM98, Beijing, 1998 (Am. Math. Soc., Providence, RI, 2001), AMS/IP Stud. Adv. Math. 20, pp. 335–343.

    Article  MathSciNet  MATH  Google Scholar 

  90. X.-Y. Zhou, “Extension theorems for special holomorphic functions,” in Geometry and Nonlinear Partial Differential Equations: Dedicated to Prof. Buqing Su in Honor of His 100th Birthday. Proc. Conf., Zhejiang Univ., 2001 (Am. Math. Soc., Providence, RI, 2002), AMS/IP Stud. Adv. Math. 29, pp. 235–237.

    Article  MathSciNet  MATH  Google Scholar 

  91. X. Zhou, “Some results related to group actions in several complex variables,” in Proc. Int. Congr. Math. 2002, Vol. II: Invited Lectures (High. Educ. Press, Beijing, 2002), pp. 743–753.

    MathSciNet  MATH  Google Scholar 

  92. X. Zhou, “Invariant holomorphic extension in several complex variables,” Sci. China, Ser. A 49 (11), 1593–1598 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  93. X. Zhou, “A survey on \(L^2\) extension problem,” in Complex Geometry and Dynamics: The Abel Symposium 2013 (Springer, Cham, 2015), Abel Symp. 10, pp. 291–309.

    Article  MathSciNet  MATH  Google Scholar 

  94. X. Zhou, “Roles of plurisubharmonic functions,” Proc. Steklov Inst. Math. 306, 288–295 (2019).

    Article  MATH  Google Scholar 

  95. X. Zhou and L. Zhu, “A generalized Siu’s lemma,” Math. Res. Lett. 24 (6), 1897–1913 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  96. X. Zhou and L. Zhu, “An optimal \(L^2\) extension theorem on weakly pseudoconvex Kähler manifolds,” J. Diff. Geom. 110 (1), 135–186 (2018).

    Article  MATH  Google Scholar 

  97. X. Zhou and L. Zhu, “Siu’s lemma, optimal \(L^2\) extension and applications to twisted pluricanonical sheaves,” Math. Ann. 377 (1–2), 675–722 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  98. X. Zhou and L. Zhu, “Extension of cohomology classes and holomorphic sections defined on subvarieties,” J. Algebr. Geom. (in press); arXiv: 1909.08822 [math.CV].

    Google Scholar 

  99. 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., Sér. 9, 97 (6), 579–601 (2012).

    Article  MathSciNet  MATH  Google Scholar 

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The work was supported in part by the National Natural Science Foundation of China (NSFC).

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Correspondence to Xiangyu Zhou.

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Dedicated to the 70th birthday of A. G. Sergeev

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Zhou, X. Recent Results in Several Complex Variables and Complex Geometry. Proc. Steklov Inst. Math. 311, 245–260 (2020). https://doi.org/10.1134/S0081543820060164

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