A Note on ξ-Bergman Kernels

Abstract

In the present note, we introduce the ξ-complex singularity exponents, which come from the asymptotic property of ξ-Bergman kernels on sub-level sets of plurisubharmonic functions; give some relations (including a closedness property) among ξ-complex singularity exponents, complex singularity exponents, and jumping numbers; generalize some properties of complex singularity exponents (such as the restriction formula and subadditivity property) to ξ-complex singularity exponents.

Keywords

ξ-Bergman kernel / generalized Lelong number / jumping number / complex singularity exponent

References

[1] BaoSJ, GuanQA. L² extension and effectiveness of strong openness property. Acta Math. Sin. (Engl. Ser.), 2022, 38(11): 1949-1964

[2] BaoSJ, GuanQA. L² extension and effectiveness of L^p strong openness property. Acta Math. Sin. (Engl. Ser.), 2023, 39(5): 814-826

[3] Bao S.J., Guan Q.A., Modules at boundary points, fiberwise Bergman kernels, and logsubharmonicity. Peking Math. J., 2023, https://doi.org/10.1007/s42543-023-00070-8.

[4] BerndtssonB. Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier (Grenoble), 2006, 56(6): 1633-1662

[5] Berndtsson B., The openness conjecture for plurisubharmonic functions. 2013, arXiv: 1305.5781

[6] BłockiZ. Suita conjecture and the Ohsawa-Takegoshi extension theorem. Invent. Math., 2013, 193(1): 149-158

[7] BoucksomS, FavreC, JonssonM. Valuations and plurisubharmonic singularities. Publ. Res. Inst. Math. Sci., 2008, 44(2): 449-494

[8] DemaillyJ-PAnalytic Methods in Algebraic Geometry, 2010, Beijing, Higher Education Press

[9] Demailly J.-P., Complex Analytic and Differential Geometry, 2012, https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

[10] DemaillyJ-P, EinL, LazarsfeldR. A subadditivity property of multiplier ideals. Michigan Math. J., 2000, 48: 137-156

[11] DemaillyJ-P, KollárJ. Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup. (4)., 2001, 34(4): 525-556

[12] Deng F.S., Wang Z.W., Zhang, L.Y., Zhou X.Y., New characterizations of plurisubharmonic functions and positivity of direct image sheaves. 2018, arXiv:1809.10371

[13] GuanQA. A sharp effectiveness result of Demailly’s strong openness conjecture. Adv. Math., 2019, 348: 51-80

[14] GuanQA, YuanZ. Effectiveness of strong openness property in L^p. Proc. Amer. Math. Soc., 2023, 151(10): 4331-4339

[15] GuanQA, ZhouXY. A proof of Demailly’s strong openness conjecture. Ann. of Math. (2), 2015, 182(2): 605-616

[16] GuanQA, ZhouXY. Effectiveness of Demailly’s strong openness conjecture and related problems. Invent. Math., 2015, 202(2): 635-676

[17] GuanQA, ZhouXY. Restriction formula and subadditivity property related to multiplier ideal sheaves. J. Reine Angew. Math., 2020, 769: 1-33

[18] GuenanciaH. Toric plurisubharmonic functions, analytic adjoint ideal sheaves. Math. Z., 2012, 271(3–4): 1011-1035

[19] OhsawaTAnalysis of Several Complex Variables, 2002, Providence, RI, American Mathematical Society

[20] OhsawaT, TakegoshiK. On the extension of L² holomorphic functions. Math. Z., 1987, 195(2): 197-204

[21] TianG. On Kähler–Einstein metrics on certain Kähler manifolds with C₁(M) > 0. Invent. Math., 1987, 89(2): 225-246

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