Analyticity theorems for parameter-dependent plurisubharmonic functions


  • Bojie He



In this paper, we first show that a union of upper-level sets associated to fibrewise Lelong numbers of plurisubharmonic functions is in general a pluripolar subset. Then we obtain analyticity theorems for a union of sub-level sets associated to fibrewise complex singularity exponents of some special (quasi-)plurisubharmonic functions. As a corollary, we confirm that, under certain conditions, the logarithmic poles of relative Bergman kernels form an analytic subset when the (quasi-)plurisubharmonic weight function has analytic singularities. In the end, we give counterexamples to show that the aforementioned sets are in general non-analytic even if the plurisubharmonic function is supposed to be continuous.


Angehrn, U., and Siu, Y.-T., Effective freeness and point separation for adjoint bundles, Invent. Math. 122 (1995), no. 2, 291–308.

Berndtsson, B., Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1633–1662.

Berndtsson, B., Curvature of vector bundles associated to holomorphic fibrations, Ann. of Math. (2) 169 (2009), no. 2, 531–560.

Berndtsson, B., The openness conjecture for plurisubharmonic functions, arXiv:1305.5781.

Berndtsson, B., and Păun, M., Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J. 145 (2008), no. 2, 341–378.

Bierstone, E., and Milman, P. D., A simple constructive proof of canonical resolution of singularities, Effective methods in algebraic geometry (Castiglioncello, 1990), 11–30, Progr. Math., 94, Birkhäuser Boston, Boston, MA, 1991.

Demailly, J.-P., Analytic methods in algebraic geometry, Surveys of Modern Mathematics, 1. International Press, Somerville, MA; Higher Education Press, Beijing, 2012.

Demailly, J.-P., Complex analytic and differential geometry, demailly/books.html.

Demailly, J.-P., Hwang, J.-M., and Peternell, T., Compact manifolds covered by a torus, J. Geom. Anal. 18 (2008), no. 2, 324–340.

Demailly, J.-P., and Kollár, J., 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.

Guan, Q., and Li, Z., Analytic adjoint ideal sheaves associated to plurisubharmonic functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 1, 391–395.

Guan, Q., Li, Z., and Zhou, X., Estimation of weighted $L^2$ norm related to Demailly's strong openness conjecture, arXiv:1603.05733.

Guan, Q., and Zhou, X., A proof of Demailly's strong openness conjecture, Ann. of Math. (2) 182 (2015), no. 2, 605–616.

Guan, Q., and Zhou, X., Lelong numbers, complex singularity exponents, and Siu's semicontinuity theorem, C. R. Math. Acad. Sci. Paris 355 (2017), no. 4, 415–419.

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

Guenancia, H., Toric plurisubharmonic functions and analytic adjoint ideal sheaves, Math. Z. 271 (2012), no. 3-4, 1011–1035.

Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.

Hironaka, H., Resolutions of singlarities of an algebraic variety over a field of characteristic zero, I and II, Ann. of Math. (2) 79 (1964), 109–203; 79 (1964), 205–326.

Hörmander, L., An Introduction to Complex Analysis in Several Variables, Third edition. North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, 1990.

Kiselman, C. O., The partial legendre transformation for plurisubharmonic functions, Invent. Math. 49 (1978), no. 2, 137–148.

Kiselman, C. O., Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math. 60 (1994), no. 2, 173–197.

Lazarsfeld, R., Positivity in Algebraic Geometry I and II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge (2004), Springer-Verlag, Berlin, 48-49.

Li, C., Analytical approximations and Monge Ampère masses of plurisubharmonic singularities, Int. Math. Res. Not. IMRN 2024, no. 1, 359–381.

Ohsawa, T., and Takegoshi, K., On the extension of $L^2$ holomorphic functions, Math. Z. 195 (1987), no. 2, 197–204.

Phong, D. H., and Sturm, J., Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions, Ann. of Math. (2) 152 (2000), no. 1, 277–329.

Siu, Y.-T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974), 53–156.

Siu, Y.-T., Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661–673.

Varchenko, A. N., The complex singularity index of a singularity do not change along the stratum $mu =$ constant, Funktsional. Anal. i Prilozhen. 16 (1982), no. 1, 1–12.

Varouchas, J., Stabilité de la classe des variété kähleriénnes par certains morphisms propre, Invent. Math. 77 (1984), no. 1, 117–127.

Wang, X. Q., Analyticity theorems for parameter-dependent currents, Math. Scand. 69 (1991), no. 2, 179–198.

Xia, M. C., Analytic Bertini theorem, Math. Z. 302 (2022), no. 2, 1171–1176.

Zhou, X., and Zhu, L., A generalized Siu's lemma, Math. Res. Lett. 24 (2017), no. 6, 1897–1913.

Zhou, X., and Zhu, L., An optimal $L^2$ extension theorem on weakly pseudoconvex Kähler manifolds, J. Differential Geom. 110 (2018), no. 1, 135–186.

Zhou, X., and Zhu, L., Siu's lemma, optimal $L^2$ extension and applications to twisted pluricanonical sheaves, Math. Ann. 377 (2020), no. 1-2, 675–722.



How to Cite

He, B. (2024). Analyticity theorems for parameter-dependent plurisubharmonic functions. MATHEMATICA SCANDINAVICA, 130(2).