Abstract
Let T be a positive closed current of bidegree (1, 1) on a multiprojective space \(X={\mathbb P}^{n_1}\times \cdots \times {{\mathbb {P}}}^{n_k}\). For certain values of \(\alpha \), which depend on the cohomology class of T, we show that the set of points of X where the Lelong numbers of T exceed \(\alpha \) have certain geometric properties. We also describe the currents T that have the largest possible Lelong number in a given cohomology class, and the set of points where this number is assumed.
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Coman, D.: Entire pluricomplex Green functions and Lelong numbers of projective currents. Proc. Am. Math. Soc. 134, 1927–1935 (2006)
Coman, D., Guedj, V.: Quasiplurisubharmonic Green functions. J. Math. Pure Appl. 92(9), 456–475 (2009)
Coman, D., Truong, T.T.: Geometric properties of upper level sets of Lelong numbers on projective spaces. Math. Ann. 361, 981–994 (2015)
Demailly, J.P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361–409 (1992)
Demailly, J.P.: Monge-Ampère operators, Lelong numbers and intersection theory, in complex analysis and geometry, pp. 115–193. Plenum, New York (1993)
Favre, C., Guedj, V.: Dynamique des applications rationnelles des espaces multiprojectifs. Indiana Univ. Math. J. 50, 881–934 (2001)
Fornæss, J.E., Sibony, N.: Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992), 135–182, Ann. of Math. Stud., 137, Princeton University Press, Princeton, (1995)
Guedj, V., Zeriahi, A.: Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15, 607–639 (2005)
Heffers, J.J.: A property of upper level sets of Lelong numbers of currents on \({\mathbb{P}}^2\). Int. J. Math. 28(14), 1750110, 18 (2017)
Heffers, J.J.: On Lelong numbers of positive closed currents on \({{\mathbb{P}}}^n\). Complex Var. Elliptic Equ. 64, 352–360 (2019)
Hörmander, L.: Notions of convexity. Birkhäuser, Basel (1994)
Sibony, N.: Dynamique des applications rationnelles de \({{\mathbb{P}}}^k\), Dynamique et géométrie complexes (Lyon, 1997), 97–185, Panor. Synthèses, 8, Soc. Math. France, Paris, (1999)
Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)
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D. Coman is partially supported by the NSF Grant DMS-1700011.
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Coman, D., Heffers, J. Lelong numbers of bidegree (1, 1) currents on multiprojective spaces. Math. Z. 295, 1569–1582 (2020). https://doi.org/10.1007/s00209-019-02427-1
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DOI: https://doi.org/10.1007/s00209-019-02427-1