Article
| Title: | Continuity of the complex Monge-Ampère operator on compact Hermitian manifolds (English) |
| Author: | Le, Mau Hai |
| Author: | Nguyen, Van Phu |
| Author: | Trinh, Tung |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 3 |
| Year: | 2026 |
| Pages: | 947-962 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We establish several results concerning the continuity (or weak convergence) of the complex Monge-Ampère operator on compact Hermitian manifolds. At the end of this note, we find a weak solution of the complex Monge-Ampère equation on a compact Hermitian manifold under the assumption of the existence of a smooth subsolution. (English) |
| Keyword: | $\omega $-plurisubharmonic function |
| Keyword: | weakly convergent |
| Keyword: | complex Monge-Ampère operator |
| Keyword: | compact Hermitian manifold |
| Keyword: | convergence in capacity |
| Keyword: | weak solution of complex Monge-Ampère equation |
| MSC: | 32Q15 |
| MSC: | 32U05 |
| MSC: | 32W20 |
| DOI: | 10.21136/CMJ.2026.0476-25 |
| . | |
| Date available: | 2026-09-11T13:15:37Z |
| Last updated: | 2026-09-14 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153737 |
| . | |
| Reference: | [1] Bedford, E., Taylor, B. A.: A new capacity for plurisubharmonic functions.Acta Math. 149 (1982), 1-40. Zbl 0547.32012, MR 0674165, 10.1007/BF02392348 |
| Reference: | [2] Cegrell, U.: Discontinuité de l'opérateur de Monge-Ampère complexe.C.R. Acad. Sci., Paris, Sér. I 296 (1983), 869-871 French. Zbl 0541.31008, MR 0715323 |
| Reference: | [3] Cegrell, U.: Pluricomplex energy.Acta Math. 180 (1998), 187-217. Zbl 0926.32042, MR 1638768, 10.1007/BF02392899 |
| Reference: | [4] Cegrell, U.: The general definition of the complex Monge-Ampère operator.Ann. Inst. Fourier 54 (2004), 159-179. Zbl 1065.32020, MR 2069125, 10.5802/aif.2014 |
| Reference: | [5] Cegrell, U.: Convergence in capacity.Available at https://arxiv.org/abs/math/0505218 (2005), 5 pages. 10.48550/arXiv.math/0505218 |
| Reference: | [6] Cegrell, U.: Convergence in capacity.Can. Math. Bull. 55 (2012), 242-248. Zbl 1251.32028, MR 2957240, 10.4153/CMB-2011-078-6 |
| Reference: | [7] Dinew, S., Hiep, P. H.: Convergence in capacity on compact Kähler manifolds.Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 11 (2012), 903-919. Zbl 1268.32008, MR 3060705, 10.2422/2036-2145.201003_008 |
| Reference: | [8] Dinew, S., Kołodziej, S.: Pluripotential estimates on compact Hermitian manifolds.Advances in Geometric Analysis Advanced Lectures in Mathematics 21. International Press, Beijing (2012), 69-86. Zbl 1317.32066, MR 3077248 |
| Reference: | [9] Guedj, V., Zeriahi, A.: Intrinsic capacities on compact Kähler manifolds.J. Geom. Anal. 15 (2005), 607-639. Zbl 1087.32020, MR 2203165, 10.1007/BF02922247 |
| Reference: | [10] Guedj, V., Zeriahi, A.: The weighted Monge-Ampère energy of quasiplurisubharmonic functions.J. Funct. Anal. 250 (2007), 442-482. Zbl 1143.32022, MR 2352488, 10.1016/j.jfa.2007.04.018 |
| Reference: | [11] Hiep, P. H.: Convergence in capacity.Ann. Pol. Math. 93 (2008), 91-99. Zbl 1142.32021, MR 2383345, 10.4064/ap93-1-8 |
| Reference: | [12] Hiep, P. H.: On the convergence in capacity on compact Kähler manifolds and its applications.Proc. Am. Math. Soc. 136 (2008), 2007-2018. Zbl 1169.32010, MR 2383507, 10.1090/S0002-9939-08-09043-6 |
| Reference: | [13] Hiep, P. H.: Convergence in capacity and applications.Math. Scand. 107 (2010), 90-102. Zbl 1202.32028, MR 2679394, 10.7146/math.scand.a-15144 |
| Reference: | [14] Kołodziej, S., Nguyen, N. C.: Weak solutions to the complex Monge-Ampère equation on Hermitian manifolds.Analysis, Complex Geometry, and Mathematical Physics Contemporary Mathematics 644. AMS, Providence (2015), 141-158. Zbl 1343.32031, MR 3372464, 10.1090/conm/644/12775 |
| Reference: | [15] Kołodziej, S., Nguyen, N. C.: The Dirichlet problem for the Monge-Ampère equation on Hermitian manifolds with boundary.Calc. Var. Partial Differ. Equ. 62 (2023), Article ID 1, 39 pages. Zbl 1517.32122, MR 4505144, 10.1007/s00526-022-02336-y |
| Reference: | [16] Kołodziej, S., Nguyen, N. C.: Weak convergence of Monge-Ampère measures on compact Hermitian manifolds.Complex Geometric Analysis Springer Proceedings in Mathematics & Statistics 481. Springer, Singapore (2025), 113-123. Zbl 08130020, MR 4898338, 10.1007/978-981-96-0447-0_10 |
| Reference: | [17] Tosatti, V., Weinkove, B.: The complex Monge-Ampère equation on compact Hermitian manifolds.J. Am. Math. Soc. 23 (2010), 1187-1195. Zbl 1208.53075, MR 2669712, 10.1090/S0894-0347-2010-00673-X |
| Reference: | [18] Xing, Y.: Continuity of the complex Monge-Ampère operator.Proc. Am. Math. Soc. 124 (1996), 457-467. Zbl 0849.31010, MR 1322940, 10.1090/S0002-9939-96-03316-3 |
| Reference: | [19] Xing, Y.: Convergence in capacity.Ann. Inst. Fourier 58 (2008), 1839-1861. Zbl 1152.32021, MR 2445835, 10.5802/aif.2400 |
| Reference: | [20] Xing, Y.: Continuity of the complex Monge-Ampère operator on compact Kähler manifolds.Math. Z. 263 (2009), 331-344. Zbl 1179.32015, MR 2534121, 10.1007/s00209-008-0420-8 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
