Convergence in capacity for Cegrell classes on complex varieties in domains of $C^n$
Main Article Content
Abstract
Let $V$ be a complex variety embedded in a bounded domain $ D \subset C^n.$ Assuming that the singularities of $ V$ are not too severe — for example, if $V$ is locally irreducible — we prove a convergence in capacity for the complex Monge-Ampère operator acting on plurisubharmonic functions in the Cegrell class on $V$.
Article Details
Keywords
plurisubharmonic function, complex varieties, Monge-Ampere operator, Capacity
References
[1] E. Bedford, The operator (ddc)n on complex spaces, S´eminaire Lelong-Skoda, Springer Lecture Notes 919 (1981), 294-323.
[2] Z. B locki, The complex Monge-Amp`ere operators in hyperconvex domains, Annali della Scuola Normale Superiore di Pisa 23 (1996), 721-747.
[3] E. Bedford and A. Taylor, The Dirichlet problem for the complex Monge-Amp`ere equation, Invent. Math. 37 (1976), 1-44.
[4] E. Bedford and A. Taylor, A new capacity for plurisubharmonic
functions, Acta. Math. 149 (1982), 1-40.
[5] U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998), 187-217.
[6] U. Cegrell, The general definition of the complex Monge-Amp`ere operator, Ann. Inst. Fourier (Grenoble) 54 (2004), 159-179.
[7] U. Cegrell, Convergence in capacity, Canad. Math. Bull., 55 (2012),
242-248.
[8] E. Chirka, Complex Analytic Sets, Kluwer, Dordecht, 1989.
[9] J.-P. Demailly, Mesures de Monge-Amp`ere et caract´erisation g´eom´etrique des vari´et´es alg´ebriques affines, M´em. Soc. Math. France
(N.S.) 19 (1985) 124 pp.
[10] N. Q. Dieu and T. V. Long, Plurisubharmonic functions and Monge-Amp`ere operators on complex varieties in bounded domains of Cn, J. Math. Anal. Appl. 505 (2022) 125477.
[11] N. Q. Dieu, T.V. Long and S. Ounhean, Approximation of plurisubharmonic functions on complex varieties, International Journal of Math. 28, No. 14, 1750107 (2017).
[12] N.Q. Dieu and S. Ounhean, A comparison principle for bounded plurisubharmonic functions on complex varieties of Cn, Proc. of the Amer. Math. Soc., 146 (2018), 309-323.
[13] J. E. Fornaess and R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann., 248 (1980) 47-72.
[14] Pham Hoang Hiep, Convergence in capacity and applications, Mathematica Scandinavica, 107 (2010), 90-102.
[15] Nguyen Van Khue, Pham HoangHiep, A comparison principle for the complex Monge-Amp`ere operator in Cegrell’s classes and applications, Trans. Am. Math. Soc. 361 (2009) 5539–5554.
[16] V. Vˆajˆaitu, Stein spaces with plurisubharmonic bounded exhaustion functions, Math. Ann., 336 (2006), 539-550.
[17] Y. Xing, Continuity of the complex Monge-Amp`ere operator, Proc. Amer. Math. Soc. 124 (1996), 457-467.
[2] Z. B locki, The complex Monge-Amp`ere operators in hyperconvex domains, Annali della Scuola Normale Superiore di Pisa 23 (1996), 721-747.
[3] E. Bedford and A. Taylor, The Dirichlet problem for the complex Monge-Amp`ere equation, Invent. Math. 37 (1976), 1-44.
[4] E. Bedford and A. Taylor, A new capacity for plurisubharmonic
functions, Acta. Math. 149 (1982), 1-40.
[5] U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998), 187-217.
[6] U. Cegrell, The general definition of the complex Monge-Amp`ere operator, Ann. Inst. Fourier (Grenoble) 54 (2004), 159-179.
[7] U. Cegrell, Convergence in capacity, Canad. Math. Bull., 55 (2012),
242-248.
[8] E. Chirka, Complex Analytic Sets, Kluwer, Dordecht, 1989.
[9] J.-P. Demailly, Mesures de Monge-Amp`ere et caract´erisation g´eom´etrique des vari´et´es alg´ebriques affines, M´em. Soc. Math. France
(N.S.) 19 (1985) 124 pp.
[10] N. Q. Dieu and T. V. Long, Plurisubharmonic functions and Monge-Amp`ere operators on complex varieties in bounded domains of Cn, J. Math. Anal. Appl. 505 (2022) 125477.
[11] N. Q. Dieu, T.V. Long and S. Ounhean, Approximation of plurisubharmonic functions on complex varieties, International Journal of Math. 28, No. 14, 1750107 (2017).
[12] N.Q. Dieu and S. Ounhean, A comparison principle for bounded plurisubharmonic functions on complex varieties of Cn, Proc. of the Amer. Math. Soc., 146 (2018), 309-323.
[13] J. E. Fornaess and R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann., 248 (1980) 47-72.
[14] Pham Hoang Hiep, Convergence in capacity and applications, Mathematica Scandinavica, 107 (2010), 90-102.
[15] Nguyen Van Khue, Pham HoangHiep, A comparison principle for the complex Monge-Amp`ere operator in Cegrell’s classes and applications, Trans. Am. Math. Soc. 361 (2009) 5539–5554.
[16] V. Vˆajˆaitu, Stein spaces with plurisubharmonic bounded exhaustion functions, Math. Ann., 336 (2006), 539-550.
[17] Y. Xing, Continuity of the complex Monge-Amp`ere operator, Proc. Amer. Math. Soc. 124 (1996), 457-467.