A remark on covering of compact Kähler manifolds and applications

V. V. Hung; H. N.  Quy

doi:10.37863/umzh.v73i1.6038

V. V. Hung Tay Bac Univ., Univ. Danang – Univ. Sci. and Education, Vietnam
H. N. Quy Tay Bac Univ., Univ. Danang – Univ. Sci. and Education, Vietnam

DOI: https://doi.org/10.37863/umzh.v73i1.6038

Keywords: Complex Monge-Amp`ere operator, ω-plurisubharmonic functions, compact K¨ahler manifolds

Abstract

UDC 517.9

Recently, Kolodziej proved that, on a compact Kähler manifold $M,$ the solutions to the complex Monge – Ampére equation with the right-hand side in $L^p,$ $p>1,$ are Hölder continuous with the exponent depending on $M$ and $\|f\|_p$ (see [Math. Ann., 342, 379-386 (2008)]).
Then, by the regularization techniques in
[J. Algebraic Geom., 1, 361-409 (1992)], the authors in [J. Eur. Math. Soc., 16, 619-647 (2014)] have found the optimal exponent of the solutions.
In this paper, we construct a cover of the compact Kähler manifold $M$ which only depends on curvature of $M.$ Then, as an application, base on the arguments in
[Math. Ann., 342, 379-386 (2008)], we show that the solutions are Hölder continuous with the exponent just depending on the function $f$ in the right-hand side and upper bound of curvature of $M.$

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