A remark on covering of compact Kähler manifolds and applications

• Дата: 2021
• Автори: Hung, V. V., Quy, H. N.
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Institute of Mathematics, NAS of Ukraine 2021
• Онлайн доступ: https://umj.imath.kiev.ua/index.php/umj/article/view/6038
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

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 $M.$