Curvature estimates for the continuity method

Abstract

We obtain curvature estimates for long-time solutions of the continuity method on compact Kähler manifolds with semi-ample canonical line bundles. In this setting, initiated in [G. La Nave and G. Tian, A continuity method to construct canonical metrics, Math. Ann. 365(3) (2016) 911–921; Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math. 218(5) (2008) 1526–1565], we adapt arguments from [F. T.-H. Fong and Y. Zhang, Local curvature estimates of long-time solutions to the Kähler–Ricci flow, Adv. Math. 375 (2020) 107416] for the Kähler–Ricci flow to this setup. As an application, we derive curvature bounds for general metrics on product manifolds.