Continuity method with movable singularities for classical Monge-Ampère equations

Abstract

On a compact Kähler manifold $(X,\omega )$, we study the strong continuity of solutions with prescribed singularities of complex Monge-Ampère equations with integrable Lebesgue densities. Moreover, we give sufficient conditions for the strong continuity of solutions when the right-hand sides are modified to include all (log) Kähler-Einstein metrics with prescribed singularities. Our findings can be interpreted as closedness of new continuity methods in which the densities vary together with the prescribed singularities. For Monge-Ampère equations of Fano type, we also prove an openness result when the singularities decrease. As an application, we deduce a strong stability result for (log-)Kähler Einstein metrics on semi-Kähler classes given as modifications of $\{\omega \}$.

Proof of Theorem C changed, some typos corrected

References in corpus (2)

• Integration by parts formula for non-pluripolar product
• Monge-Ampère measures on contact sets

Cited by in corpus (1)

• Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting