Abstract
Let $$(X,\omega )$$ be a compact Kahler manifold. We prove the existence and uniqueness of solutions to complex Monge–Ampere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the Kahler case as well. As an application we confirm a conjecture by Boucksom–Eyssidieux–Guedj–Zeriahi concerning log-concavity of the volume of closed positive (1, 1)-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn–Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and P-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.