[Paper Review] Log-concavity of volume and complex Monge-Amp\\`ere equations with prescribed singularity

Tamás Darvas, Eleonora Di Nezza+1 more|arXiv (Cornell University)|Jul 1, 2018

Geometry and complex manifolds40 references39 citations

TL;DR

This paper establishes the existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularity type on compact Kähler manifolds, removing the prior restriction of small unbounded locus. It confirms the log-concavity of volume for closed positive (1,1)-currents and identifies a deep correspondence between relative pluripotential theory and Brunn-Minkowski theory in convex geometry.

ABSTRACT

Let $(X,\\omega)$ be a compact K\\"ahler manifold. We prove the existence and uniqueness of solutions to complex Monge-Amp\\`ere 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 K\\"ahler 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.

Motivation & Objective

• To resolve the long-standing conjecture by Boucksom-Eyssidieux-Guedj-Zeriahi on the log-concavity of the volume functional for closed positive (1,1)-currents.
• To establish the existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularity type, without requiring small unbounded locus.
• To generalize previous results by Kołodziej and Boucksom-Eyssidieux-Guedj-Zeriahi to the setting of big cohomology classes and general model-type singularities.
• To reveal a precise dictionary between relative pluripotential theory in complex geometry and P-relative convex geometry, particularly linking to the Brunn-Minkowski inequality.

Proposed method

• Introduces a relative Kołodziej estimate (Theorem 3.3) as a key technical tool to control the $L^p$-integrability of solutions.
• Uses the relative envelope $P_{ heta}[ullet]$ to define and characterize model-type singularities, ensuring the singularity type is well-behaved.
• Applies pluripotential theory to define the non-pluripolar Monge-Ampère measure $\theta_u^n$ for $u \in \textup{PSH}(X,\theta)$, enabling the formulation of the equation in the sense of currents.
• Establishes a correspondence between convex bodies in $\mathbb{R}^n$ and singularities of positive currents via the support function and the toric model potential $\phi_P$.
• Translates the Brunn-Minkowski inequality in convex geometry into a complex geometric inequality via the mixed Monge-Ampère product on $\mathbb{CP}^n$.
• Uses invariance under the torus action $(S^1)^n$ to reduce the problem to the toric setting, enabling the use of convex analysis and capacity estimates.

Experimental results

Research questions

• RQ1 Can complex Monge-Ampère equations with prescribed singularity type be solved without assuming small unbounded locus?
• RQ2 Is the volume functional log-concave on the cone of closed positive (1,1)-currents on a compact Kähler manifold?
• RQ3 To what extent does the relative pluripotential theory of singular metrics mirror the Brunn-Minkowski theory of convex bodies?
• RQ4 How does the solution to the complex Monge-Ampère equation behave under the Aubin-Yau type equation with exponential nonlinearity?
• RQ5 What is the precise relationship between the singularity type of a metric and the integrability of its associated Monge-Ampère measure?

Key findings

• Theorem A(i) establishes the existence and uniqueness (up to a constant) of a solution $u$ to $\theta_u^n = f\omega^n$ with $[u] = [\phi]$, provided $[\phi]$ is a model-type singularity and $f \in L^p(\omega^n)$, $p > 1$, with $\int_X f\omega^n = \int_X \theta_\phi^n > 0$.
• Theorem A(ii) extends this to the Aubin-Yau type equation $\theta_u^n = e^{\lambda u}f\omega^n$, proving existence and uniqueness for any $\lambda > 0$ under the same conditions.
• The log-concavity conjecture of Boucksom-Eyssidieux-Guedj-Zeriahi is fully confirmed: the volume functional $[\phi] \mapsto \int_X \theta_\phi^n$ is log-concave on the space of singularity types.
• The paper establishes a precise correspondence: the Brunn-Minkowski inequality in convex geometry corresponds to the log-concavity of volume in complex geometry, with mixed Monge-Ampère products mirroring mixed volumes.
• The solution to the Monge-Ampère equation on $\mathbb{CP}^n$ with toric symmetry is shown to be bounded if the data satisfy a certain $L^{n+\delta}$ integrability condition, via a volume-capacity comparison.
• The mixed Monge-Ampère product $\int_{\mathbb{CP}^n} \prod_{j=1}^n (r\omega_{FS} + i\partial\bar{\partial}\phi_{P_j})$ equals $\frac{n!}{2^n} \textup{MV}(P_1,\ldots,P_n)$, proving the complex analog of the Brunn-Minkowski inequality.