Estimates on Monge-Amp\`ere operators derived from a local algebra inequality

TL;DR

This paper establishes a link between the integrability of exponential functions of plurisubharmonic functions with singularities and bounds on their Monge-Ampère mass, using local algebra inequalities applicable locally or globally in complex space.

Contribution

It introduces a new inequality derived from local algebra that relates Monge-Ampère mass bounds to exponential integrability, extending previous results to arbitrary dimensions.

Findings

If the Monge-Ampère mass is less than n^n, then the exponential of the plurisubharmonic function is integrable on compact subsets.

The inequality is valid both locally and globally in complex space.

The result generalizes previous work by extending it to higher dimensions.

Abstract

The goal of this short note is to relate the integrability property of the exponential $e^{-2\phi}$ of a plurisubharmonic function $\phi$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Amp\`ere mass of $(dd^c\phi)^n$. The inequality is valid locally or globally on an arbitrary open subset $\Omega$ in $\bC^n$. We show that $\int_\Omega(dd\phi)^n<n^n$ implies $\int_Ke^{-2\phi}<+\infty$ for every compact subset $K$ in $\Omega$, while functions of the form $\phi(z)=n\log|z-z_0|$, $z_0\in\Omega$, appear as limit cases. The result is derived from an inequality of pure local algebra, which turns out a posteriori to be equivalent to it, proved by A.Corti in dimension $n=2$, and later extended by L.Ein, T.De Fernex and M.Musta\c{t}\v{a} to arbitrary dimensions.