[Paper Review] Estimates on Monge-Ampère operators derived from a local algebra inequality

Jean-Pierre Demailly|ArXiv.org|Sep 21, 2007

Geometry and complex manifolds15 references21 citations

TL;DR

This paper establishes an a priori bound on the $L^1$-integrability of $e^{-2\varphi}$ for plurisubharmonic functions $\varphi$ with compactly supported singularities, derived from a uniform upper bound on the total Monge-Ampère mass $\int_\Omega (dd^c\varphi)^n < n^n$. The key result shows that if the Monge-Ampère mass is strictly less than $n^n$, then $\int_K e^{-2\varphi} < \infty$ for every compact $K \subset \Omega$, with extremal behavior achieved by functions of the form $\varphi(z) = n\log|z - z_0|$. The proof relies on a deep inequality from local algebra, originally due to Corti and extended by Ein, De Fernex, and Mustaća, which links the log-canonical threshold to the Hilbert-Samuel multiplicity of an ideal.

ABSTRACT

The goal of this short note is to relate the integrability property of the exponential $e^{-2ϕ}$ of a plurisubharmonic function $ϕ$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Ampère mass of $(dd^cϕ)^n$. The inequality is valid locally or globally on an arbitrary open subset $Ω$ in $\bC^n$. We show that $\int_Ω(ddϕ)^n

Motivation & Objective

• To relate the integrability of $e^{-2\varphi}$ for plurisubharmonic functions $\varphi$ with isolated or compactly supported singularities to bounds on the total Monge-Ampère mass $\int_\Omega (dd^c\varphi)^n$.
• To prove that if $\int_\Omega (dd^c\varphi)^n < n^n$, then $\int_K e^{-2\varphi} < \infty$ for every compact $K \subset \Omega$, independent of $\varphi$.
• To show that the bound $n^n$ is sharp, with functions $\varphi(z) = (n - \varepsilon)\log|z - z_0|$ approaching the threshold as $\varepsilon \to 0$.
• To establish that the result follows from a fundamental inequality in local algebra, specifically relating the log-canonical threshold of an ideal to its Hilbert-Samuel multiplicity.
• To demonstrate that the analytic estimate is equivalent a posteriori to the algebraic inequality, proven by Corti in dimension 2 and extended by Ein, De Fernex, and Mustaća to arbitrary dimensions.

Proposed method

• The proof reduces the analytic problem to a known inequality in local algebra, specifically the bound $\mathop{\rm lc}(\mathcal{J}) \leq n / e(\mathcal{J})^{1/n}$ for the log-canonical threshold of a zero-dimensional ideal $\mathcal{J}$ in $\mathcal{O}_{\mathbb{C}^n,0}$.
• The argument uses the equivalence between the analytic Monge-Ampère mass bound and the algebraic multiplicity condition, leveraging the fact that equality in the algebraic inequality occurs precisely when the integral closure of $\mathcal{J}$ is a power of the maximal ideal.
• A key step involves the use of the Ohsawa-Takegoshi $L^2$ extension theorem and approximation techniques for singularities, to control the growth of $e^{-2\varphi}$ near the singular locus.
• The compactness of the class $\mathcal{P}_{0,M}(\Omega)$ in $L^1_{\rm loc}(\Omega)$ is used to ensure uniform bounds, relying on Cegrell’s class $\mathcal{F}(\Omega)$ and weak convergence of Monge-Ampère measures.
• A subextension theorem from [CZ03] is applied to extend $\varphi$ from $\Omega$ to a larger hyperconvex domain $\tilde{\Omega}$, preserving the Monge-Ampère mass bound.
• The final estimate is derived by combining the uniform $L^1$-boundedness of $e^{-2\varphi}$ on compact subsets with the subextension and compactness arguments, yielding a constant $C'(\Omega, M)$ independent of $\varphi$.

Experimental results

Research questions

• RQ1 Under what conditions on the Monge-Ampère mass $\int_\Omega (dd^c\varphi)^n$ is the function $e^{-2\varphi}$ guaranteed to be locally integrable for plurisubharmonic $\varphi$ with compactly supported singularities?
• RQ2 Is the threshold $n^n$ for the Monge-Ampère mass sharp, and what happens when this bound is approached?
• RQ3 Can the integrability of $e^{-2\varphi}$ be controlled uniformly across a family of plurisubharmonic functions with bounded Monge-Ampère mass?
• RQ4 How is the analytic condition $\int_\Omega (dd^c\varphi)^n < n^n$ related to algebraic invariants such as the Hilbert-Samuel multiplicity and log-canonical threshold?
• RQ5 Is there an equivalence between the analytic estimate on $e^{-2\varphi}$ and a fundamental inequality in local algebra, and if so, under what conditions does equality hold?

Key findings

• If $\int_\Omega (dd^c\varphi)^n < n^n$, then $\int_K e^{-2\varphi} < \infty$ for every compact subset $K \subset \Omega$, with the bound depending only on $\Omega$, $K$, and the mass bound $M < n^n$.
• The bound $n^n$ is optimal: for $\varphi_\varepsilon(z) = (n - \varepsilon)\log|z - z_0|$, the Monge-Ampère mass is $(n - \varepsilon)^n < n^n$, but $\int_K e^{-2\varphi_\varepsilon} \to \infty$ as $\varepsilon \to 0^+$, showing the threshold is sharp.
• Functions of the form $\varphi(z) = n\log|z - z_0|$ are extremal cases, achieving the critical mass $n^n$ and corresponding to the threshold of integrability.
• The result is equivalent a posteriori to a deep inequality in local algebra: $\mathop{\rm lc}(\mathcal{J}) \leq n / e(\mathcal{J})^{1/n}$, with equality if and only if $\overline{\mathcal{J}} = \mathfrak{m}^k$ for some $k$, where $\mathfrak{m}$ is the maximal ideal.
• The proof relies on the compactness of the class of plurisubharmonic functions with bounded Monge-Ampère mass in $L^1_{\rm loc}$, and the use of Cegrell’s class $\mathcal{F}(\Omega)$ to extend the definition of the Monge-Ampère operator to limits of such functions.
• The uniform bound $\int_K e^{-2\varphi} \leq C'(\Omega, M)$ is established via subextension and truncation techniques, ensuring independence from the specific choice of $\varphi$.