[Paper Review] Integration by parts formula for non-pluripolar product

Mingchen Xia|arXiv (Cornell University)|Jul 15, 2019

Geometry and complex manifolds5 references14 citations

TL;DR

This paper establishes the integration by parts formula for the non-pluripolar Monge–Ampère product on compact Kähler manifolds, generalizing a result from [BEGZ10] that only applied to potentials with small unbounded loci. The authors use Witt Nyström's construction to lift potentials to products with $<math xmlns="http://www.w3.org/1998/Math/MathML">\mathbb{P}^N$ and reduce the general case to the known bounded case via convergence of Monge–Ampère measures.

ABSTRACT

In this paper, we prove the integration by parts formula for the non-pluripolar product on a compact Kähler manifold. Our result generalizes the special case of potentials with small unbounded loci proved in [BEGZ10].

Motivation & Objective

• To extend the integration by parts formula for non-pluripolar Monge–Ampère products beyond potentials with small unbounded loci.
• To resolve a gap in the literature where the general case was not previously established.
• To provide a generalization of Theorem 1.14 in [BEGZ10], which only applied to potentials with small unbounded loci.
• To establish a direct link between integrals on the original manifold and those on $X \times \mathbb{P}^N$ via a limiting construction.
• To prove that the integration by parts formula holds under the condition that $[\varphi_1] = [\varphi_2]$ and $[\psi_1] = [\psi_2]$.

Proposed method

• Leverage Witt Nyström's construction to associate each $\varphi \in \mathrm{PSH}(X,\theta)$ with a family of potentials $\Phi_N[\varphi]$ on $X \times \mathbb{P}^N$ having small unbounded loci.
• Use the convergence of Monge–Ampère measures of $\Phi_N[\varphi]$ to the measure of $\varphi$ as $N \to \infty$ to reduce the general case to the bounded case.
• Apply the known integration by parts formula for potentials with small unbounded loci (from [BEGZ10]) to the lifted potentials on $X \times \mathbb{P}^N$.
• Use polarization and homogenization techniques to extend the formula to differences of potentials with the same cohomology class.
• Establish a polynomial identity in coefficients $a_0, \dots, a_n$ to equate coefficients of $a_2 \cdots a_n$, thereby proving the formula on $X$.
• Use quadratic optimization on the simplex $\Sigma_N$ to analyze the behavior of the closest point projection and derive asymptotic expansions for the error function $g_N$.

Experimental results

Research questions

• RQ1 Can the integration by parts formula for the non-pluripolar Monge–Ampère product be extended beyond potentials with small unbounded loci?
• RQ2 Is it possible to reduce the general integration by parts problem on a compact Kähler manifold to the classical Bedford–Taylor theory using a geometric lifting construction?
• RQ3 What is the precise relationship between integrals involving $u = \varphi_1 - \varphi_2$ and $v = \psi_1 - \psi_2$ under the non-pluripolar product when $[\varphi_1] = [\varphi_2]$ and $[\psi_1] = [\psi_2]$?
• RQ4 How does Witt Nyström's construction facilitate the convergence of Monge–Ampère measures in the context of integration by parts?
• RQ5 Can the integration by parts formula be recovered via a polynomial coefficient comparison argument after lifting to $X \times \mathbb{P}^N$?

Key findings

• The integration by parts formula holds for all $\varphi_1, \varphi_2, \psi_1, \psi_2 \in \mathrm{PSH}(X,\theta)$ with $[\varphi_1] = [\varphi_2]$ and $[\psi_1] = [\psi_2]$, generalizing [BEGZ10] Theorem 1.14.
• The formula is proven by lifting potentials to $X \times \mathbb{P}^N$ via Witt Nyström's construction, where the unbounded loci become small, allowing application of the known bounded case.
• The Monge–Ampère measures of the lifted potentials $\Phi_N[\varphi]$ converge to the original Monge–Ampère measure of $\varphi$ in a suitable sense, enabling the reduction.
• A direct formula is established relating the integrals on $X$ to those on $X \times \mathbb{P}^N$, as stated in Corollary 2.14.
• The proof relies on a polynomial identity in coefficients $a_0, \dots, a_n$, showing that the coefficients of $a_2 \cdots a_n$ in the two sides of the integration by parts formula are equal.
• The function $g_N(x) = \min_{\alpha \in \Sigma_N} (x - \alpha)^2 - x^2$ is shown to be continuous and piecewise linear with bounded slope, enabling asymptotic analysis in the proof.