Pluripotential Product Structure

• Pluripotential product structure is a framework that defines wedge products and establishes rigorous foundations for mixed Monge–Ampère operators in various settings.
• It employs properties like graded-commutativity, associativity, and monotone continuity to extend classical Bedford–Taylor theory to tropical, P-pluripotential, and quaternionic environments.
• The structure facilitates explicit computations for extremal functions, capacity estimates, and non-archimedean Monge–Ampère equations by linking complex and convex analysis.

The pluripotential product structure refers to the precise way in which wedge products (or exterior products) of currents and forms are defined, manipulated, and interpreted in various extensions of classical pluripotential theory. These include the complex, tropical, non-archimedean, and quaternionic settings. The concept underpins the ability to construct and analyze non-linear operators such as the (mixed) Monge–Ampère operators, yielding powerful results for extremal functions, capacity calculations, and the structure of solutions to complex and non-archimedean Monge–Ampère equations.

1. Foundations of Pluripotential Product Structures

The classical Bedford–Taylor product structure for wedge products of closed positive currents—particularly those constructed from plurisubharmonic (psh) functions—establishes graded-commutativity, associativity, multilinearity, positivity, closedness under $\partial$ and $\overline\partial$ (or their analogs), and continuity under monotone approximation. This structure extends to a broader context in several settings:

• Tropical toric pluripotential theory: Bedford–Taylor products are defined for Lagerberg currents on open subsets of tropical toric varieties, constructed via pullback along the tropicalization map and pushforward back to the tropical space (Gil et al., 2021).
• $P$-pluripotential theory: The structure manifests in the product property for $P$-extremal functions, which are built from classes of psh functions with prescribed logarithmic growth determined by support functions of convex bodies in $\mathbb{R}^d$ (Levenberg et al., 2018).
• Quaternionic pluripotential theory: The wedge product of closed positive $2$-forms relies on linear algebraic data (Moore determinant), with product structure mirroring the complex situation, but adapted to hyperhermitian matrices and Baston operators (Wang, 2018).

The product structure enables the explicit computation of non-linear measures and operators central to pluripotential theory in each context.

2. Tropical Toric Bedford–Taylor Product

Given a smooth rational fan $\Sigma \subset N \simeq \mathbb{Z}^n$, with $N_{\mathbb{R}}$ its support and $N_\Sigma$ the associated tropical toric variety, the Bedford–Taylor product for Lagerberg currents is formulated as follows:

Let $U \subset N_\Sigma$ be open, $\overline\partial$0 be plurisubharmonic (tropical-psh), and $\overline\partial$1 a closed positive Lagerberg current. There exists a unique $\overline\partial$2, $\overline\partial$3, with $\overline\partial$4 for $\overline\partial$5. The product is defined by: $\overline\partial$6 with $\overline\partial$7. This extends the classical Bedford–Taylor product and is uniquely characterized by locality and monotone regularization (Gil et al., 2021).

Properties include graded-commutativity, associativity, multilinearity, positivity, closedness under $\overline\partial$8, continuity in the psh-factors, and a projection formula for equivariant maps. In top degree and $\overline\partial$9, this yields the Monge–Ampère measure $P$0, generalizing Alexandrov–Bedford–Taylor theory to tropical settings.

3. Product Structure in $P$1-Pluripotential Theory

For a convex body $P$2 with nonempty interior and $P$3 for some $P$4, the $P$5-extremal function $P$6 for a compact set $P$7 is: $P$8 where $P$9 is the Lelong-type class defined by prescribed $P$0-support function growth (Levenberg et al., 2018).

The main product formula asserts that for $P$1 compact nonpolar sets and support function $P$2,

$P$3

holds exactly, generalizing the classical case. This structure unifies and extends all known product-type results in weighted and unweighted pluripotential theory, and the wedge product (Monge–Ampère measure) supports factorization in certain cases.

Table: Product Formula Examples in $P$4-Pluripotential Theory

Convex body $P$5	$P$6	Product formula for $P$7
Standard simplex $P$8	$P$9	$\mathbb{R}^d$0
$\mathbb{R}^d$1-ball $\mathbb{R}^d$2	$\mathbb{R}^d$3	$\mathbb{R}^d$4
Unit torus $\mathbb{R}^d$5	$\mathbb{R}^d$6	$\mathbb{R}^d$7

4. Linear Algebraic Product Structure in Quaternionic Pluripotential Theory

On the quaternionic vector space $\mathbb{R}^d$8 (viewed as $\mathbb{R}^d$9), the real 2-form

$2$0

is said to be "real" if it is invariant under the anti-complex involution associated with $2$1. This is equivalent to the existence of a unique hyperhermitian matrix $2$2 such that $2$3, with $2$4 the complex matrix embedding.

Under quaternionic unitary equivalence, any such $2$5 can be reduced to block-diagonal form: $2$6 and the $2$7-th wedge power relates to the Moore determinant as

$2$8

Thus, the Moore determinant is the coefficient of the top wedge power of the real 2-form. The quaternionic Monge–Ampère operator is defined by the Baston wedge product, yielding

$2$9

and in general, the mixed wedge product gives

$\Sigma \subset N \simeq \mathbb{Z}^n$0

(Wang, 2018). This identifies the precise algebraic role of the product structure in defining quaternionic Monge–Ampère measures.

5. Applications and Capacity-Type Estimates

Product structures are central in translating between extremal functions, capacity calculations, and equilibrium measures:

• For $\Sigma \subset N \simeq \mathbb{Z}^n$1-extremal functions and equilibrium measures, factorization properties and sharp capacity bounds are obtainable directly from the product formula:

$\Sigma \subset N \simeq \mathbb{Z}^n$2

• In the tropical toric setting, the product structure enables solution of non-archimedean Monge–Ampère equations by lifting to a real Monge–Ampère problem on $\Sigma \subset N \simeq \mathbb{Z}^n$3 and pulling back solutions to an invariant semipositive metric on the analytification of a toric variety (Gil et al., 2021).

The structure-enhanced regularity, explicitness, and compatibility with monotone limits are critical for both existence and uniqueness results in non-classical Monge–Ampère equations.

6. Connections Across Complex, Tropical, and Non-Archimedean Theories

The plural product structure forms the link across complex, tropical, and non-archimedean settings:

• Complex: The classical Bedford–Taylor product for positive currents and psh functions serves as the starting point.
• Tropical: The Bedford–Taylor calculus is transferred to the tropical toric setting through tropicalization maps, establishing a canonical correspondence.
• Non-archimedean: Via Chambert–Loir–Ducros theory, the wedge product of first Chern currents of semipositive metrics is reduced to tropical data by pushforward along the tropical skeleton. Solutions to non-archimedean Monge–Ampère equations on toric and abelian varieties are then constructed (Gil et al., 2021).
• Quaternionic: The algebraic structure ensures that results analogous to those in the complex case hold, with the Moore determinant playing the role of the complex determinant in the Monge–Ampère operator (Wang, 2018).

A plausible implication is that further extensions to other non-classical contexts may be realized via transfer and reinterpretation of the product structure, as the underlying algebraic and convex-analytic principles are sufficiently robust to carry over with appropriate modifications.

7. Illustrative Examples

• Tropical-psh on $\Sigma \subset N \simeq \mathbb{Z}^n$4, $\Sigma \subset N \simeq \mathbb{Z}^n$5: A tropical-psh function is a continuous convex function $\Sigma \subset N \simeq \mathbb{Z}^n$6. Then $\Sigma \subset N \simeq \mathbb{Z}^n$7 is the distributional second derivative. For piecewise linear $\Sigma \subset N \simeq \mathbb{Z}^n$8, $\Sigma \subset N \simeq \mathbb{Z}^n$9 is supported on breakpoints with weights equal to slope jumps. The twofold product (for $N_{\mathbb{R}}$0 convex) yields a Radon measure $N_{\mathbb{R}}$1, vanishing unless $N_{\mathbb{R}}$2 (Gil et al., 2021).
• $N_{\mathbb{R}}$3-pluripotential with $N_{\mathbb{R}}$4-balls: The explicit formula $N_{\mathbb{R}}$5 makes apparent the direct incorporation of convex information from $N_{\mathbb{R}}$6 into the pluripotential product structure (Levenberg et al., 2018).

These examples underscore the essential role of convexity, multilinearity, and the algebraic backbone of product structures in determining the analytic and measure-theoretic character of pluripotential theory in a variety of advanced mathematical settings.