Lelong classes of plurisubharmonic functions on an affine variety

TL;DR

This paper investigates classes of plurisubharmonic functions on affine varieties, introduces a new notion of polynomial degree via pluripotential theory, and establishes an affine Bézout's theorem, with explicit computations and examples.

Contribution

It defines the Lelong degree of polynomials on affine varieties using pluripotential theory and derives an affine Bézout's theorem, providing explicit calculations and examples.

Findings

Computed the Monge-Ampère mass of Lelong classes.

Defined the Lelong degree of polynomials on affine varieties.

Established an affine version of Bézout's theorem.

Abstract

We study the Lelong classes $\mathcal{L}(V),\mathcal{L}^+(V)$ of psh functions on an affine variety $V$. We compute the Monge-Amp\`ere mass of these functions, which we use to define the degree of a polynomial on $V$ in terms of pluripotential theory (the Lelong degree). We compute the Lelong degree explicitly in a specific example. Finally, we derive an affine version of B\'ezout's theorem.