On piecewise pluriharmonic functions

TL;DR

This paper generalizes results on piecewise linear functions to piecewise pluriharmonic functions on complex manifolds, constructing a ring of currents and exploring their algebraic properties and dependencies.

Contribution

It introduces a ring generated by currents from piecewise pluriharmonic functions and extends derivation properties within this ring.

Findings

Constructed a ring generated by currents h and dd^ch.

Proved the extension of certain maps to derivations on the ring.

Showed that specific wedge products depend only on the product of functions.

Abstract

We extend some results on piecewise linear functions on $\C^n$ to piecewise pluriharmonic functions on any complex manifold. We construct a ring generated by currents $h$ and $dd^ch$, where $\{h\}$ is a finite set of piecewise pluriharmonic functions. We prove that, with some restrictions on the set $\{h\}$, the map $\{h\mapsto dd^ch,\ dd^ch\mapsto0\}$ can be continued to the derivation on the ring. As a corollary, the current $dd^cg_1\wedge...\wedge dd^cg_k$ depends on the product of piecewise pluriharmonic functions $g_1,...,g_k$ only.