On piecewise pluriharmonic 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 $d{d}^{c}h$, where $\{h\}$ is a finite set of piecewise pluriharmonic functions. We prove that, with some restrictions on the set $\{h\}$, the map $\{h\mapsto d{d}^{c}h,d{d}^{c}h\mapsto 0\}$ can be continued to the derivation on the ring. As a corollary, the current $d{d}^c{g}_1\wedge ...\wedge d{d}^c{g}_k$ depends on the product of piecewise pluriharmonic functions $g_1,...,g_k$ only.

Four tipos corrected