Holomorphic last multipliers on complex manifolds

TL;DR

This paper explores the theory of holomorphic last multipliers on complex manifolds, linking them to real last multipliers and applying the concepts to various holomorphic vector fields and structures.

Contribution

It introduces the concept of holomorphic last multipliers, establishes their relation to real last multipliers, and applies the theory to holomorphic gradient, Hamiltonian, and Poisson structures.

Findings

Holomorphic last multipliers are characterized and related to real last multipliers.

Applications to holomorphic gradient vector fields on Riemannian manifolds.

Applications to holomorphic Hamiltonian and Poisson vector fields.

Abstract

The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic last multipliers with real last multipliers. Also, we consider some applications of our study for holomorphic gradient vector fields on holomorphic Riemannain manifolds as well as for holomorphic Hamiltonian vector fields and holomorphic Poisson bivector fields on holomorphic Poisson manifolds.