  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.
