  In this paper we propose and analyze a virtual element method for the two
dimensional non-symmetric diffusion-convection eigenvalue problem in order to
derive a priori and a posteriori error estimates. Under the classic assumptions
of the meshes, and with the aid of the classic theory of compact operators, we
prove error estimates for the eigenvalues and eigenfunctions. Also, we develop
an a posteriori error estimator which, in one hand, results to be reliable and
on the other, with standard bubble functions arguments, also results to be
efficient. We test our method on domains where the complex eigenfunctions are
not sufficiently regular, in order to assess the performance of the estimator
that we compare with the uniform refinement given by the a priori analysis
