<p>The aim of this paper is to propose and analyze a stabilization-free virtual element method for the non-selfadjoint convection–diffusion eigenvalue problem. The method is based on high order harmonic polynomial projections which are used to approximate the continuous sesquilinear forms. In order to analyze the continuous and discrete eigenvalue problems, we introduce solution operators and we prove convergence in norm. Then, by using the approximation spectral theory for compact non-selfadjoint operators, we show that the scheme provides a correct approximation of the spectrum and prove optimal error estimates for the eigenfunctions and a double order for the eigenvalues. The theoretical analysis considers only the case of quadrilateral meshes. Our study is supported by a series of numerical experiments, that assess the robustness of the method.</p>