  In the present paper, we investigate the underlying Stokes complex structure
of the Virtual Element Method for Stokes and Navier--Stokes introduced in
previous papers by the same authors, restricting our attention to the two
dimensional case. We introduce a Virtual Element space $\Phi_h \subset
H^2(\Omega)$ and prove that the triad $\{\Phi_h, V_h, Q_h\}$ (with $V_h$ and
$Q_h$ denoting the discrete velocity and pressure spaces) is an exact Stokes
complex. Furthermore, we show the computability of the associated differential
operators in terms of the adopted degrees of freedom and explore also a
different discretization of the convective trilinear form. The theoretical
findings are supported by numerical tests.
