  The lowest-order nonconforming virtual element extends the Morley triangular
element to polygons for the approximation of the weak solution $u\in
V:=H^2_0(\Omega)$ to the biharmonic equation. The abstract framework allows
(even a mixture of) two examples of the local discrete spaces $V_h(P)$ and a
smoother allows rough source terms $F\in V^*=H^{-2}(\Omega)$. The a priori and
a posteriori error analysis in this paper circumvents any trace of second
derivatives by some computable conforming companion operator $J:V_h\to V$ from
the nonconforming virtual element space $V_h$. The operator $J$ is a
right-inverse of the interpolation operator and leads to optimal error
estimates in piecewise Sobolev norms without any additional regularity
assumptions on $u\in V$. As a smoother the companion operator modifies the
discrete right-hand side and then allows a quasi-best approximation. An
explicit residual-based a posteriori error estimator is reliable and efficient
up to data oscillations. Numerical examples display the predicted empirical
convergence rates for uniform and optimal convergence rates for adaptive
mesh-refinement.
