  We study the effects of off-shellness in the generalized parton distributions
of the pion. On general grounds, these distributions exhibit a richer structure
than in the on-shell case due to absence of the crossing symmetry. In
particular, their moments involve additional terms odd in the skewness
parameter, associated with new form factors. We bring up relations between the
off-shell charge and gravitational form factors, as well as the pion form
factor, and discuss their derivations based on the Ward-Takahashi identities.
We illustrate the features at the (leading-$N_c$) one-quark-loop level with the
help of the spectral quark model of the pion, constructed to embed the vector
meson dominance. Simple analytic expressions for the form factors and the
distributions follow. Thus obtained off-shell generalized parton distributions
are evolved from the quark model scale to higher scales with the LO DGLAP
equations. We evaluate the corresponding Compton amplitudes which enter the
cross-section for the electroproduction of the pion off the proton (the
Sullivan process). It is found in our model that the effects of off-shellness
in the generalized parton distribution are substantial, however, they can be
largely canceled by the corresponding off-shell corrections to the pion
propagator. In particular, this is the case of the Compton form factors
entering the deeply virtual Compton scattering amplitude. As a result, we
expect small off-shellness effects in electroproduction reactions, such as the
Sullivan process.
