  Off-shell effects in generalized parton distributions (GPDs) of the pion,
appearing, e.g., in the Sullivan process, are considered. Due to the lack of
crossing symmetry, the moments of GPDs involve also odd powers of the skewness
(longitudinal momentum transfer) parameter, which results in emergence of new
off-shell form factors. With current-algebra techniques, we derive exact
relations between the four off-shell gravitational form factors of the pion, in
analogy to the electromagnetic case. Our results place stringent constraints on
the off-shell GPDs of the pion. We provide an explicit realization in terms of
a chiral quark model, where we show that the off-shell effects in GPDs are
potentially significant in modeling physical processes and should not be
neglected.
