  In complex general relativity, Lorentzian space-time is replaced by a
four-complex-dimensional complex-Riemannian manifold, with holomorphic
connection and holomorphic curvature tensor. A multisymplectic analysis shows
that the Hamiltonian constraint is replaced by a geometric structure linear in
the holomorphic multimomenta, providing some boundary conditions are imposed on
two-complex-dimensional surfaces. On studying such boundary conditions, a link
with the Penrose twistor programme is found. Moreover, in the case of real
Riemannian four-manifolds, the local theory of primary and secondary potentials
for gravitino fields, recently proposed by Penrose, has been applied to
Ricci-flat backgrounds with boundary. The geometric interpretation of the
differential equations obeyed by such secondary potentials is related to the
analysis of integrability conditions in the theory of massless fields, and
might lead to a better understanding of twistor geometry. Thus, new tools are
available in complex general relativity and in classical field theory in real
Riemannian backgrounds.
