Gauge invariance of complex general relativity

In this paper it is implemented how to make compatible the boundary conditi ons and the gauge fixing condition, for complex general relativity written in terms of Ashtekar variables using the approach of Ref. [1]. Moreover, it is found that at first order in the gauge parameters, the Hamiltonian acti on is (on shell) fully gauge-invariant under the gauge symmetry generated b y the first class constraints in the case when spacetime M has the topology M = R x Sigma and Sigma has no boundary. Thus, the statement that the cons traints linear in the momenta do riot contribute to the boundary terms is r ight, but only in the case when Sigma has no boundary.