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Starting from a Hamiltonian description of four dimensional generalrelativity in presence of a cosmological constant we perform the program ofcanonical quantisation. This is done using complex Ashtekar variables whilekeeping the Barbero--Immirzi parameter real. Introducing the SL(2,C) Kodamastate formally solving all first class constraints we propose a spinfoam vertexamplitude. We construct SL(2,C) boundary spinnetwork functions coloured byfinite dimensional representations of the group, and derive the skein relationsneeded to calculate the amplitude. The space of boundary states is shown tocarry a representation of the holonomy flux algebra and can naturally beequipped with a non-degenerate inner product. It fails to be positive definite,but cylindrical consistency is perfectly satisfied.

Complex Ashtekar variables, the Kodama state and spinfoam gravity