  We introduce a holomorphic representation for the Lorentzian EPRL spinfoam on
arbitrary 2-complexes. The representation is obtained via the
Ashtekar-Lewandowski-Marolf-Mour\~ao-Thiemann heat kernel coherent state
transform. The new variables are classical holonomy-flux phase space variables
$(h,X)\simeq \mathcal T^*SU(2)$ of Hamiltonian loop quantum gravity prescribing
the holonomies of the Ashtekar connection $A=\Gamma + \gamma K$, and their
conjugate gravitational fluxes. For small heat kernel `time' the spinfoam
amplitude is peaked on classical space-time geometries, where at most countably
many curvatures are allowed for non-zero Barbero-Immirzi parameter. We briefly
comment on the possibility to use the alternative flipped classical limit.
