  Given a Heegaard splitting of a three-manifold Y, we consider the SL(2,C)
character variety of the Heegaard surface, and two complex Lagrangians
associated to the handlebodies. We focus on the smooth open subset
corresponding to irreducible representations. On that subset, the intersection
of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi
associates to such an intersection a perverse sheaf of vanishing cycles. We
prove that in our setting, the perverse sheaf is an invariant of Y, i.e., it is
independent of the Heegaard splitting. The hypercohomology of this sheaf can be
viewed as a model for (the dual of) SL(2,C) instanton Floer homology. We also
present a framed version of this construction, which takes into account
reducible representations. We give explicit computations for lens spaces and
Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations
and Khovanov homology.
