We define pre-Kähler structures on a complex manifold and motivate their study by noting that they are naturally induced on complex submanifolds of a pseudo-Kähler manifold and also arise as symmetry reductions of CR structures of hypersurface type that may not be Levi nondegenerate. Mimicking the notion of k-nondegeneracy in the CR setting, we define k-nondegenerate pre-Kähler structures, which turn out to have finite-dimensional symmetry algebra. The correspondence between Sasakian and Kähler structures, arising either as the symmetry reduction or as the metric cone, is generalized to the pre-Kähler setting.

Lastly, as the lowest dimensional non-Kähler pre-Kähler case, we analyze 2-nondegenerate pre-Kähler structures on complex surfaces and give a solution of their equivalence problem in terms of a Cartan geometry. After providing parametric expressions for their essential invariants in terms of a pre-Kähler potential, we give examples and analyze a double fibration that they naturally define. This is a joint work with David Sykes.