Pre-Kähler structures and finite-nondegeneracy
Omid Makhmali, David Sykes
TL;DR
The paper introduces pre-Kähler geometry as a natural Levi-degenerate analogue of Kähler geometry, defined by a real closed (1,1)-form compatible with the complex structure. It develops a comprehensive framework: (i) Freeman filtration and k-nondegeneracy diagnose holomorphic degeneracy and symmetry finiteness; (ii) a precise pre-Kähler–pre-Sasakian correspondence ties leaf-space and hypersurface CR data, complemented by a presymplectification construction; and (iii) a Cartan-geometric treatment of 2-nondegenerate pre-Kähler complex surfaces yields explicit invariants in terms of a local potential, with a clear link to the twistor bundle of 2D symplectic connections. The paper then analyzes symmetry reductions of homogeneous 2-nondegenerate CR 5-manifolds, identifying special symplectic connections and proving that locally homogeneous 2-nondegenerate pre-Kähler complex surfaces are flat. Overall, the work extends CR-geometric techniques to a broader pre-symplectic context, providing concrete local invariants and Cartan-descriptions that connect to twistor theory and symplectic geometry.
Abstract
Motivated by the geometry of Levi degenerate CR hypersurfaces, we define a pre-Kähler structure on a complex manifold as a pre-symplectic structure compatible with the almost complex structure, i.e. a closed (1,1)-form. Extending Freeman filtration to the pre-Kähler setting, we define holomorphic degeneration and finite-nondegeneracy and show that the symmetry algebra of a real analytic pre-Kähler structure is finite-dimensional if and only if it is finitely nondegenerate. Concurrently, we extend the classical correspondence between Kähler and Sasakian structures to the pre-Kähler setting, i.e. a one-to-one (local) correspondence between $k$-nondegenerate CR hypersurfaces equipped with a transverse infinitesimal symmetry and $k$-nondegenerate pre-Kähler structures. We additionally formalize a second relationship between the categories, constructing a natural $k$-nondegenerate pre-Kähler structure defined on a line bundle over such CR structures via pre-symplectification. Focusing on the lowest dimensional case, we solve the equivalence problem of non-Kähler pre-Kähler complex surfaces that are 2-nondegenerate by associating a Cartan geometry to them and explicitly express their local invariants in terms of the fifth jet of a potential function. We describe the vanishing of their basic invariants in terms of a double fibration, which gives a pre-Kähler characterization of the twistor bundle of symplectic connections on surfaces. Lastly, our study of pre-Kähler complex surfaces that are symmetry reductions of homogeneous 2-nondegenerate CR 5-manifolds leads to a characterization of certain critical symplectic connections on surfaces and shows locally homogeneous 2-nondegenerate pre-Kähler complex surfaces are flat.