Pre-K\"ahler structures and finite-nondegeneracy

Abstract

Motivated by the geometry of Levi degenerate CR hypersurfaces, we define a \emph{pre-K\"ahler structure} on a complex manifold as a pre-symplectic structure compatible with the almost complex structure, i.e. a closed (1,1)-form. Extending \emph{Freeman filtration} to the pre-K\"ahler setting, we define holomorphic degeneration and finite-nondegeneracy and show that the symmetry algebra of a real analytic pre-K\"ahler structure is finite-dimensional if and only if it is finitely nondegenerate. Concurrently, we extend the classical correspondence between K\"ahler and Sasakian structures to the pre-K\"ahler setting, i.e. a one-to-one (local) correspondence between $k$-nondegenerate CR hypersurfaces equipped with a transverse infinitesimal symmetry and $k$-nondegenerate pre-K\"ahler structures. Focusing on the lowest dimensional case, we solve the equivalence problem of non-K\"ahler pre-K\"ahler 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-K\"ahler characterization of the twistor bundle of symplectic connections on surfaces. Lastly, we study the pre-K\"ahler complex surfaces arising as symmetry reductions of homogeneous $2$-nondegenerate CR 5-manifolds, which leads to a characterization of certain \emph{critical} symplectic connections on surfaces. For such pre-K\"{a}hler manifolds, their moduli space of geometrically distinct structures contain $2$-dimensional open dense subsets, and they all have nontrivial infinitesimal symmetries. Finally, we show that all locally homogeneous pre-K\"{a}hler complex surfaces are locally flat.