2025/05/14 by Omid Makhmali, Makhmali, Omid, David Sykes +1
Computer Science · Mathematics · #32L25 #32Q15 #53C10 #53C12 #53D10 #Advanced Algebra and Logic #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Primary: 32V05 #Secondary: 53D05
paper · pdf · doi:10.48550/arxiv.2505.09467
openalex publication_date 2025/05/14 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28
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. 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, we study the pre-Kähler complex surfaces arising as symmetry reductions of homogeneous 2-nondegenerate CR 5-manifolds, which leads to a characterization of certain critical symplectic connections on surfaces. For such pre-Kähler 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-Kähler complex surfaces are locally flat.