Pre-K\"ahler structures and finite-nondegeneracy

Mathematics > Differential Geometry [Submitted on 14 May 2025 (v1), last revised 16 Jun 2026 (this version, v2)] Title:Pre-Kähler structures and finite-nondegeneracy View PDF HTML (experimental)Abstract:Motivated by the geometry of Levi degenerate CR hypersurfaces, we define a \emph{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 \emph{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 \emph{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. Submission history From: David Sykes [view email][v1] Wed, 14 May 2025 15:18:38 UTC (164 KB) [v2] Tue, 16 Jun 2026 14:17:40 UTC (167 KB) Current browse context: math.DG References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.