[Paper Review] Principles of Locally Conformally Kahler Geometry

Liviu Ornea, Misha Verbitsky|arXiv (Cornell University)|Aug 15, 2022

Geometry and complex manifolds4 citations

TL;DR

This paper presents a comprehensive theoretical framework for locally conformally Kähler (LCK) geometry, unifying multiple definitions through Kähler covers, weight bundles, and automorphic forms. It establishes foundational results in Hodge theory and Vaisman geometry, proving that compact Vaisman manifolds admit a canonical foliation and decompose via the structure theorem, with key applications to holomorphic foliations and flat affine structures on LCK manifolds.

ABSTRACT

An LCK (locally conformally Kahler) manifold is a complex manifold admitting a Kahler covering with monodromy acting by homotheties. Hopf manifolds and their submanifolds are the prime examples. This book presents an introduction to the principles of LCK geometry (the first two parts) and its current situation (the last part). It is supposed to be accessible to master and graduate students of complex geometry. The book contains many exercises of different levels of difficulty. We finish it by a list of open questions.

Motivation & Objective

• To unify and systematize the multiple definitions of locally conformally Kähler (LCK) manifolds using Kähler covers, weight bundles, and automorphic forms.
• To establish foundational results in Hodge theory and cohomology for LCK and Vaisman manifolds, particularly focusing on harmonic forms and the Lee form.
• To investigate the geometric and topological structure of compact Vaisman manifolds, culminating in the structure theorem for such spaces.
• To explore open problems related to holomorphic foliations, flat affine structures, and the monodromy of LCK manifolds, especially in relation to Beauville-type theorems and split tangent bundles.

Proposed method

• Utilizes Kähler covers via Galois theory and deck transformation groups to define LCK manifolds as quotients of Kähler manifolds by discrete group actions.
• Introduces the weight bundle and homothety character to relate LCK structures to automorphic forms and conformal changes of Kähler metrics.
• Applies Ehresmann connections and Frobenius theorem to study foliations and integrability in the context of LCK geometry.
• Employs the Riemann–Hilbert correspondence to relate flat connections and local systems, particularly in the study of holomorphic vector bundles.
• Applies the Calabi formula and Chern connection to analyze curvature and positivity in holomorphic line bundles over LCK manifolds.
• Uses the de Rham splitting theorem and conical Riemannian metrics to analyze holonomy and local structure of Vaisman manifolds.

Experimental results

Research questions

• RQ1 Does every logarithmic foliation on a Vaisman manifold admit a transversally Kähler structure, and is it always taut?
• RQ2 Can a Vaisman manifold admit a smooth 1-dimensional holomorphic foliation without any compact leaves?
• RQ3 Are there non-Vaisman LCK manifolds with split tangent bundles (TM = ⊕Li), and what conditions ensure that each Li admits a flat holomorphic connection?
• RQ4 Is the monodromy of the flat connection on a complex affine LCK manifold always abelian?
• RQ5 Do all complex, complete affine manifolds admitting an LCK metric arise as Vaisman nilmanifolds of Heisenberg type?

Key findings

• The structure theorem for compact Vaisman manifolds establishes a canonical foliation and a decomposition of the manifold into a product-like structure involving a Kähler base and a circle action.
• Harmonic 1-forms on Vaisman manifolds decompose into the direct sum of the Lee form and a cohomology class in the orthogonal complement, providing a refined Hodge decomposition.
• Rank 1 Vaisman structures are characterized by the existence of a global Killing vector field dual to the Lee form, and such manifolds are locally isometric to a Kähler cone.
• The tangent bundle of OT manifolds and diagonal Hopf manifolds splits into flat holomorphic line bundles, and these admit flat connections.
• All known examples of complex, complete affine LCK manifolds are Vaisman nilmanifolds of Heisenberg type, suggesting a possible classification constraint.
• The monodromy of the flat connection on known LCK manifolds (e.g., Vaisman nilmanifolds, Hopf, OT manifolds) is abelian, raising the open question of whether this holds universally.