[Paper Review] Locally conformally Kähler manifolds. A selection of results

Liviu Ornea|ArXiv.org|Nov 23, 2004

Geometry and complex manifolds39 references16 citations

TL;DR

This paper surveys recent advances in locally conformally Kähler (l.c.K.) geometry post-1997, focusing on Vaisman structures, l.c.K. manifolds with potential, and their relations to Sasakian geometry. It establishes embedding theorems, reduction procedures, and stability results, while highlighting open problems in moduli, singularities, curvature, and fundamental groups of compact l.c.K. manifolds.

ABSTRACT

I present a selection of results on locally conformally Kähler geometry published after 1997. The proofs are mainly sketched, some of them are even omitted. Several open problems are indicated in the end.

Motivation & Objective

• To synthesize and summarize key developments in locally conformally Kähler geometry from 1997 onward, particularly focusing on Vaisman and l.c.K. manifolds with potential.
• To clarify the relationship between l.c.K. geometry and Sasakian geometry, especially through the cone construction and reduction theory.
• To present new structural results, including embedding theorems and stability under small deformations, for compact l.c.K. manifolds.
• To identify and frame open problems in moduli spaces, singularities, curvature, and fundamental groups of l.c.K. manifolds.
• To explore the compatibility of l.c.K. reduction with minimal presentations and the preservation of geometric invariants like rank and Vaisman structure.

Proposed method

• Utilizes the Gauduchon metric on compact Vaisman manifolds to develop stability theory and analyze harmonic forms and vector fields.
• Applies symplectic and Kähler reduction techniques to l.c.K. structures via the cone construction, lifting reductions from Sasakian to Vaisman manifolds.
• Employs the minimal presentation of l.c.K. manifolds using the weight bundle and the Lee form to generalize reduction and embedding theorems.
• Applies algebro-geometric techniques, including Dolbeault cohomology and vanishing theorems, to derive Kodaira-type embedding results for Vaisman manifolds.
• Uses the notion of l.c.K. structures with potential to generalize Hopf manifolds and prove embedding theorems into linear Hopf manifolds in complex dimension ≥3.
• Introduces a new equivalent definition of l.c.K. structures via presentations, enhancing the understanding of the Kähler-l.c.K. duality and providing a new invariant.

Experimental results

Research questions

• RQ1 What is the structure of compact Vaisman manifolds, and how does it relate to Sasakian geometry through the cone construction?
• RQ2 Can compact l.c.K. manifolds with potential of dimension ≥3 be embedded into linear Hopf manifolds, and what are the conditions for such embeddings?
• RQ3 Which compact complex surfaces with non-zero Euler-Poincaré characteristic admit l.c.K. metrics, especially those not covered by Belgun’s classification?
• RQ4 Is the class of l.c.K. manifolds with potential preserved under l.c.K. reduction, and does the reduction of a non-Vaisman manifold yield a Vaisman quotient?
• RQ5 Can a convexity theory be developed for l.c.K. manifolds, given the triviality of $d^\theta$-cohomology and the limitations of Morse theory?

Key findings

• Compact Vaisman manifolds are completely classified topologically and geometrically, with their structure reducible to Sasakian geometry via the cone construction.
• Any compact l.c.K. manifold with potential of complex dimension at least 3 admits a holomorphic embedding into a linear Hopf manifold.
• The l.c.K. reduction procedure preserves the minimal presentation and the rank of the l.c.K. structure, with the reduced manifold inheriting the minimal presentation from the original.
• The automorphism group of a l.c.K. manifold is studied via Hamiltonian actions, and the reduction process is compatible with the structure theorem for Vaisman manifolds.
• The Ricci-like curvature and Weyl-Ricci tensor of l.c.K. manifolds impose topological restrictions, particularly in the Einstein-Weyl case.
• A new equivalent definition of l.c.K. structures via presentations clarifies the duality with Kähler geometry and provides a new invariant for l.c.K. structures.