[Paper Review] On formality of Kahler orbifolds and Sasakian manifolds

Indranil Biswas, Marisa Fernández+2 more|arXiv (Cornell University)|Feb 27, 2014

Geometry and complex manifolds19 references3 citations

TL;DR

This paper proves that compact Kähler orbifolds are formal, leading to key topological consequences for Sasakian manifolds. It establishes that all higher Massey products vanish on simply connected Sasakian manifolds, proving they obstruct Sasakian structures, and constructs the first examples of simply connected non-formal regular Sasakian manifolds of dimension 2n+1 for n > 2 via non-vanishing triple Massey products.

ABSTRACT

We prove that compact Kahler orbifolds are formal, and derive applications of it to the topology of compact Sasakian manifolds. In particular, answering questions raised by Boyer and Galicki, we prove that all higher Massey products on any simply connected Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this we produce a method of constructing simply connected K-contact non-Sasakian manifolds. On the other hand, for every n>2 we exhibit the first examples of simply connected compact regular Sasakian manifolds of dimension 2n+1 which are non-formal. They are non-formal because they have a non-zero triple Massey product. We also prove that arithmetic lattices in some simple Lie groups cannot be the fundamental group of a compact Sasakian manifold.

Motivation & Objective

• To establish the formality of compact Kähler orbifolds and extend its implications to Sasakian geometry.
• To resolve open questions by Boyer and Galicki concerning the role of Massey products in Sasakian structures.
• To construct simply connected K-contact non-Sasakian manifolds using formality obstructions.
• To produce the first examples of simply connected compact regular Sasakian manifolds that are non-formal, specifically due to non-vanishing triple Massey products.
• To determine which arithmetic lattices in simple Lie groups cannot arise as fundamental groups of compact Sasakian manifolds.

Proposed method

• Utilizes the theory of formality in rational homotopy theory to analyze the cohomology ring structure of compact Kähler orbifolds.
• Applies formality of Kähler orbifolds to derive constraints on the cohomology of compact Sasakian manifolds.
• Employs the existence of non-vanishing triple Massey products as an obstruction to formality, constructing non-formal Sasakian manifolds in odd dimensions 2n+1 for n > 2.
• Uses the fact that formality implies vanishing of all higher Massey products to show that non-vanishing Massey products obstruct Sasakian structures.
• Applies results from geometric group theory and arithmetic lattices in simple Lie groups to rule out certain groups as fundamental groups of compact Sasakian manifolds.
• Combines techniques from complex geometry, contact topology, and rational homotopy theory to analyze the interplay between Kähler, Sasakian, and K-contact structures.

Experimental results

Research questions

• RQ1 Do higher Massey products vanish on simply connected Sasakian manifolds, and if so, does this imply formality or obstruction to such structures?
• RQ2 Can non-formal compact Sasakian manifolds be constructed in odd dimensions 2n+1 for n > 2, and what cohomological invariants detect their non-formality?
• RQ3 Can the formality of Kähler orbifolds be used to construct simply connected K-contact manifolds that are not Sasakian?
• RQ4 Which arithmetic lattices in simple Lie groups cannot be realized as fundamental groups of compact Sasakian manifolds?
• RQ5 What is the role of Massey products in obstructing Sasakian structures, and how do they relate to formality in the context of Kähler orbifolds?

Key findings

• All higher Massey products vanish on any simply connected Sasakian manifold, confirming that such products obstruct Sasakian structures.
• The first examples of simply connected compact regular Sasakian manifolds of dimension 2n+1 for n > 2 are constructed that are non-formal due to a non-zero triple Massey product.
• A method is provided to construct simply connected K-contact non-Sasakian manifolds by exploiting the formality obstruction from Massey products.
• Arithmetic lattices in certain simple Lie groups cannot be the fundamental group of any compact Sasakian manifold, as shown via cohomological obstructions.
• Compact Kähler orbifolds are proven to be formal, extending formality results from smooth Kähler manifolds to the orbifold setting.
• The non-vanishing of the triple Massey product serves as a definitive invariant distinguishing non-formal Sasakian manifolds from formal ones.