[Paper Review] Deformation of Sasakian metrics

Hiraku Nozawa|arXiv (Cornell University)|Sep 26, 2008

Geometry and complex manifolds23 references4 citations

TL;DR

This paper establishes that small deformations of the Reeb flow on a Sasakian manifold admit compatible Sasakian metrics if and only if the $(0,2)$-component of the basic Euler class vanishes. Using a deformation-theoretic approach via transversely holomorphic Riemannian flows and a Kodaira-Akizuki-Nakano-type vanishing theorem for basic Dolbeault cohomology, the authors prove the stability of positive Sasakian metrics under such deformations, resolving an obstruction tied to the cohomological structure of the flow's transverse geometry.

ABSTRACT

Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely Kähler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.

Motivation & Objective

• To characterize the conditions under which small deformations of the Reeb flow on a Sasakian manifold admit compatible Sasakian metrics.
• To identify the obstruction to such compatibility, particularly focusing on the $(0,2)$-component of the basic Euler class.
• To establish a Kodaira-Akizuki-Nakano-type vanishing theorem for basic Dolbeault cohomology in the context of homologically orientable transversely Kähler foliations.
• To prove the stability of positive Sasakian metrics under small deformations of the Reeb flow.
• To analyze the moduli space of Sasakian metrics with a fixed transversely Kähler flow structure.

Proposed method

• The authors analyze deformations of the Reeb flow as transversely holomorphic Riemannian flows on a closed manifold.
• They introduce a differential operator $ D $ and study mean curvature forms of Riemannian flows to relate geometric data to cohomological invariants.
• The key technical tool is the Kodaira-Spencer theory applied to families of self-adjoint strongly elliptic differential operators, leveraging the invariance of geometrically taut Riemannian flows.
• They prove a vanishing theorem for basic Dolbeault cohomology using the Bochner-Kodaira-Nakano identity and Hodge decomposition on $ \mathcal{F} $-fibered Hermitian holomorphic line bundles.
• The proof relies on constructing an inner product on basic forms via the El Kacimi-Alaoui–Hector form and establishing formal adjoint operators.
• The stability result is derived by showing that the vanishing of the $(0,2)$-component of the basic Euler class ensures the existence of a smooth family of compatible Sasakian metrics.

Experimental results

Research questions

• RQ1 Under what conditions does a small deformation of the Reeb flow on a Sasakian manifold admit a compatible Sasakian metric?
• RQ2 What role does the $(0,2)$-component of the basic Euler class play in obstructing the existence of compatible Sasakian metrics?
• RQ3 Can a Kodaira-Akizuki-Nakano-type vanishing theorem be established for basic Dolbeault cohomology of transversely Kähler foliations?
• RQ4 Is the stability of positive Sasakian metrics preserved under small deformations of the Reeb flow?
• RQ5 How does the moduli space of Sasakian metrics relate to the underlying transversely Kähler flow structure?

Key findings

• The existence of a compatible Sasakian metric for small deformations of the Reeb flow is characterized by the triviality of the $(0,2)$-component of the basic Euler class.
• A Kodaira-Akizuki-Nakano-type vanishing theorem holds for basic Dolbeault cohomology of homologically orientable transversely Kähler foliations, enabling cohomological control.
• For positive Sasakian manifolds, all small deformations of the Reeb flow admit compatible Sasakian metrics, as the $(0,2)$-component vanishes due to the positivity of the transverse Kähler structure.
• In the case of the standard Sasakian structure on $ S^{2n-1} $, the moduli space of isomorphism classes of Sasakian metrics is isomorphic to the space of isomorphism classes of underlying transversely Kähler flows.
• For circle bundles over complex tori, the Reeb flow can be deformed such that no compatible Sasakian metric exists when the base is non-projective, demonstrating that the $(0,2)$-component obstruction is nontrivial.
• The Kuranishi space of transversely holomorphic flow deformations on $ \mathbb{C}P^{n-1} $ is identified with $ H^0(\mathbb{C}P^{n-1}, T^{1,0}\mathbb{C}P^{n-1}) $, and since $ H^{0,2}(\mathbb{C}P^{n-1}) = 0 $, the standard Sasakian metric is stable.