An Aubin-Yau theorem for transversally Kähler foliations

Technical Summary: An Aubin-Yau theorem for transversally Kähler foliations

Problem Statement
The paper addresses the existence and uniqueness of special Hermitian metrics on transversally Kähler foliations, a class of geometric structures that generalizes Kähler manifolds to the complex non-Kähler setting. Specifically, the author investigates the analogue of the Aubin-Yau theorem in this context. While the classical Aubin-Yau theorem guarantees the existence of a unique Kähler-Einstein metric with negative Einstein constant on compact Kähler manifolds with negative first Chern class, the corresponding result for transversally Kähler foliations had not been rigorously established in the literature. Previous references to such a result (e.g., El Kacimi-Alaoui) claimed it followed "as in the classical case" without providing a detailed proof. Furthermore, the paper notes that the transversal version of the $\partial\bar{\partial}$-lemma, essential for such proofs, relies on the condition of homological orientability, a nuance often overlooked in brief claims.

Methodology
The author employs a self-contained analytical approach, adapting the classical Schauder continuity method and elliptic regularity theory to the transversal setting of foliations. The methodology proceeds through several key stages:

1. Geometric Framework: The paper utilizes Molino theory to define transversally Kähler foliations. It establishes the necessary geometric objects, including the transverse frame bundle, transverse $G$-structures, and the basic Laplacian. A critical distinction is made between the transverse Laplace-Beltrami operator ($\Delta_T$) and the basic Laplacian ($\Delta_{bas}$), noting that under the condition of homological orientability, their difference is a zero-order operator.
2. Functional Analysis: The author defines Hölder spaces of basic functions ($C^{k,\alpha}_{bas}$) and Sobolev spaces ($W^{k,p}_{bas}$) on the foliated manifold. A compact embedding theorem for these spaces (Proposition 4.5) is established, which is crucial for the continuity method.
3. Elliptic Operators: The core of the analysis focuses on the fully nonlinear second-order differential operator $\mathcal{C}al^-_T$, which encodes the transversally Kähler Aubin-Yau problem. The equation takes the form:
   $$\mathcal{C}al^-_T(u) = \log \left( \frac{(\omega_0 + \sqrt{-1}\partial_T \bar{\partial}_T u)^q}{\omega_0^q} \right) - u = f$$
   where $\omega_0$ is the initial transversally Kähler form.
4. Continuity Method: To prove existence, the author applies the Schauder continuity method. This involves:
   • Openness: Proving that the set of parameters $t \in [0,1]$ for which the equation $\mathcal{C}al^-_T(u) = tf$ has a solution is open. This relies on the Implicit Function Theorem in Banach spaces, requiring the linearization of the operator to be an isomorphism. The author demonstrates that the linearized operator is essentially a perturbed transverse Laplacian, which is an isomorphism due to the spectral properties of the Laplacian on compact manifolds.
   • Closedness: Proving the set is closed by establishing uniform a priori estimates ($C^0$, $C^1$, and $C^{2,\alpha}$) for the solutions. The author adapts Yau's classical estimates (specifically the $C^2$ estimate) to the transversal setting, relying on local arguments around maximum points of the solution.
5. Regularity: Once a weak solution is obtained via the continuity method, elliptic regularity theory is invoked to show that the solution is smooth ($C^\infty_{bas}$).

Key Contributions and Results

• Theorem 5.1 (Main Result): The paper proves that for a compact, homologically orientable, transversally Kähler foliation $(M, \mathcal{F})$ where the first Chern class of the transverse bundle $c_1(\nu\mathcal{F})$ is negative (represented by a transversally negative form), there exists a unique transversally Kähler, transversally Einstein metric with Einstein constant $-1$.
• Transversal $\partial\bar{\partial}$-Lemma: The paper explicitly utilizes and relies on a transversal version of the $\partial\bar{\partial}$-lemma (Lemma 3.3), which holds under the homological orientability condition. This allows the reduction of the geometric problem to a scalar PDE on basic functions.
• Uniqueness Proof: A detailed proof of uniqueness is provided, utilizing the maximum principle on the difference of two potential solutions. The argument diagonalizes the Hessian of the difference function with respect to the background metric to show that the difference must be zero.
• Application to Vaisman Manifolds: The paper applies the main theorem to Vaisman manifolds. It provides a new, simpler proof of the Aubin-Yau theorem for Vaisman manifolds (Theorem 6.12). Unlike previous proofs that relied on Weitzenböck-type formulas to establish quasi-regularity, this approach directly applies the transversal result. The author clarifies that the existence of a transversally Einstein metric on the canonical foliation of a Vaisman manifold implies the existence of a Vaisman metric with specific Chern-Ricci curvature properties, provided the cohomology class of the weight bundle satisfies specific conditions ($c_1(L) = c_1(K_M^{\otimes 2})$).

Significance and Claims
The author claims that this work fills a gap in the literature where the existence of transversally Einstein metrics for transversally Kähler foliations was previously only asserted without proof. The paper emphasizes that the adaptation of classical techniques to the foliated setting is non-trivial and requires careful handling of the basic cohomology and the specific properties of the transverse Laplacian.

The significance of the work is twofold:

1. Theoretical: It provides a rigorous, self-contained proof of the Aubin-Yau theorem in the context of transversally Kähler foliations, establishing the necessary analytical framework (elliptic operators on foliations, a priori estimates).
2. Applied: It offers a streamlined alternative proof for the Vaisman Aubin-Yau theorem, bypassing more complex machinery used in prior literature. The author notes that while the result for Vaisman manifolds was known, the method of proof presented here is distinct and simpler.

The paper remains modest regarding broader implications, stating that the result is missing from the literature and that the application to Vaisman manifolds serves as a demonstration of the main theorem's utility. It does not propose new experimental applications or future directions beyond the scope of the established theorems.