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The Hard Lefschetz Theorem on Kähler Lie Algebroids

Shane Rankin

TL;DR

This work extends the Hard Lefschetz Theorem from classical Kähler manifolds to a class of Kähler Lie Algebroids by introducing a Hodge-admissible framework defined via ellipticity and an integrating section. It demonstrates that under these conditions, harmonic forms admit a Dolbeault-type bigrading, cohomology decomposes accordingly, and the $ A$-Hard Lefschetz isomorphisms hold, mirroring the classical theory. The results connect HL to symplectic harmonicity, Hodge-Riemann relations, and the $dd^{\star}$-lemma within the algebroid setting, while also providing obstructions and explicit examples, including a product construction and a non-elliptic failure case on the $b$-sphere. The work yields Betti-number constraints and shows how HL can fail without ellipticity, emphasizing the role of ellipticity in transferring Kähler geometry to Lie algebroids and enriching the landscape of generalized complex geometry.

Abstract

Compact Kähler manifolds classically satisfy the Hard Lefschetz Theorem, which gives strong control on the underlying topology of the manifold. One expects a similar theorem to be true for Kähler Lie Algebroids, and we show for a certain class of them that this is indeed true, with an added ellipticity requirement. We provide examples of Lie Algebroids satisfying this, as well as an example of a Kähler Lie Algebroid that does not meet this Ellipticity requirement, and consequently fails to satisfy the Hard Lefschetz condition.

The Hard Lefschetz Theorem on Kähler Lie Algebroids

TL;DR

This work extends the Hard Lefschetz Theorem from classical Kähler manifolds to a class of Kähler Lie Algebroids by introducing a Hodge-admissible framework defined via ellipticity and an integrating section. It demonstrates that under these conditions, harmonic forms admit a Dolbeault-type bigrading, cohomology decomposes accordingly, and the -Hard Lefschetz isomorphisms hold, mirroring the classical theory. The results connect HL to symplectic harmonicity, Hodge-Riemann relations, and the -lemma within the algebroid setting, while also providing obstructions and explicit examples, including a product construction and a non-elliptic failure case on the -sphere. The work yields Betti-number constraints and shows how HL can fail without ellipticity, emphasizing the role of ellipticity in transferring Kähler geometry to Lie algebroids and enriching the landscape of generalized complex geometry.

Abstract

Compact Kähler manifolds classically satisfy the Hard Lefschetz Theorem, which gives strong control on the underlying topology of the manifold. One expects a similar theorem to be true for Kähler Lie Algebroids, and we show for a certain class of them that this is indeed true, with an added ellipticity requirement. We provide examples of Lie Algebroids satisfying this, as well as an example of a Kähler Lie Algebroid that does not meet this Ellipticity requirement, and consequently fails to satisfy the Hard Lefschetz condition.

Paper Structure

This paper contains 15 sections, 28 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Kähler Lie Algebroids
  5. Complex Lie Algebroids
  6. Kähler Identities
  7. Kähler Hodge Theory
  8. Ellipticity
  9. Symplectic Lie Algebroids
  10. Symplectic Harmonicity
  11. Adjoint Relations
  12. Consequences
  13. Betti Control
  14. Examples
  15. Failure of the Kähler $b$-Sphere

Key Result

Theorem 1.1

Let $(\mathcal{A},\rho, [ \cdot, \cdot],g_\mathcal{A},\omega_\mathcal{A},J_\mathcal{A})$ be a Hodge-admissible Kähler Lie Algebroid. Then the harmonic forms are bigraded, that is where $\mathcal{H}^{p,q} = \ker(\Delta) \cap\Omega_\mathcal{A}^{p,q}$ is finite-dimensional. We also have that where $H_\mathcal{A}^k(M;\mathbb{C})$ is the complexified Lie Algebroid Cohomology.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Theorem 5: MR3483864,Theorem 2.1
  • Definition 6
  • Definition 7
  • ...and 68 more