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On the spaces of $(d+d^c)$-harmonic forms and $(d+d^Λ)$-harmonic forms on almost Hermitian manifolds and complex surfaces

Lorenzo Sillari, Adriano Tomassini

TL;DR

This work develops a general theory of $(d+d^c)$- and $(d+d^Λ)$-harmonic forms on compact almost Hermitian manifolds via elliptic, self-adjoint Laplacians, extending Bott-Chern and symplectic harmonic theory beyond integrable complex and symplectic cases. It establishes metric-independence results and four-dimensional decompositions that link harmonic dimensions to topological invariants such as Betti numbers, and it demonstrates that Bott-Chern and Aeppli numbers of compact complex surfaces depend only on topology, giving explicit diamonds determined by $b_1$, $b^+$, and $b^-$. The paper further computes these invariants for solvmanifolds with invariant compatible triples and for Hopf-type manifolds, producing data tables that are invariant under the choice of invariant structures. Collectively, the results extend cohomological and harmonic perspectives to a broad class of almost Hermitian manifolds, offering new tools for relating geometric structures to underlying topology and to Lie-group quotients.

Abstract

We study the spaces of $(d + d^c)$-harmonic forms and $(d + d^Λ)$-harmonic forms, the natural generalization of the spaces of Bott-Chern harmonic forms, resp. symplectic harmonic forms from complex, resp. symplectic, manifolds to almost Hermitian manifolds. With the same techniques, we also prove that Bott-Chern and Aeppli numbers of compact complex surfaces depend only on the topology of the underlying manifold, a fact that was well-known for Hodge numbers of compact complex surfaces. We give several applications to compact quotients of Lie groups by a lattice.

On the spaces of $(d+d^c)$-harmonic forms and $(d+d^Λ)$-harmonic forms on almost Hermitian manifolds and complex surfaces

TL;DR

This work develops a general theory of - and -harmonic forms on compact almost Hermitian manifolds via elliptic, self-adjoint Laplacians, extending Bott-Chern and symplectic harmonic theory beyond integrable complex and symplectic cases. It establishes metric-independence results and four-dimensional decompositions that link harmonic dimensions to topological invariants such as Betti numbers, and it demonstrates that Bott-Chern and Aeppli numbers of compact complex surfaces depend only on topology, giving explicit diamonds determined by , , and . The paper further computes these invariants for solvmanifolds with invariant compatible triples and for Hopf-type manifolds, producing data tables that are invariant under the choice of invariant structures. Collectively, the results extend cohomological and harmonic perspectives to a broad class of almost Hermitian manifolds, offering new tools for relating geometric structures to underlying topology and to Lie-group quotients.

Abstract

We study the spaces of -harmonic forms and -harmonic forms, the natural generalization of the spaces of Bott-Chern harmonic forms, resp. symplectic harmonic forms from complex, resp. symplectic, manifolds to almost Hermitian manifolds. With the same techniques, we also prove that Bott-Chern and Aeppli numbers of compact complex surfaces depend only on the topology of the underlying manifold, a fact that was well-known for Hodge numbers of compact complex surfaces. We give several applications to compact quotients of Lie groups by a lattice.
Paper Structure (8 sections, 30 theorems, 110 equations, 2 tables)

This paper contains 8 sections, 30 theorems, 110 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Almost complex and almost symplectic Laplacians on almost Hermitian manifolds
  4. The spaces of -harmonic forms
  5. The spaces of -harmonic forms
  6. Relations between -harmonic forms and -harmonic forms
  7. Bott-Chern and Aeppli numbers of compact complex surfaces
  8. The numbers of invariant compatible triples

Key Result

Theorem A

Let $(M,J,\omega,g)$ be an almost Hermitian $4$-manifold and let $(\tilde{\omega},\tilde{g})$ be a Gauduchon metric in the same conformal class of $(\omega,g)$. Then

Theorems & Definitions (61)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • ...and 51 more