Bott-Chern and ∂ Harmonic forms on Almost Hermitian 4-manifolds

2021/10/31 by Tom Holt, Holt, Tom
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2111.00518

Abstract

We prove that on a compact almost Hermitian 4-manifold the space of ∂-harmonic (1,1)-forms always has dimension h_∂^1,1 = b_- +1 or b_-, whilst the space of Bott-Chern harmonic (1,1)-forms always has dimension h_BC^1,1 = b_- +1. We also perform calculations of h^2,1_BC and h^1,2_BC on the Kodaira-Thurston manifold, thereby providing a full account of when h^p,q_BC is or is not invariant of the choice of almost Hermitian metric. Finally, we introduce a decomposition of the space of L² functions on all torus bundles over S¹, which has proven useful for solving linear PDEs, and we demonstrate its use in the calculation of h^p,q_∂.

Related

• ∂-Harmonic forms on 4-dimensional almost-Hermitian manifolds
• On the spaces of (d+d^c)-harmonic forms and (d+d^Λ)-harmonic forms on almost Hermitian manifolds and complex surfaces
• Primitive decomposition of Bott-Chern and Dolbeault harmonic (k,k)-forms on compact almost Kähler manifolds
• Harmonic Forms on the Kodaira-Thurston Manifold
• Bott-Chern harmonic forms and primitive decompositions on compact almost Kähler manifolds