Harmonic Forms on the Kodaira-Thurston Manifold

Abstract

We introduce an effective method to solve the $\bar{\partial }$-harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic PDE system to countably many linear ODE systems. By solving a fundamental problem on linear ODE systems, the problem of finding $\bar{\partial }$-harmonic forms is equivalent to a generalised Gauss circle problem. We demonstrate two remarkable applications. First, the dimension of the almost complex $\bar{\partial }$-Hodge numbers on the Kodaira-Thurston manifold could be arbitrarily large. Second, Hodge numbers vary with different choices of Hermitian metrics. This answers a question of Kodaira and Spencer in Hirzebruch's 1954 problem list.

28 pages. v4: presentation improved. v2 and v3: presentation improved, mistakes corrected, references added

References in corpus (2)

• Kodaira dimensions of almost complex manifolds II
• Almost Kähler Kodaira-Spencer problem

Cited by in corpus (7)

• Kodaira dimensions of almost complex manifolds II
• Almost Kähler Kodaira-Spencer problem
• $\overline{\partial }$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds
• Leafwise flat forms on Inoue-Bombieri surfaces
• On the spectral sets of Inoue surfaces
• Bott-Chern Laplacian on almost Hermitian manifolds
• From Smooth to Almost Complex