Bott–Chern and $$\bar{\partial }$$ harmonic forms on almost Hermitian 4-manifolds

Abstract

We prove that on a compact almost Hermitian 4-manifold the space of $${\bar{\partial }}$$ ∂ ¯ -harmonic (1, 1)-forms always has dimension $$h_{{\bar{\partial }}}^{1,1} = b_- +1$$ h ∂ ¯ 1 , 1 = b - + 1 or $$b_-$$ b - , whilst the space of Bott–Chern harmonic (1, 1)-forms always has dimension $$h_{BC}^{1,1} = b_- +1$$ h BC 1 , 1 = b - + 1 . We also perform calculations of $$h^{2,1}_{BC}$$ h BC 2 , 1 and $$h^{1,2}_{BC}$$ h BC 1 , 2 on the Kodaira–Thurston manifold, thereby providing a full account of when $$h^{p,q}_{BC}$$ h BC p , q is or is not invariant of the choice of almost Hermitian metrics. Finally, we introduce a decomposition of the space of $$L^2$$ L 2 functions on all torus bundles over $$S^1$$ S 1 , which has proven useful for solving linear PDEs, and we demonstrate its use in the calculation of $$h^{p,q}_{{\bar{\partial }}}$$ h ∂ ¯ p , q .

Primitive Decomposition of Bott-Chern and Dolbeault Harmonic (k,k)-Forms on Compact almost Kähler Manifolds

“…∂ is either equal to b − or b − + 1, depending on the choice of metric, see [4,5,12]. Similarly, for Bott-Chern harmonic forms, it yields h 1,1 BC ∶= dim C H 1,1 BC = b − + 1 for all metrics, see [4,10].…”

“…By elliptic theory, if M is compact, then the space of Dolbeault harmonic (p, q)-formshas finite dimension h p,q ∂ , and if moreover J is integrable, then H p,q ∂ is isomorphic to the Dolbeault cohomology group H p,q ∂ ∶= ker ∂ im∂ , which is an invariant of the complex structure.In Problem 20 of Hirzebruch's 1954 Problem List [10], Kodaira and Spencer asked if the number h p,q ∂ is independent on the choice of the almost Hermitian metric g. The first author and Zhang just recently gave a negative answer to this problem [13,14], building a family of almost complex structures on the 4dimensional Kodaira-Thurston manifold, namely the product between the circle S 1 and the compact quotient H 3 (Z) H 3 (R) of the Heisenberg group, showing that h 0,1 ∂ varies with different choices of the almost Hermitian metric. We refer to [11,12,19,20,22] for further studies of Dolbeault harmonic forms on almost Hermitian manifolds. We remark that, at the current state of the art, the only known example of a non integrable almost Hermitian structure where it is possible to compute h p,q ∂ completely for all 0 ≤ p, q ≤ n is just the 4-dimensional Kodaira-Thurston manifold.…”

Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds

“…Indeed they construct on the Kodaira-Thurston manifold an almost-complex structure that, with respect to different almost-Hermitian metrics, has varying dim H 0,1 ∂ . With different techniques, in [16] it was shown that also the dimension of the space of ∂-harmonic (1, 1)-forms depends on the metric on 4-dimensional manifolds (for other results in this direction, see [13] and [10]).…”

Left-invariant almost complex structures on the higher dimensional Kodaira–Thurston manifolds

“…In Problem 20 of Hirzebruch's 1954 Problem List [11], Kodaira and Spencer asked if the number h p,q ∂ is independent on the choice of the almost Hermitian metric g. The first author and Zhang just recently gave a negative answer to this problem [14,15], building a family of almost complex structures on the 4-dimensional Kodaira-Thurston manifold, namely the product between the circle S 1 and the compact quotient H 3 (Z)\H 3 (R) of the Heisenberg group, showing that h 0,1 ∂ varies with different choices of the almost Hermitian metric. We refer to [12,13,20,21,24] for further studies of Dolbeault harmonic forms on almost Hermitian manifolds. We remark that, at the current state of the art, the only known example of a non integrable almost Hermitian structure where it is possible to compute h p,q ∂ completely for all 0 ≤ p, q ≤ n is just the 4-dimensional Kodaira-Thurston manifold.…”