论文标题
Bott-Chern和Dolbeault Harmonic $（k，k）$的原始分解 - 紧凑的形式几乎是Kähler歧管
Primitive decomposition of Bott-Chern and Dolbeault harmonic $(k,k)$-forms on compact almost Kähler manifolds
论文作者
论文摘要
我们考虑$ \ bar \ partial，\ partial $，bott-chern和aeppli-harmonic $（k，k）$的原始分解 - 紧凑型几乎是kähler歧管$（m，j，ω）$。对于任何$ d \ in \ {\ bar \ partial，\ partial，bc，a \} $，我们证明$ l^k p^0 $ compongent in \ in \ mathcal {h Mathcal {h} _ {h} _ {d} _ {d} _关注尺寸8，我们给出了空间的完整描述$ \ MATHCAL {h} _ {bc}^{2,2} $和$ \ MATHCAL { $ \ MATHCAL {H}^{2,2} _ {BC} \ subseteq \ Mathcal {H}^{2,2} _ {\ partial} $ and $ \ MATHCAL {H}^{2,2} _ {a} \ subseteq \ Mathcal {h}^{2,2} _ {\ bar \ partial} $。 We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of an harmonic form $ψ\in \mathcal{H}_{D}^{k,k}$ are not $D$-harmonic, showing that the primitive decomposition of $(k,k)$-forms in general does not descend to harmonic forms.
We consider the primitive decomposition of $\bar \partial, \partial$, Bott-Chern and Aeppli-harmonic $(k,k)$-forms on compact almost Kähler manifolds $(M,J,ω)$. For any $D \in \{\bar\partial, \partial, BC, A\}$, we prove that the $L^k P^0$ component of $ψ\in \mathcal{H}_{D}^{k,k}$, is a constant multiple of $ω^k$. Focusing on dimension 8, we give a full description of the spaces $\mathcal{H}_{BC}^{2,2}$ and $\mathcal{H}_{A}^{2,2}$, from which follows $\mathcal{H}^{2,2}_{BC}\subseteq\mathcal{H}^{2,2}_{\partial}$ and $\mathcal{H}^{2,2}_{A}\subseteq\mathcal{H}^{2,2}_{\bar\partial}$. We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of an harmonic form $ψ\in \mathcal{H}_{D}^{k,k}$ are not $D$-harmonic, showing that the primitive decomposition of $(k,k)$-forms in general does not descend to harmonic forms.
