论文标题
几乎复杂而几乎是Kähler流形的不变性
Invariants of almost complex and almost Kähler manifolds
论文作者
论文摘要
令$（m^{2n}，j）$为紧凑的几乎复杂的歧管。几乎复杂不变的$ h^{p，q} _j $定义为同胞空间的复杂维度$ \ left \ weft \ {\ left [α\ right] \ in H^{p+q} _ {dr} _ {dr}（m^{2n}; a^{p，q}（m^{2n}），\，dα= 0 \ right \} $。当$ 2N = 4 $时，它具有许多有趣的属性。 eng $（m^{2n}，j）$，带有几乎Hermitian公制$ g $。数字$ h^{p，q} _d $，即，当$ 2n = 4 $时，hodge-de rham谐波$（p，q）$（p，q）$（p，q）$的复杂维度几乎是kähler不变的。在本文中，我们研究了$ h^{p，q} _j $和$ h^{p，q} _d $ in Dimension $ 2N \ ge4 $之间的关系。我们证明$ h^{n，0} _j = 0 $如果$ j $是不可集成的，并证明$ h^{p，0} _d $几乎是kähler不变的。如果$ m^{2n} $是完全可解决的谎言组的紧凑商，而$（j，g，ω）$不变，我们也会在$ h^{1,1} _d $上找到信息。最后，我们研究$ \ MATHCAL {C}^\ infty $ -pure和$ \ Mathcal {C}^\ infty $ -full $ j $的$ j $ on $ n $ - 特殊尺寸$ 2N = 4M $。
Let $(M^{2n},J)$ be a compact almost complex manifold. The almost complex invariant $h^{p,q}_J$ is defined as the complex dimension of the cohomology space $\left\{\left[α\right]\in H^{p+q}_{dR}(M^{2n};\mathbb{C}) \,\vert\,α\in A^{p,q}(M^{2n}),\, dα= 0 \right\}$. When $2n=4$, it has many interesting properties. Endow $(M^{2n},J)$ with an almost Hermitian metric $g$. The number $h^{p,q}_d$, i.e., the complex dimension of the space of Hodge-de Rham harmonic $(p,q)$-forms, is almost Kähler invariant when $2n=4$. In this paper we study the relationship between $h^{p,q}_J$ and $h^{p,q}_d$ in dimension $2n\ge4$. We prove $h^{n,0}_J=0$ if $J$ is non integrable and show that $h^{p,0}_d$ is almost Kähler invariant. If $M^{2n}$ is a compact quotient of a completely solvable Lie group and $(J,g,ω)$ is left invariant, we find information also on $h^{1,1}_d$. Finally we study the $\mathcal{C}^\infty$-pure and $\mathcal{C}^\infty$-full properties of $J$ on $n$-forms for the special dimension $2n=4m$.
