高阶页雅可比与阿尔巴内塞环面
Higher-Page Jacobian and Albanese Tori
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中文总结 AI 辅助
本文通过霍奇理论为紧致复流形构造高阶页雅可比环面、阿尔巴内塞环面及映射，并应用于结构定理、代数维数恒等式及六维球面的复结构研究，证明其无强高登度量。
中文摘要 AI 辅助
我们通过霍奇理论方法，为任意紧致复流形构造了所谓的$E_r$-雅可比环面、$E_r$-阿尔巴内塞环面及$E_r$-阿尔巴内塞映射，该流形要么是{\it page-(r-1)-$\partial\bar\partial$}型，要么是{\it $E_r$-sGG}型。这两类流形（前者包含于后者）分别由两位作者与J. Stelzig共同引入，以及由第一作者单独引入。我们还为后一类流形发展了霍奇理论。随后，我们将结果应用于给出结构定理以及用$E_r$-阿尔巴内塞映射和环面表示的代数维数恒等式。其他应用则导致关于$6$维球面的上同调与度量定理，该球面要么配备假定的复结构，要么配备文献中最近声称存在的复结构。例如，我们证明在后一种情形下，$S^6$上不存在{\it 强高登}度量。
英文摘要
We construct, through Hodge-theoretical methods, what we call the $E_r$-Jacobian torus, the $E_r$-Albanese torus and the $E_r$-Albanese map of any compact complex manifold that is either {\it page-(r-1)-$\partial\bar\partial$} or {\it $E_r$-sGG}. These two classes of manifolds, the former of which is contained in the latter, have been introduced recently by both authors jointly with J. Stelzig, respectively by the first-named author. A Hodge theory is also developed for the latter class of manifolds. We then apply our results to give structure theorems and an identity of algebraic dimensions in terms of the $E_r$-Albanese map and torus. Other applications result in cohomological and metrical theorems for the $6$-dimensional sphere when it is equipped either with a hypothetical complex structure or with the complex structures very recently claimed to exist in the literature. For example, we show that in the latter case no {\it strongly Gauduchon} metric exists on $S^6$.
