Spin(7)流形的$dd^Φ$-引理与Bott–Chern型上同调
A $dd^Φ$-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds
中文总结 AI 辅助
该文研究8维无挠Spin(7)-流形上的$dd^Φ$-算子，证明其Hodge分解与类似Kähler几何的$dd^Φ$-引理，定义Bott–Chern型上同调并建立其与模空间、校准几何的关联，同时给出具有独立价值的Spin(7)结构相关恒等式。
中文摘要 AI 辅助
我们研究了具有无挠Spin(7)-结构$Φ$的8维Spin(7)-流形上$dd^Φ$-算子的性质。这类算子由Harvey和Lawson首次提出（《校准几何中的位势理论导论》，Am.J. Math. 131.4 (2009)，arXiv:0710.3920）。我们证明了$dd^Φ$-算子的Hodge分解定理，并得到了Kähler几何中$\bar{\bar{}}$-引理的类似结果。基于此，我们定义了Spin(7)-流形的Bott–Chern型上同调。我们将Bott–Chern型上同调空间与无挠Spin(7)-结构的模空间以及Spin(7)-流形的校准几何联系起来。这些联系自然引出了Cayley正锥的概念。在证明结果的过程中，我们陈述并证明了外微分及其分解为不可约Spin(7)-表示的各类恒等式，以及二阶导数和拉普拉斯算子的恒等式。我们证明的恒等式适用于所有Spin(7)-结构，其中针对无挠结构的特殊形式是Kähler恒等式以及$\text{G}_2$情形下Bryant–Harvey恒等式的Spin(7)类似物（R. Bryant，《关于$\text{G}_2$-结构的若干注记》，第11、12届Gökova几何拓扑会议论文集，arXiv:math/0305124），这些恒等式本身也具有独立的研究价值。
英文摘要
We study the properties of the $dd^Φ$-operator on $8$-dimensional Spin(7)-manifolds with torsion-free Spin(7)-structures $Φ$. These operators were first introduced by Harvey and Lawson (An introduction to potential theory in calibrated geometry. Am.J. Math. 131.4 (2009), arXiv:0710.3920). We prove a Hodge decomposition theorem for the $dd^Φ$-operator and obtain an analogue of the $\partial \bar{\partial}$-lemma in Kähler geometry. Using this, we define Bott--Chern-type cohomologies for Spin(7)-manifolds. We relate the Bott-Chern-type cohomology spaces to the moduli space of torsion-free Spin(7)-structures and calibrated geometry of Spin(7)-manifolds. These relations naturally give rise to the notion of Cayley-positive cones. In the course of proving the results, we state and prove various identities for the exterior derivative and its decompositions into irreducible Spin(7)-representations as well as identities for second order derivatives and Laplacians. The identities we prove are for any Spin(7)-structures and the specialized torsion-free ones are Spin(7)-analogoues of Kähler identities and Bryant--Harvey's identities in the $\mathrm{G}_2$-case (R. Bryant, Some remarks on $\mathrm{G}_2$-structures, Proceedings of the 11th and 12th Gökova geometry-topology conference, arXiv:math/0305124) and are results of independent interest.
