Banach非交换几何中的紧Kähler流形上的Dolbeault-Dirac算子
Dolbeault-Dirac operators on compact Kähler manifolds in Banach noncommutative geometry
AI总结：
本文研究了紧Kähler流形上带有Hermite全纯向量丛系数的Dolbeault-Dirac算子的L^p理论，建立了闭L^p实化算子的双侧性，证明了其具有有界H^∞函数分析，获得了Gaffney型估计和L^p-Hodge分解，并证明了该算子生成紧Banach谱三元组，其索引与全纯欧拉特征数一致且与p无关。
摘要：
我们发展了紧Kähler流形上带有Hermite全纯向量丛E系数的Dolbeault-Dirac算子的L^p理论。对于每个p∈(1,∞)，我们考虑Dolbeault-Dirac算子D_E在Banach空间L^p(Ω^{0,•}(M,E))上的闭L^p实化D_{E,p}。我们证明D_{E,p}是双侧的，并且具有有界H^∞函数分析。我们建立了Gaffney型估计，控制协变导数在L^p中的行为，并获得了L^p-Hodge分解。作为应用，我们证明闭算子D_{E,p}生成紧Banach谱三元组，并且识别所关联的Fredholm算子的索引与全纯欧拉特征数一致，特别地，证明其与p无关。本文在复几何、L^p分析和Banach非交换几何之间建立了联系，超越了Hilbert空间设定。
英文摘要：
We develop an $\mathrm{L}^p$-theory for Dolbeault-Dirac operators on compact Kähler manifolds with coefficients in a Hermitian holomorphic vector bundle $E$. For each $p \in (1,\infty)$ we consider the closed $\mathrm{L}^p$-realization $\mathcal{D}_{E,p}$ of the Dolbeault-Dirac operator $\mathcal{D}_{E}$ on the Banach space $\mathrm{L}^p(Ω^{0,\bullet}(M,E))$. We prove that $\mathcal{D}_{E,p}$ is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus. We establish a Gaffney-type estimate controlling covariant derivatives in $\mathrm{L}^p$, and also obtain $\mathrm{L}^p$-Hodge decompositions. As an application, we show that the closed operator $\mathcal{D}_{E,p}$ yields a compact Banach spectral triple, and we identify the index of the associated Fredholm operator with the holomorphic Euler characteristic, proving in particular that it is independent of $p$. This work initiates a connection between complex geometry, $\mathrm{L}^p$-analysis and Banach noncommutative geometry, beyond the Hilbert space setting.
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