Dolbeault-Dirac operators on compact K\"ahler manifolds in Banach noncommutative geometry

TL;DR

This paper develops an $L^p$-theory for Dolbeault-Dirac operators on compact K"ahler manifolds, establishing functional calculus, Hodge decompositions, and linking to Banach noncommutative geometry.

Contribution

It introduces a novel $L^p$-framework for Dolbeault-Dirac operators, extending analysis and geometry beyond the Hilbert space setting.

Findings

Proves bisectoriality and bounded $H^$ calculus of $_{E,p}$

Establishes $L^p$-Hodge decompositions and Gaffney estimates

Identifies the index with the holomorphic Euler characteristic, independent of $p$

Abstract

We develop an $\mathrm{L}^p$-theory for Dolbeault-Dirac operators on compact K\"ahler 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 $\mathrm{L}^p(\Omega^{0,\bullet}(M,E))$. 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.