Dolbeault-Dirac operators on compact Kähler manifolds in Banach noncommutative geometry

Abstract

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 ${\mathrm{L}}^p(\hat{I}{\text{c◯}}^{0,\bullet }(M,E))$. 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.

This paper has been substantially revised and merged into "Curvature, Dolbeault-Dirac operators, and an $\mathring{A}{}^p$ index theorem on compact Kähler manifolds", arXiv:2401.04203