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

Cédric Arhancet

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 $\mathrm{L}^p(Ω^{0,\bullet}(M,E))$. 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.

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

Abstract

We develop an -theory for Dolbeault-Dirac operators on compact Kähler manifolds with coefficients in a Hermitian holomorphic vector bundle . For each we consider the closed -realization of the Dolbeault-Dirac operator on the Banach space . We prove that is bisectorial and admits a bounded functional calculus. We establish a Gaffney-type estimate controlling covariant derivatives in , and also obtain -Hodge decompositions. As an application, we show that the closed operator 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 . This work initiates a connection between complex geometry, -analysis and Banach noncommutative geometry, beyond the Hilbert space setting.
Paper Structure (39 sections, 36 theorems, 280 equations, 1 figure)

This paper contains 39 sections, 36 theorems, 280 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Preliminaries on operator theory and functional calculus
  4. Sectorial operators
  5. Sectorial operators
  6. Bisectorial operators
  7. Functional calculus
  8. Complex geometry and analysis
  9. Background and notations
  10. $\mathrm{L}^p$-spaces of sections
  11. Complex geometry
  12. Closability of the Dolbeault-Dirac operator on the Banach space $\mathrm{L}^p(\Omega^{0,\bullet}(M,E))$
  13. Canonical grading by form degree
  14. $\mathrm{H}^\infty$ calculus of the Dolbeault-Dirac operator on a compact Kähler manifold
  15. Some Sobolev spaces and actions of pseudo-differential operators
  16. ...and 24 more sections

Key Result

Lemma 3.1

Let $E$ be a holomorphic vector bundle over a compact complex manifold $M$. We have

Figures (1)

  • Figure :

Theorems & Definitions (44)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 4.1
  • Proposition 4.2
  • Theorem 4.3
  • ...and 34 more