Curvature, Dolbeault-Dirac operators, and an $\mathrm{L}^p$ index theorem on compact Kähler manifolds
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914582851747840 |
|---|---|
| author | Arhancet, Cédric |
| author_facet | Arhancet, Cédric |
| contents | We establish an $\mathrm{L}^p$-index theorem for Dolbeault--Dirac operators on compact Kähler manifolds with coefficients in a Hermitian holomorphic vector bundle $E$. For every $p \in (1,\infty)$, we prove that the closed $\mathrm{L}^p$-realization $\mathcal{D}_{E,p}$ of the Dolbeault-Dirac operator is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on $\mathrm{L}^p(Ω^{0,\bullet}(M,E))$. We also show an $\mathrm{L}^p$-Gaffney-type estimate, obtain $\mathrm{L}^p$-Hodge decompositions, and prove that $\mathcal{D}_{E,p}$ gives rise to an even compact Banach spectral triple over the algebra $\mathrm{C}(M)$, graded by form parity. The index of the associated Fredholm operator is equal to the holomorphic Euler characteristic $χ(M,E)$. In particular, it is independent of $p$. A central tool is an abstract notion of Ricci curvature lower bound for strongly continuous semigroups on reflexive Banach spaces, formulated as a semigroup-level intertwining relation. Under this condition, together with natural Riesz equivalences and bounded $\mathrm{H}^\infty$ functional calculi for the relevant generators, the associated Hodge--Dirac operator is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus. The framework also applies to heat semigroups on Riemannian manifolds, $q$-Ornstein-Uhlenbeck semigroups and semigroups of Schur multipliers. This provides a unified Banach-space approach to curvature, functional calculus and index theory beyond the Hilbert space setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04203 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curvature, Dolbeault-Dirac operators, and an $\mathrm{L}^p$ index theorem on compact Kähler manifolds Arhancet, Cédric Functional Analysis Operator Algebras We establish an $\mathrm{L}^p$-index theorem for Dolbeault--Dirac operators on compact Kähler manifolds with coefficients in a Hermitian holomorphic vector bundle $E$. For every $p \in (1,\infty)$, we prove that the closed $\mathrm{L}^p$-realization $\mathcal{D}_{E,p}$ of the Dolbeault-Dirac operator is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on $\mathrm{L}^p(Ω^{0,\bullet}(M,E))$. We also show an $\mathrm{L}^p$-Gaffney-type estimate, obtain $\mathrm{L}^p$-Hodge decompositions, and prove that $\mathcal{D}_{E,p}$ gives rise to an even compact Banach spectral triple over the algebra $\mathrm{C}(M)$, graded by form parity. The index of the associated Fredholm operator is equal to the holomorphic Euler characteristic $χ(M,E)$. In particular, it is independent of $p$. A central tool is an abstract notion of Ricci curvature lower bound for strongly continuous semigroups on reflexive Banach spaces, formulated as a semigroup-level intertwining relation. Under this condition, together with natural Riesz equivalences and bounded $\mathrm{H}^\infty$ functional calculi for the relevant generators, the associated Hodge--Dirac operator is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus. The framework also applies to heat semigroups on Riemannian manifolds, $q$-Ornstein-Uhlenbeck semigroups and semigroups of Schur multipliers. This provides a unified Banach-space approach to curvature, functional calculus and index theory beyond the Hilbert space setting. |
| title | Curvature, Dolbeault-Dirac operators, and an $\mathrm{L}^p$ index theorem on compact Kähler manifolds |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2401.04203 |