Curvature, Dolbeault-Dirac operators, and an ${\mathrm{L}}^p$-index theorem on compact Kähler manifolds

Abstract

We develop an ${\mathrm{L}}^p$-Banach noncommutative-geometric framework 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 ${\mathrm{L}}^p({\Omega }^{0,\bullet }(M,E))$. 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 $\chi (M,E)$. In particular, it is independent of $p$. A central tool is an abstract notion of Ricci curvature lower bound for strongly continuous semigroups on $\mathrm{UMD}$ 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, Riesz transforms and index theory beyond the Hilbert space setting.

85 pages, revision