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

We establish an $\mathrm{L}^p$-index theorem for Dolbeault--Dirac operators on compact K\"ahler 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 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.