Complex Analytic and Differential Geometry 2025, 17 - Hodge Theory

A linear diff operator with degree δ from E to F is 𝕂-linear, P: C^∞(M, E) → C^∞(M, F), u → Pu, Pu(x) = Σ_|α|≤δ a_α(x)D^αu(x) with E_↑Ω ≅ Ω × 𝕂^r, F_↑Ω ≅ Ω × 𝕂^r' locally trivialized on an open chart equipped with local coords, and where a(x) are r' × r matrices with C^∞ coefficients on the chart. If P is a diff operator and E, F are euclidean or hermitian there is a unique diff operator adjoint to P P*: C^∞(M, F) → C^∞(M, E). A diff. operator is elliptic if σ_P(x, ξ) ∈ Hom(E_x, F_x) is injective for all x ∈ M and ξ ∈ T*_{M, x} \ {0}.

Define the Sobolev space W^k(M, F) of sections s: M → F with derivatives up to order k in L². For k > l + m/2, W^k(M, F) ⊂ C^l(M, F). For all integers k the k+1 → k inclusion of Sobolev spaces are compact linear operators. An extension P' of P to sections with distribution coefficients, any u ∈ W⁰(M, F): P'u ∈ W^k(M, F), then u ∈ W^k+d(M, F), and the k+d-norm of u is upper-bound to a positive constant depending only on k. The dimension of ker P is finite, and P(C^∞(M, F)) is closed and of finite codimension. If P* is the formal adjoint of P, the decomposition C^∞(M, F) = P(C^∞(M, F)) ⊕ ker P* as an orthogonal direct sum in W⁰(M, F) = L²(M, F) exists.

Use ☆ as the Hodge-star operator. Refer to the String Theory stuff for definition. d^☆ = (-1)^mp+1☆d☆ is the formal adjoint of the exterior derivative d acting on C^∞(M, Λ^pT^☆_M⊗E). Δ = dd^☆ + d^☆d is the Laplace-Beltrami operator of M. The one associated to D_E is the second order Δ_E = D_ED_E^☆ + D_E^☆D_E. It's self-adjoint and elliptic. For all p ∃ C^∞(M,Λ^pT^☆_M⊗E) = H^p(M, E) ⊕ Im D_E ⊕ Im D_E^☆ Im D_E = D_E(C^∞(M,Λ^pT^☆_M⊗E) ) Im D_E^☆ = D_E^☆(C^∞(M,Λ^pT^☆_M⊗E) ) The De-Rahm cohomology group is finite-dim and isomorphic to H^p(M, E)