Complex Analytic and Differential Geometry 2025, 19 - Serre Duality

For a holomorphic hermitian vector bundle E of rank r over X, note D_E as the Chern connection of E, with formal adjoints and D', D'' the components of type (-1, 0), (0, -1). For bidegree (p, q) there is an orthogonal decomposition

C^∞(X, Λ^p,qT*_X ⊗ E) = H^p,q(X, E) ⊕ Im D''_E ⊕ Im D''_E*

where H^p,q is the space of Δ''_E-harmonic forms in C^∞(X, Λ^p,qT_X* ⊗ E). The subspace of d''-cocycles in C^∞(X, Λ^p,qT_X* ⊗ E) is H^p,q(X, E) ⊕ Im D''_E. The Dolbeault cohomology group H^p,q(X, E) is finite dim, and there is an isomorphism H^p,q(X, E) ≅ H^p,q(X, E). The bilinear pairing between H^p,q(X, E) × H^{n-p, n-q}(X, E*) → ℂ, (s, t) → ∫_Ms∧t is a non-degenerate duality.

H^p,q_BC(X, ℂ) = (C^∞(X, Λ^p,qT*_X) ∩ ker d) / d' d'' C^∞(X, Λ^{p-1, q-1} T*_X) is the Bott-Chern Cohomology Group of X. We gain

For a d-closed (p, q)-form u, d is d, d', d'' and d'd''-exact and orthogonal to H^p,q(X, ℂ). If the above isomorphism exists, then it's equivalent to the Hodge-Froelicher spectral sequence at E₁. It canonically defines the isomorphism and X is a Hodge decomposition. A bounded complex of finite-dim vector spaces over a field, then the Euler characteristic χ(C^○) = Σ(-1)^qdim C^q is equal to χ(H^○(C^○)) of its cohomology module. For any compact complex manifold X, χ_top(X) = Σ_0≤k≤2n(-1)^kb_k.