By Demailly J.-P.
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Additional info for Applications of the theory of L2 estimates and positive currents in algebraic geometry
19) Lemma. In Cn , we have [d′′⋆ , L] = i d′ . Proof. 17, we get [d′′⋆ , L]u = d′′⋆ (ω ∧ u) − ω ∧ d′′⋆ u ∂ ∂ (ω ∧ u) + ω ∧ =− ∂z k ∂zk k Since ω has constant coefficients, we have [d′′⋆ , L] u = − k =− k by the derivation property of ∂ ∂z k ω= ∂ ∂z k k ∂u . ∂zk ∂ ∂u (ω ∧ u) = ω ∧ and therefore ∂zk ∂zk ∂ ∂z k ω∧ ∂u ∂zk ∂ ∂z k ω ∧ ∂u ∂zk −ω∧ ∂ ∂z k ∂u ∂zk . Clearly ∂ ∂z k dzj ∧ dz j = −i dzk , i 1≤j≤n hence [d′′⋆ , L] u = i dzk ∧ k ∂u = i d′ u. ∂zk The final step is to extend these results to an arbitrary K¨ahler manifold (X, ω) and an arbitrary hermitian vector bundle (F, h).
E. is also a symplectic 2-form. 15) Examples. n • C equipped with its canonical metric ω = i 1≤j≤n dzj ∧ dz j (or any other hermitian metric ω = i 1≤j,k≤n ωjk dzj ∧ dz k with constant coefficients) is K¨ahler. n • A complex torus is a quotient X = C /Γ by a lattice (closed discrete subgroup) Γ of rank 2n. Then X is a compact complex manifold. Any positive definite hermitian form ω = i ωjk dzj ∧ dz k with constant coefficients defines a K¨ahler metric on X. n • The complex projective space P is K¨ ahler.
24) now follows from the Hodge isomorphisms for De Rham and Dolbeault groups. The decomposition is canonical since H p,q (X) coincides with the set of classes in H k (X, C) which can be represented by d-closed (p, q)-forms. 5. 25) Remark. The decomposition formula shows that the so called Betti numbers k bk := dim HDR (X, C) satisfy the relation hp,q bk = p+q=k in terms of the Hodge numbers hp,q := dim H p,q (X, C). In addition to this, the complex conjugation u → u takes (p, q)-harmonic forms to (q, p)-harmonic forms, hence there is a canonical conjugate linear isomorphism H q,p (X, C) ≃ H p,q (X, C) and hq,p = hp,q .