Applications of the theory of L2 estimates and positive by Demailly J.-P.

By Demailly J.-P.

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Now, J is generated by any Hilbert basis of H (Ω, ϕ), because it is wellknown that the space of sections of any coherent sheaf is a Fr´echet space, therefore closed under local L2 convergence. The multiplier ideal sheaves satisfy the following basic functoriality property with respect to direct images of sheaves by modifications. 8) Proposition. e. a proper generically 1:1 holomorphic map), and let ϕ be a psh function on X. Then µ⋆ O(KX ′ ) ⊗ I(ϕ ◦ µ) = O(KX ) ⊗ I(ϕ). Proof. Let n = dim X = dim X ′ and let S ⊂ X be an analytic set such that µ : X ′ S ′ → X S is a biholomorphism.

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 . In particular, if X is compact K¨ahler, the Betti numbers bk of odd index are even. The Serre duality theorem gives the further relation H n−p,n−q (X, C) ≃ H p,q (X, C)∗ (actually, this is even true for an arbitrary compact complex manifold X). This observation implies the existence of compact complex manifolds X which are non K¨ahler (hence non projective) : for instance the Hopf surface defined by X = C2 {0}/Γ , where Γ ≃ Z is the discrete group generated by a contraction z → λz, 0 < λ < 1, is easily seen to be diffeomorphic to S 1 × S 3 .

One can choose local coordinates (x1 , . . , xn ) such that (dx1 , . . , dxn ) is an ω-orthonormal basis of Tx⋆0 X. 16 d) ωlm = δlm + O(|x|) = δlm + (ajlm xj + a′jlm xj ) + O(|x|2 ). 1≤j≤n Since ω is real, we have a′jlm = ajml ; on the other hand the K¨ahler condition ∂ωlm /∂xj = ∂ωjm /∂xl at x0 implies ajlm = aljm . Set now 4. K¨ ahler identities and Hodge Theory zm = xm + 1 2 ajlm xj xl , 33 1 ≤ m ≤ n. j,l Then (zm ) is a coordinate system at x0 , and dzm = dxm + ajlm xj dxl , j,l dzm ∧ dz m = i i dxm ∧ dxm + i m m ajlm xj dxl ∧ dxm j,l,m ajlm xj dxm ∧ dxl + O(|x|2 ) +i j,l,m ωlm dxl ∧ dxm + O(|x|2 ) = ω + O(|z|2 ).

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