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Hodge Theory and Complex Algebraic Geometry I by Claire Voisin, Leila Schneps

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8

Holomorphic de Rham Complexes and Spectral Sequences

In this chapter, we develop the results obtained in the preceding chapters in various directions. First of all, we propose another definition of the Hodge filtration on the cohomology of a compact Kähler manifold, which makes it possible to generalise this filtration to the cohomology of a complex manifold without the Kähler hypothesis. For this, we introduce the holomorphic de Rham complex , and we show (via the holomorphic Poincaré lemma) that this complex is a resolution of the constant sheaf. This resolution is not a resolution by acyclic sheaves, but by sheaves of free -modules. We ...

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