Download PDF by Voisin C.: Hodge theory and complex algebraic geometry 1

By Voisin C.

ISBN-10: 0521802601

ISBN-13: 9780521802604

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Let U be an open set of X such that there exists a non-zero section χ of E over U, and a submersive map φ : U → Rn−1 whose fibres are the trajectories of χ . By the flow-box theorem, we may assume that U is diffeomorphic to V × (0, 1) where V is an open set of Rn−1 , that φ is the first projection and that χ is identified with ∂t∂ . We will show the following result. 21 The integrability condition implies that there exists a distribution F of rank k − 1 on V such that E = (φ∗ )−1 (F). Moreover, E satisfies the integrability condition if and only if F does.

5 at every point of U . 7 If f is holomorphic and does not vanish on U , then 1f is holomorphic. Similarly, if f, g are holomorphic, f g and f + g and g ◦ f (when g is defined on the image of f ) are all holomorphic. Proof The map z → 1z is holomorphic on C∗ , so that the first assertion follows from the last one. Furthermore, if g and f are C 1 and g is defined on the image of f , then g ◦ f is C 1 and we have d(g ◦ f )u = dg f (u) ◦ d f u . 24 1 Holomorphic Functions of Many Variables If dg f (u) and d f u are both C-linear for the natural identifications of TC,u , TC, f (u) and TC,g◦ f (u) with C, then d(g ◦ f )u is also C-linear, and the last assertion is proved.

E. a holomorphic vector subbundle of rank k of the holomorphic tangent bundle TX . Then E is integrable in the holomorphic sense if and only if we have the integrability condition [E, E] ⊂ E. Here, the integrability in the holomorphic sense means that X is covered by open sets U such that there exists a holomorphic submersive map φU : U → Cn−k satisfying E u = Ker φ∗ : TU,u → TCn−k ,φ(u) for every u ∈ U . Proof We first reduce to the real Frobenius theorem, by noting that the conditions that E is holomorphic and that [E, E] ⊂ E imply that the real distribution E ⊂ TX,R also satisfies the Frobenius integrability condition, and thus is integrable.

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Hodge theory and complex algebraic geometry 1 by Voisin C.

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Categories: Algebraic Geometry