By Voisin C.
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Additional resources for Hodge theory and complex algebraic geometry 1
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.
Hodge theory and complex algebraic geometry 1 by Voisin C.
Categories: Algebraic Geometry