By Claire Voisin, Leila Schneps

ISBN-10: 0521802601

ISBN-13: 9780521802604

This can be a smooth advent to Kaehlerian geometry and Hodge constitution. insurance starts with variables, complicated manifolds, holomorphic vector bundles, sheaves and cohomology conception (with the latter being handled in a extra theoretical means than is common in geometry). The booklet culminates with the Hodge decomposition theorem. In among, the writer proves the Kaehler identities, which results in the tough Lefschetz theorem and the Hodge index theorem. the second one a part of the e-book investigates the that means of those leads to numerous instructions.

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**Sample text**

D xk . e. of local orientationpreserving coordinates. e. if we compose all the charts with local diffeomorphisms of Rk with negative Jacobians, the integrals ∗ M φ α change sign. This follows from the change of variables formula for multiple integrals, which uses only the absolute value of the Jacobian, whereas the change of variables formula for differential forms of maximal degree uses the Jacobian itself. Suppose now that α is a C 1 (k − 1)-form on U . Then, as φ|∂ M is differentiable and ∂ M is a compact oriented manifold of dimension k − 1, we can compute the integral ∂ M φ ∗ α.

As a real vector bundle, TX1,0 is naturally isomorphic to TX,R via the application (real part) which to a complex ﬁeld u +iv associates its real part u. Moreover, this identiﬁcation identiﬁes the operators i on TX1,0 and I on TX,R . Clearly TX1,0 is generated by the u − i I u, u ∈ TX,R . Furthermore, in the case where X = Cn , consider the isomorphism TCn ,R ∼ = n C ×R2n given by the sections ∂∂x j , ∂∂y j of the tangent bundle of Cn , where z k = xk + i yk and the z k are complex linear coordinates on Cn .

N )| |ζ1 − z 1| ≤ 1, |z i | ≤ ri , i ≥ 2} is contained in D − {ζ1 = 0}, so that Cauchy’s formula gives f (z) = 1 2iπ n f (ζ ) ∂D 1 dζ1 dζn ∧ ··· ∧ , ζ1 − z 1 ζn − z n where ∂ D 1 := {(ζ1 , . . , ζn )| |ζ1 − z 1| = 1, |ζi | = ri , i ≥ 2}. 2 Holomorphic functions of several variables 33 Consider, also, the product of circles ∂ D := {(ζ1 , . . , ζn )| |ζ1| = , |ζi | = ri , i ≥ 2}. is sufﬁciently small, ∂ D − ∂ D 1 − ∂ D is the boundary of the Then when manifold M = {(ζ1 , . . , ζn )| |ζ1 − z 1| ≥ 1, |ζ1| ≥ , |ζi | = ri , i ≥ 2}, which is contained in D and intersects neither the hypersurface {ζ1 = 0} nor the hypersurfaces {ζi = z i }.

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

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