By Claire Voisin, Leila Schneps
It is a smooth advent to Kaehlerian geometry and Hodge constitution. assurance starts with variables, complicated manifolds, holomorphic vector bundles, sheaves and cohomology conception (with the latter being taken care of in a extra theoretical approach than is common in geometry). The ebook culminates with the Hodge decomposition theorem. In among, the writer proves the Kaehler identities, which ends up in the challenging Lefschetz theorem and the Hodge index theorem. the second one a part of the e-book investigates the which means of those ends up in numerous instructions.
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Extra info for Hodge Theory and Complex Algebraic Geometry I
I): the J1l(E) of global holomorphic differentials on E. )) ~ a: • (the lattice of "integral" holomorphic differential forms) which can be identified with L ~ a: under the preceding isomorphism. ", Z because L has no torsion, acts freely on the contractible space a: with quotient precisely E : Hurewitz Theorem). This group can also be interpreted as space of Z-bilinear alternate integral forms on L. B· ~ 7l The space HI(E,~~) is the most interesting one. It can be computed by taking a covering of E with open discs U.
B· ~ 7l The space HI(E,~~) is the most interesting one. It can be computed by taking a covering of E with open discs U. in ([ (mod L), each U~ being sufficiently small to avoid pairs z, z+ 4J (w E L - (OJ). Since all the Uo( and their finite intersections U ~ ... r(U"•... ef') ). p ) CoC, (3) ~ : (d,P) In other words, (o(,p Soft ~s = -1 '1) ~ s~1 so(l so(~ = = se(t1" s(31 f ¢ of the open intersecting discs U« . e. nu, · (when this triple intersection uo(~y is non-empty). ~ defined on the non-empty .
These line bundles would be the ~-powers of the canonical one (cotangent bundle), so that a modular form of u(-J. 7) expresses the degree of their divisors. 7): By hypothesis, n. (f) zero nor pole when Im(z» > - co , so that f has no Rand R is large enough. Let DR be the intersection of the fundamental domain D with is to integrate f'lf on the boundary ~DR Im(z)~ R. The idea of DR. In case f has zeros or poles on the sides, f'lf will have simple poles on the sides, and the usual modification of the contour has to be made.
Hodge Theory and Complex Algebraic Geometry I by Claire Voisin, Leila Schneps
Categories: Algebraic Geometry