Dr. Katz DEq Homework Solutions 26

Dr. Katz DEq Homework Solutions 26 - 26 N.M. Katz: I n...

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26 N.M. Katz: In order to explain the transcendental analogue of (2.3.3.6), let Y be a a proper and smooth C-scheme, and denote by Y"" the "underlying" complex manifold. By GAGA ([39, 36]) and Poincar6's lemma ([14]), we have isomorphisms (2.3.3.7) H"(Y,,O~,/c ) ~, H"(Y~",f2~/~), ~ H"(Y~",C), ~ H"(Y'~",Z)| by means of which any automorphism of the field C operates on H"(Y, O~,/c ) (by transporting by (2.2.3.6) its action on H"(Y a", Z)| through the second factor). In particular, the automorphism "complex conjugation ", denoted (2.3.3.8) F~bs: C ---} C, operates on H"(Y, I2~,/c), furnishing a canonical (albeit transcendental) isomorphism (2.3.3.9) Fa* s H" (Y, f2~/c) ~ H" (Y, f2~./c ) . The complex conjugate of the Hodge filtration, noted ,onF i, is by definition the image under (2.3.3.8) of the filtration F*s(U) of F*sH"(Y,O~,/c ). According to Hodge theory ([42, 5]), we have, for i= 1 ..... n, a direct sum decomposition (2.3.3.10) H"(Y, f2~,/c)=FiH"(Y, ,+1-, , Or/c) G Fdo, H (Y, Or/c) or, what is the same, a bigraduation (Hodge decomposition)
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