26
N.M. Katz:
In order to explain the transcendental analogue of (2.3.3.6), let Y be
a a proper and smooth Cscheme, 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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 Fall '11
 NormanKatz
 Hilbert space, Group isomorphism, Hodge, automorphism, complex conjugate, transcendental analogue

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