Algebraic Solutions of Differential Equations
69
According to the "primitive decomposition" ([43], pp. 7779), for n < N
the mapping
(4.3.1.7)
t~
Prim.2if, n(z)(_i)
~L,
,R"f,"(Z)
0 < i < [n/21
is an
isogeny.
Thus it remains to polarize Prim"f,"(Z) for
n<N.
Ac
cording to the Hodge Index Theorem (cf. [43]), the pairing
Prim"f," (Z) x Prim"f," (Z) * R 2 nf,, (Z) ~ Z ( N)
(4.3.1.8)
(x,y)~xwLN"y,
w = cupproduct
is
a polarization. This concludes the proof of (4.3.1.6) and of (4.3.1).
(4.3.2)
Interpretation. When S=Spec(C), the mixed Hodge structure
(4.3.1) on H"(U an, Z) depends only on U, and
not
on the compactification
U~,X
chosen to represent U as the complement in a proper smooth
variety
X/C
of a divisor with normal crossings. This mixed Hodge
structure is
fimctorial
in U, in the sense that if h: U ~ V is a morphism
of smooth Cschemes, the induced morphisms h*: H"( V an , Z) ~ H"( U a", Z)
are morphisms of mixed Hodge structures. (4.3.2.1) By Hironaka [18],
any
quasiprojective smooth Cscheme U may be compactified as above,
and thus its integral cohomology
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 Fall '11
 NormanKatz
 Algebraic geometry, Hodge, hodge theory, Hodge structure, mixed Hodge structures

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