Smooth Varieties
The main goal of this chapter is to show that there exists a functorial mixed Hodge structure on any of the cohomology groups of a smooth variety and which coincides with the classical Hodge structure if the variety is smooth and projective. To define thi
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The main goal of this chapter is to show that there exists a functorial mixed Hodge structure on any of the cohomology groups of a smooth variety and which coincides with the classical Hodge structure if the variety is smooth and projective. To define this mixed Hodge structure, we first compactify the variety by a divisor whose singularities locally look like the crossing of coordinate hyperplanes. In § 4.1 we study the cohomology with respect to this compactification and we shall show in § 4.1–4.3 how to put weight and Hodge filtrations on the cohomology groups defining a mixed Hodge structure. The rational component of the Hodge De Rham complex which gives this Hodge structure can be given using so-called log structures which are treated in § 4.4 and which will be used in a decisive way in Chapter 11. In § 4.5 we check that the mixed Hodge structure does not depend on our chosen compactification and that the construction is functorial. We also prove the theorem on the fixed part and show that for projective families over a smooth curve the Leray spectral sequence degenerates at E2 .
4.1 Main Result Let U be a smooth complex algebraic variety. By [Naga] U is Zariski open in some compact algebraic variety X, which by [Hir64] one can assume to be smooth and for which D = X − U locally looks like the crossing of coordinate hyperplanes. It is called a normal crossing divisor. If the irreducible components Dk of D are smooth, we say that D has simple or strict normal crossings. Definition 4.1. We say that X is a good compactification of U = X − D if X is smooth and D is a simple normal crossing divisor. We return for the moment to the situation where D ⊂ X is a hypersurface (possibly with singularities and reducible) inside a smooth n-dimensional complex manifold X and as above, we set j : U = X − D ,→ X
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4 Smooth Varieties
A holomorphic differential form ω on U is said to have logarithmic poles along D if ω and dω have at most a pole of order one along D. It follows that • these holomorphic differential forms constitute a subcomplex ΩX (log D) ⊂ • j∗ ΩU , the logarithmic de Rham complex Suppose now that D has simple normal crossings, p ∈ D and V ⊂ X is an open neighbourhood with coordinates (z1 , . . . , zn ) in which D has equation z1 · · · zk = 0. On can show [Grif-Ha, p. 449] 1 ΩX (log D)p = OX,p p ΩX (log D)p =
p ^
dz1 dzk ⊕ · · · OX,p ⊕ OX,p dzk+1 ⊕ · · · ⊕ OX,p dzn , z1 zk
1 ΩX (log D)p .
An essential ingredient in the proof of the following theorem is the residue map which is defined as follows. We set Dk = {zk = 0} and we let D0 be the divisor on Dk traced out by D. Then writing ω = η ∧ (dzk /zk ) + η 0 with η, η 0 not containing dzk , the residue map can be defined as p p−1 0 res : ΩX (log D) → ΩDk (log D ) ω 7→ η D . k
1 As a special case we have the Poincar´e residues Rk : ΩX (log D) → ODk which we shall use in § 11.1.1. As an aside, in § 4.2 we iterate this procedure to get residues for multiple intersections. We can now formulate the main result of this chapter:
Theorem 4.2. Let U be a compl
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