By Weil A.
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Additional info for Collected papers. Vol.2 (1951-1964)
Moreover, Z is supported in T × X with codim T ≥ c. Let l : T → X be a desingularization of T , and let Z in CHn (T ×X) be such that (l, Id)∗ (Z ) = Z . The above decomposition gives a decomposition of the corresponding morphisms of Hodge structures for every k: k k m[∆X ]∗ = m Id = [Z0 ]∗ + · · · + [Zk ]∗ + [Z ]∗ : HB (X, Q) → HB (X, Q). 17) j where , ∗ (X, Q). In particular, we have is the intersection form on HB [Zi ]∗ (α) = 0 ∀ α ∈ H p,q (X), p = q. For α satisfying this hypothesis, we thus have mα = [Z ]∗ (α) = l∗ ([Z ]∗ (α)) in H p,q (X).
This is the integral Betti cycle class of Z. In many places we will use the 2k rational cycle class [Z] ∈ HB (X, Q). We extend the above cycle class by linearity to any cycle Z = i ni Zi ∈ Z k (X). 16. If Z is rationally equivalent to 0, then [Z] = 0 in HB (X, Z). The map Z → [Z] thus gives the “cycle class” map 2n−2l cl : CHl (X) → HB (X, Z). The cycle class map is compatible with the intersection product · : CHl (X) × CHk (X) → CHk+l (X), weyllecturesformat September 3, 2013 6x9 REVIEW OF HODGE THEORY AND ALGEBRAIC CYCLES 23 and the cup-product 2k+2l 2l 2k ∪ : HB (X, Z) × HB (X, Z) → HB (X, Z).
This correspondence can be spread out over a finite cover of a Zariski open set of Y , and this can be done as well for the curve Cη and the cycle z, giving rise to families over a finite cover of a Zariski dense open set Y 0 of the base. The problem is that the spread-out curve C is not in general isotrivial on Y 0 , that is, isomorphic to C0 × Y 0 , maybe up to passing to a finite ´etale cover of Y 0 . Even if it was isotrivial, the codimension 1 cycle z would spread to a codimension 1 cycle Z on C0 × Y 0 , which might not be cohomologous to 0 over any Zariski open set of Y 0 , if g(C0 ) > 0.
Collected papers. Vol.2 (1951-1964) by Weil A.
Categories: Algebraic Geometry