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Algebraic cycles and Hodge theoretic connectivity. (English) Zbl 0822.14008
We prove the following generalization of a basic result due to P. A. Griffiths [Ann. Math., II. Ser. 90, 496-541 (1969; Zbl 0215.081)]:
Theorem 1. Let $$X \subset \mathbb{P}^ m$$ be a smooth projective variety, defined over the field of complex numbers. Let $$D_ 1$$, $$D_ 2, \dots, D_ h$$ be general hypersurfaces of sufficiently large degrees. Let $$Y = X \cap D_ 1 \cap \cdots \cap D_ h$$. Assume that $$\dim X = n + h$$ so that $$Y$$ is smooth and $$\dim Y = n$$. Let $$\xi$$ be a codimension $$d$$ algebraic cycle on $$X$$, whose cohomology class in $$H^{2d} (X; \mathbb{Q})$$ is nonzero. Assume also that $$d < n$$. Then the restriction of $$\xi$$ to $$Y$$ is not algebraically equivalent to zero. For $$h = 1$$, this result is due to Griffiths (loc. cit.) using which he provided the first examples of homologically trivial cycles that are not algebraically equivalent to zero. In general, there is an increasing filtration $0 \subset A_ 0 \text{CH}^ d(Y) \subset A_ 1 \text{CH}^ d (Y) \subset \cdots \subset \text{CH}^ d (Y)$ where $$\text{CH}^ d (Y)$$ denotes the Chow group of codimension $$d$$ algebraic cycles on a smooth projective variety $$Y$$, modulo rational equivalence. Roughly speaking, $$A_ r \text{CH}^ d (Y)$$ consists of those cycles induced by correspondence from homologically trivial cycles of dimension $$\leq r$$ on other varieties. If $$I_ r^{2d - 1} (Y)$$ denotes the intermediate Jacobian of the largest integral Hodge structure contained in $$F^{d - r - 1} H^{2d - 1} (Y)$$, then it is easy to see that $$\theta_ d (\eta) \in I_ r^{2d - 1} (Y)$$, where $$F^ k$$ denotes the $$k$$-th level of Hodge filtration.
Theorem 2. Let $$X \subset \mathbb{P}^ m$$ be smooth, projective, and let $$Y$$ be the intersection of $$X$$ with $$h$$ general hypersurfaces of sufficiently large degrees. Let $$\xi \in \text{CH}^ d (X)$$ and put $$\eta = \xi | Y$$. Assume $$r + d <$$ dimension of $$Y$$. If $$\eta \in A_ r \text{CH}^ d (Y) \otimes \mathbb{Q}$$, then
(a) The cohomology class of $$\xi$$ vanishes in $$H^{2d} (X; \mathbb{Q})$$, and
(b) the Abel-Jacobi invariant of a nonzero multiple of $$\xi$$ belongs to $$I_ r^{2d - 1} (X)$$.

MSC:
 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14C25 Algebraic cycles 14D07 Variation of Hodge structures (algebro-geometric aspects)
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References:
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