Abstract
This paper studies monodromy-invariant cycles in the cohomology of fibers of a family of algebraic varieties. It is shown that the localization of invariant cycles in a neighborhood of a degeneration of the family is a morphism of Hodge structures. An application of this result is the geometric analogue of the Mumford-Tate conjecture for families with strong degenerations. A large class of nonconstant abelian schemes for which the geometric analogue of the Mumford-Tate conjecture holds is constructed. Bibliography: 29 titles.
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