Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Motivic decompositions of families with Tate fibers: smooth and singular
  cases   We apply Wildeshaus's theory of motivic intermediate extensions to the
motivic decomposition conjecture, formulated by Deninger-Murre and
Corti-Hanamura. We first obtain a general motivic decomposition for the Chow
motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose
geometric fibers are Tate. Using Voevodsky's motives with rational
coefficients, the formula is valid for an arbitrary regular base $S$, without
assuming the existence of a base field or even of a prime integer $\ell$
invertible on $S$. This result, and some of Bondarko' ideas, lead us to a
generalized formulation of Corti-Hanamura's conjecture. Secondly we establish
the existence of the motivic decomposition when $f:X \rightarrow S$ is a
projective quadric bundle over a characteristic $0$ base, which is either
sufficiently general or whose discriminant locus is a normal crossing divisor.
This provides a motivic lift of the Bernstein-Beilinson-Deligne decomposition
in this setting.

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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci