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Mikhail Bondarko
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Motives and(and examples) of projective bundles over projective spaces

If a projective bundle $Y$ over a variety $X$ is obtained from a vector bundle $E/X$ then the cohomology and the motif of $Y$ is known to be closely related to that of $X$. Now, what can one say in the case where $Y$ doesn't come from an $X$-vector bundle and $X$ is just a projective space?

More generally, if $X$ is smooth projective, its integral Chow motif is a Tate one, and there is a fibration $Y\to X$ whose fibre (is smooth projective and) has a Tate motif, is the motif of $Y$ a Tate one in general?

Upd. I am interested in "positive" statements of this sort over fields that are (perfect but) not algebraically closed. So, I would start from the most restrictive notion of a bundle: $Y$ is Zariski-locally isomorphic to $P^n(X)$. If this question is "too easy", then what about $P^n\times P^m$-bundles? Do there exist any papers on bundles of this sort?

Motives and examples of projective bundles over projective spaces

If a projective bundle $Y$ over a variety $X$ is obtained from a vector bundle $E/X$ then the cohomology and the motif of $Y$ is known to be closely related to that of $X$. Now, what can one say in the case where $Y$ doesn't come from an $X$-vector bundle and $X$ is just a projective space?

More generally, if $X$ is smooth projective, its integral Chow motif is a Tate one, and there is a fibration $Y\to X$ whose fibre (is smooth projective and) has a Tate motif, is the motif of $Y$ a Tate one in general?

Motives (and examples) of projective bundles over projective spaces

If a projective bundle $Y$ over a variety $X$ is obtained from a vector bundle $E/X$ then the cohomology and the motif of $Y$ is known to be closely related to that of $X$. Now, what can one say in the case where $Y$ doesn't come from an $X$-vector bundle and $X$ is just a projective space?

More generally, if $X$ is smooth projective, its integral Chow motif is a Tate one, and there is a fibration $Y\to X$ whose fibre (is smooth projective and) has a Tate motif, is the motif of $Y$ a Tate one in general?

Upd. I am interested in "positive" statements of this sort over fields that are (perfect but) not algebraically closed. So, I would start from the most restrictive notion of a bundle: $Y$ is Zariski-locally isomorphic to $P^n(X)$. If this question is "too easy", then what about $P^n\times P^m$-bundles? Do there exist any papers on bundles of this sort?

Source Link
Mikhail Bondarko
  • 16.9k
  • 4
  • 34
  • 99

Motives and examples of projective bundles over projective spaces

If a projective bundle $Y$ over a variety $X$ is obtained from a vector bundle $E/X$ then the cohomology and the motif of $Y$ is known to be closely related to that of $X$. Now, what can one say in the case where $Y$ doesn't come from an $X$-vector bundle and $X$ is just a projective space?

More generally, if $X$ is smooth projective, its integral Chow motif is a Tate one, and there is a fibration $Y\to X$ whose fibre (is smooth projective and) has a Tate motif, is the motif of $Y$ a Tate one in general?