1
$\begingroup$

I was trying to find examples of schemes (preferably smooth) over $\mathbb{C}$ which have motives that aren't mixed Tate. I wasn't able to come up with anything or find an argument that's been written down. Does anyone have an example or a reference for this?

Thanks in advance!

Improve this question
$\endgroup$
4
  • 7
    $\begingroup$ Elliptic curve? Smooth genus g curve, g>0? $\endgroup$
    – rvk
    Commented Oct 7, 2021 at 1:29
  • 1
    $\begingroup$ What's the reason for this? $\endgroup$
    – Arpith
    Commented Oct 7, 2021 at 1:33
  • 4
    $\begingroup$ Just look at the Hodge structure. Or look at them in the Grothendieck group of motives (equivalently look at the Hodge E-polynomial). $\endgroup$
    – rvk
    Commented Oct 7, 2021 at 1:36
  • 1
    $\begingroup$ rvk is right. To say that the motive of some variety is mixed Tate is a very strong condition. So if you pick a variety "at random", it probably won't satisfy this. Concretely, in higher dimensions, you can take smooth hypersurface in $\mathbb{P}^n$ of degree $n+1$ or more. (Look at the Hodge numbers.) $\endgroup$
    – Donu Arapura
    Commented Oct 7, 2021 at 15:23

0

Reset to default

You must log in to answer this question.