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Let $A$ be an abelian variety over $\mathbb{C}$ and $A^*$ the dual Abelian variety. The class of $A$ and the class of $A^*$ in $\mathcal{M}_{\mathbb{C}}=K_0(Var_\mathbb{C})[\mathcal{L}^{- 1}]$ are same?

In general, is there a pair of varieties (just finite type and separated, integral over $\mathbb{C}$) $(X,Y)$, such that these bounded derived categories are equivalence, but these motivic integrations are not same?

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    $\begingroup$ By Larsen-Lunts, if two varieties have the same class in $K_0(\operatorname{Var}_{\mathbb{C}})/(\mathbb{L}) )$ they are stably birational, which for abelian varieties means that they are isomorphic. Thus the answer is: if and only if $A$ and $A^*$ are isomorphic. $\endgroup$
    – abx
    Sep 12, 2019 at 14:36
  • $\begingroup$ @abx But does this apply in the ring with $\mathbb L$ inverted, as discussed in the question? $\endgroup$
    – Will Sawin
    Sep 12, 2019 at 15:02
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    $\begingroup$ @Will Sawin: The original question was in $K_0$ mod. $\Bbb{L}$, it was edited after my answer. $\endgroup$
    – abx
    Sep 12, 2019 at 15:36
  • $\begingroup$ @abx It appears I fell victim to one of the classic blunders... $\endgroup$
    – Will Sawin
    Sep 12, 2019 at 15:48
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    $\begingroup$ For abelian varieties, in particular, for A and A dual, derived equivalence does NOT imply equality in the Grothendieck ring localized at L. This is a result proved independently by Efimov and Ito-Miura-Okawa-Ueda by taking a homomorphism to the Grothendieck ring of Hodge structures. For simply connected varieties this is unknown and goes under the name of L-equivalence and D-equivalence conjecture. It is open, most importantly, for K3 surfaces. $\endgroup$
    – Evgeny Shinder
    Sep 12, 2019 at 16:58

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