# Motivic integration of an Abelian variety and its dual are same?

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?

• 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.
– abx
Sep 12, 2019 at 14:36
• @abx But does this apply in the ring with $\mathbb L$ inverted, as discussed in the question? Sep 12, 2019 at 15:02
• @Will Sawin: The original question was in $K_0$ mod. $\Bbb{L}$, it was edited after my answer.
– abx
Sep 12, 2019 at 15:36
• @abx It appears I fell victim to one of the classic blunders... Sep 12, 2019 at 15:48
• 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. Sep 12, 2019 at 16:58
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