The log Grothendieck ring of varieties is K0(Var)[P]/(P^2+P[G_m]), and a log chi-y genus built from it is motivic even though log Hodge numbers are not.
The Log Product Formula in quantum $K$-theory
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abstract
We prove a formula expressing the $K$-theoretic log Gromov-Witten invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. The proof requires introducing log virtual fundamental classes in $K$-theory and verifying their various functorial properties. We introduce a log version of $K$-theory and prove the formula there as well.
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The log Grothendieck ring of varieties
The log Grothendieck ring of varieties is K0(Var)[P]/(P^2+P[G_m]), and a log chi-y genus built from it is motivic even though log Hodge numbers are not.