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.
Complexe de de Rham filtr\'e d'une vari\'et\'e singuli\`ere
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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.