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for questions about motives in algebraic geometry, including constructions of categories of motives and motivic sheaves, and aspects of the standard conjectures.

9 votes

Grothendieck ring of "varieties carrying a function"

I have just come across this question, and I have no idea if the OP still has any interest in it, but theories built out of pairs $(X, f)$ go under the name of "exponential motives", and there has been … Two recent papers that come to mind, emphasising different aspects, are Motivic classes of Nakajima quiver varieties, https://arxiv.org/pdf/1603.03200.pdf, and Exponential motives http://javier.fresan.perso.math.cnrs.fr …
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