Abstract:
For a singular variety $X$, one can define several “motivic” types of singularities using
the Grothendieck ring of varieties. One of them is the notion of $\mathbb L$-rational singularities,
introduced by Nicaise and Shinder in their work on the specialization of stable birationality.
The question of when quotient varieties $X/G$ have $\mathbb L$-rational singularities turns out to be
particularly interesting. I will discuss some results in this direction, in particular a proof
that symmetric powers of smooth varieties have $\mathbb L$-rational singularities. I will then explain
how one can use this result to prove the irrationality of motivic zeta functions for a large
class of varieties, generalizing a method of Larsen and Lunts to higher dimensions. If time
permits, I will also discuss some other applications and open questions.
Language: English
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