Motivic measures and stable birational geometry

By Michael Larsen, V. Lunts. 2001

167 citations recorded by the source corpus.

We study the motivic Grothendieck group of algebraic varieties from the point of view of stable birational geometry. In particular, we obtain a counter-example to a conjecture of M. Kapranov on the rationality of motivic zeta-function.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/97c6d0bad44813b17001d149ae464ae3302f986d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Integration in valued fields (2005, methodology): “u= b+ v). More generally, for any subsemigroup S′ of a semigroup S, write a∼ b(S′) for (∃u,v∈ S)(a+u= b+v). We write K(RV[≤ n])/Isp for the groupiﬁcation of K+(RV[≤ n])/Isp 2This already follows from [21], who use diﬀerent methnods. 112 EHUD HRUSHOVSKI, DAVID KAZHDAN Lemma 13.2. Let V be a smooth pr”
The Grothendieck ring of varieties and piecewise isomorphisms (2010, background): “In [25], Larsen and Lunts have obtained the following very strong result (we give here a slightly different presentation).”
The Grothendieck ring of varieties is not a domain (2002, background): “) The proof in [LL01] requires resolution of singularities and weak factorization of birational maps [AKMW00, Theorem 0.”
HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES (2005, background): “ing homomorphisms on K0(var/{pt}), because this is the case for χy = Hfc (for example by remark 3.3). Such ring homomorphism are called “characteristics” [DL, Lo] or sometimes also “motivic measures” [LL], and there are much more examples known. In this sense our transformation Ty∗ is certainly a mo”
The motivic nearby fiber and degeneration of stable rationality (2017, background): “ational to the point SpecF. It is crucial for the applications in this paper that stable birationality is detected by the Grothendieck ring K(VarF), by the following criterion due to Larsen and Lunts [LL03]. Its proof relies on the Weak Factorization Theorem for birational maps between smooth and pr”
Arc spaces, motivic integration and stringy invariants (2004, background): “Larsen and Lunts [LL1][ LL2 ] determined for which surfaces this is a rational function over K0(V arC).”
Discriminants in the Grothendieck Ring (2012, background): “es. Cheah’sargumentdealsonlywiththespecializationtotheHodge-Delignepolynomials (actually an enrichment of this), but the proof can be adapted to establish Theorem 1.9. 1.10. However, Larson and Lunts [LL1, LL2] showed that ZX(t)is not always rational in M[[t]]. But an important (if vague) question r”