# Motivic measures and stable birational geometry

@article{Larsen2001MotivicMA, title={Motivic measures and stable birational geometry}, author={Michael Larsen and Valery A. Lunts}, journal={arXiv: Algebraic Geometry}, year={2001} }

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.

## 149 Citations

### The Grothendieck ring of varieties and algebraic K-theory of spaces

- Mathematics
- 2016

Waldhausen's algebraic K-theory machinery is applied to motivic homotopy theory, producing an interesting motivic homotopy type. Over a field F of characteristic zero, its path components receive a…

### Algebraic independence in the Grothendieck ring of varieties

- Mathematics
- 2004

We give sufficient cohomological criteria for the classes of given varieties over a field k to be algebraically independent in the Grothendieck ring of varieties over k and construct some examples.

### A Refinement of the Motivic Volume, and Specialization of Birational Types

- MathematicsRationality of Varieties
- 2021

We construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version…

### Specialization of birational types

- MathematicsInventiones mathematicae
- 2019

We introduce a simplified version of the Grothendieck group of algebraic varieties and use it to show that birational types specialize in families with mild singularities of the central fiber.

### ON A RATIONALITY QUESTION IN THE GROTHENDIECK RING OF VARIETIES

- Mathematics
- 2010

We discuss elementary rationality questions in the Grothendieck ring of varieties for the quotient of a finite dimensional vector space over a characteristic 0 field by a finite group.

### The motivic nearby fiber and degeneration of stable rationality

- MathematicsInventiones mathematicae
- 2019

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization…

### The motivic nearby fiber and degeneration of stable rationality

- MathematicsInventiones mathematicae
- 2019

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization…

### Motivic invariants of birational maps

- Mathematics
- 2022

. We construct invariants of birational maps with values in the Kontsevich–Tschinkel group and in the truncated Grothendieck groups of varieties. These invariants are morphisms of groupoids and are…

## References

SHOWING 1-10 OF 10 REFERENCES

### Torification and factorization of birational maps

- Mathematics
- 1999

Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular…

### Birational cobordisms and factorization of birational maps

- Mathematics
- 1999

In this paper we develop a Morse-like theory in order to decompose birational maps and morphisms of smooth projective varieties defined over a field of characteristic zero into more elementary steps…

### Algebraic Geometry

- MathematicsNature
- 1973

Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)

### Lectures on Hilbert schemes of points on surfaces

- Mathematics
- 1999

Introduction Hilbert scheme of points Framed moduli space of torsion free sheaves on $\mathbb{P}^2$ Hyper-Kahler metric on $(\mathbb{C}^2)^{[n]}$ Resolution of simple singularities Poincare…

### The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups

- Mathematics
- 2000

We establish a relation between the generating functions appearing in the S-duality conjecture of Vafa and Witten and geometric Eisenstein series for Kac-Moody groups. For a pair consisting of a…

### The Douady Space of a Complex Surface

- Mathematics
- 1998

Abstract We prove that a standard realization of the direct image complex via the so-called Douady–Barlet (Hilbert-Chow in the algebraic case) morphism associated with a smooth complex analytic…

### Motivic measures

- Mathematics
- 2000

© Société mathématique de France, 2002, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les…

### Springer-Verlag

- New York,
- 1977

### USA E-mail address: larsen@math.indiana

- USA E-mail address: larsen@math.indiana