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On the Grothendieck ring of varieties

Published online by Cambridge University Press:  18 January 2015

AMIT KUBER
AMIT KUBER*
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL e-mail: expinfinity1@gmail.com
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Abstract

Let K0(Vark) denote the Grothendieck ring of k-varieties over an algebraically closed field k. Larsen and Lunts asked if two k-varieties having the same class in K0(Vark) are piecewise isomorphic. Gromov asked if a birational self-map of a k-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb{Z}$[$\mathfrak{B}$] where $\mathfrak{B}$ denotes the multiplicative monoid of birational equivalence classes of irreducible k-varieties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 3 , May 2015 , pp. 477 - 486
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

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