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Homological Planes in the Grothendieck Ring of Varieties

Published online by Cambridge University Press:  20 November 2018

Julien Sebag
Julien Sebag*
Affiliation:
Institut de recherche mathámatique de Rennes, UMR 6625 du CNRS, Universit´e de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex (France). e-mail: julien.sebag@univ-rennes1.fr
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Abstract

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In this note we identify the classes of $\text{Q}$-homological planes in the Grothendieck group of complex varieties ${{K}_{0}}\left( \text{Va}{{\text{r}}_{\text{C}}} \right)$. Precisely, we prove that a connected, smooth, affine, complex, algebraic surface $X$ is a $\text{Q}$-homological plane if and only if $\left[ X \right]\,=\,\left[ \text{A}_{\text{C}}^{2} \right]$ in the ring ${{K}_{0}}\left( \text{Va}{{\text{r}}_{\text{C}}} \right)$ and $\text{Pic}{{\left( X \right)}_{\text{Q}}}\,:=\,\text{Pic}\left( X \right)\,{{\otimes }_{\text{Z}}}\,\text{Q}\,\text{=}\,\text{0}$.

Keywords

14E0514R10motivic nearby cyclesmotivic Milnor fibernearby motives
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 356 - 362
Copyright
Copyright © Canadian Mathematical Society 2015

References

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