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# Homological Planes in the Grothendieck Ring of Varieties

Published:2014-10-20
Printed: Jun 2015
• Julien Sebag,
Institut de recherche mathématique de Rennes , UMR 6625 du CNRS , Université de Rennes 1 , Campus de Beaulieu , 35042 Rennes cedex, France
 Format: LaTeX MathJax PDF

## Abstract

In this note, we identify, in the Grothendieck group of complex varieties $K_0(\mathrm Var_\mathbf{C})$, the classes of $\mathbf{Q}$-homological planes. Precisely, we prove that a connected smooth affine complex algebraic surface $X$ is a $\mathbf{Q}$-homological plane if and only if $[X]=[\mathbf{A}^2_\mathbf{C}]$ in the ring $K_0(\mathrm Var_\mathbf{C})$ and $\mathrm{Pic}(X)_\mathbf{Q}:=\mathrm{Pic}(X)\otimes_\mathbf{Z}\mathbf{Q}=0$.
 Keywords: motivic nearby cycles, motivic Milnor fiber, nearby motives
 MSC Classifications: 14E05 - Rational and birational maps 14R10 - Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)

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