Abstract view
Homological Planes in the Grothendieck Ring of Varieties


Published:20141020
Printed: Jun 2015
Julien Sebag,
Institut de recherche mathÃ©matique de Rennes , UMR 6625 du CNRS , UniversitÃ© de Rennes 1 , Campus de Beaulieu , 35042 Rennes cedex, France
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 $\mathrm{Pic}(X)_\mathbf{Q}:=\mathrm{Pic}(X)\otimes_\mathbf{Z}\mathbf{Q}=0$.