The Chow motive of the Godeaux surface, by Vladimir Guletskii and Claudio Pedrini

Let $X$ be the Godeaux surface obtained as a quotient of the Fermat quintic in $\mathbb P^3_{\mathbb C}$ under the appropriate action of $\mathbb Z/5$. We show that its Chow motive $h(X)$ splits as $1\oplus 9\mathbb L\oplus \mathbb L^2$ where $\mathbb L=(\mathbb P^1,[\mathbb P^1\times pt])$ is the Lefschetz motive. This provides a purely motivic proof of the Bloch conjecture for $X$. Our results also give a motivic proof of the Bloch conjecture for all surfaces with $p_g=0$ which are not of general type.


Vladimir Guletskii <guletskii@im.bas-net.by>
Claudio Pedrini <pedrini@dima.unige.it>