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Let $S$ be a smooth projective surface over a field. We introduce the notionof integral decomposability and, respectively, the opposite notion of integralindecomposability, of the transcendental motive $M^2_{\rm tr}(S)$. If thetranscendental motive is indecomposable rationally, then it is indecomposableintegrally. For example, $M^2_{\rm tr}(S)$ is rationally, and hence integrallyindecomposable if $S$ is an algebraic $K3$-surface whose motive is known to befinite-dimensional. In the paper we prove that $M^2_{\rm tr}(S)$ is integrallyindecomposable when $S$ is the self-product of a smooth projective curve havingenough morphisms onto an elliptic curve with complex multiplication. Thisapplies, for example, when $S$ is the self-product of the Fermat sextic in$\mathbb P^2$. Some refinement of the same technique yields that $M^2_{\rmtr}(S_6)$ is integrally indecomposable, where $S_6$ is the Fermat sextic in$\mathbb P^3$. This suggests a conjecture saying that the transcendental motiveof any smooth projective surface is integrally indecomposable. We prove in thepaper that if this motivic integral indecomposability conjecture is true, andif the motive of any smooth projective surface is finite-dimensional, then avery general cubic hypersurface in $\mathbb P^5$ is not rational.

Motivic obstruction to rationality of a very general cubic hypersurface in $\mathbb P^5$