Motivic obstruction to rationality of a very general cubic hypersurface in \mathbb P⁵

2016/05/30 by Guletskii, Vladimir
#14C15 #14C25 #14E08 #14J70 #14M20 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1605.09434

Abstract

Let S be a smooth projective surface over a field. We introduce the notion of integral decomposability and, respectively, the opposite notion of integral indecomposability, of the transcendental motive M²_{\rm tr}(S). If the transcendental motive is indecomposable rationally, then it is indecomposable integrally. For example, M²_{\rm tr}(S) is rationally, and hence integrally indecomposable if S is an algebraic K3-surface whose motive is known to be finite-dimensional. In the paper we prove that M²_{\rm tr}(S) is integrally indecomposable when S is the self-product of a smooth projective curve having enough morphisms onto an elliptic curve with complex multiplication. This applies, for example, when S is the self-product of the Fermat sextic in \mathbb P². Some refinement of the same technique yields that M²_{\rm tr}(S₆) is integrally indecomposable, where S₆ is the Fermat sextic in \mathbb P³. This suggests a conjecture saying that the transcendental motive of any smooth projective surface is integrally indecomposable. We prove in the paper that if this motivic integral indecomposability conjecture is true, and if the motive of any smooth projective surface is finite-dimensional, then a very general cubic hypersurface in \mathbb P⁵ is not rational.

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