On the rationality and the finite dimensionality of a cubic fourfold

2017/01/20 by Claudio Pedrini, Pedrini, Claudio
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1701.05743

openalex publication_date 2017/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a cubic fourfold in P⁵_C. We prove that, assuming the Hodge conjecture for the product S × S, where S is a complex surface, and the finite dimensionality of the Chow motive h(S), there are at most a countable number of decomposable integral polarized Hodge structures, arising from the fibers of a family of smooth projective surfaces. According to the results in [ABB] this is related to a conjecture proving the irrationality of a very general X. If X is special, in the sense of B.Hasset, and F(X) ≃ S^[2], with S a K3 surface associated to X, then we show that the Chow motive h(X) contains as a direct summand a "transcendental motive" t(X) such that t(X)≃ t₂(S)(1). The motive of X is finite dimensional if and only if S has a finite dimensional motive, in which case t(X) is indecomposable. Similarly, if X is very general and the motive h(X) is finite dimensional, then t(X) is indecomposable

Citations

• Algebraic cycles on the Fano variety of lines of a cubic fourfold

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