Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On the rationality and the finite dimensionality of a cubic fourfold (1701.05743v1)

Published 20 Jan 2017 in math.AG

Abstract: Let $X$ be a cubic fourfold in $P5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \times S$, 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 $F(X) \simeq S{[2]}$, 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 $t(X)\simeq t_2(S)(1)$. 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

Summary

We haven't generated a summary for this paper yet.