Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the rationality and the finite dimensionality of a cubic fourfold   Let $X$ be a cubic fourfold in $P^5_{C}$. 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

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The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph