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The transcendental motive of a a cubic fourfold

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KAH - K-theory, algebraic cycles and motivic homotopy theory

 The transcendental part $t(X)$ of the motive of a cubic fourfold  $X$  is isomorphic to the (twisted) transcendental part $h_2(F(X))$ in a suitable Chow-K\”unneth decomposition for the motive of the Fano variety of lines $F(X)$. Similarly to the case of a cubic 3-fold, the transcendental motive $t(X)$ is isomorphic to the {\it Prym motive} associated to the surface $S_l \subset F(X)$ of lines meeting a general line $l$. If $X$ is a special cubic fourfold in the sense of Hodge theory,  and $F(X) \simeq S{[2]}$, with $S$ a K3 surface then     $t(X)\simeq t_2(S)(1)$, where $t_2(S)$ is the transcendental motive.  If $X$ is very general then $t(X)$ cannot be isomorphic to the (twisted) transcendental motive of a surface.  The existence  of an isomorphism $t(X) \simeq t_2(S)(1)$ is related to the  conjectures by Hassett and Kuznetsov on the rationality of a special cubic fourfold.  I will also consider the case of  other hyper-K\”alher varieties than $F(X)$ associated to a cubic fourfold $X$.




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