The Chow motive of LSV hyper-K\"alher manifolds

TL;DR

This paper investigates the Chow motive of LSV hyper-K"ahler manifolds derived from cubic fourfolds, establishing their motive as a summand of the motive of the fifth power of the fourfold, and identifying conditions for abelian type motives.

Contribution

It proves that the Chow motive of certain hyper-K"ahler manifolds is a direct summand of the motive of the fifth power of the cubic fourfold, and characterizes cases with abelian type motives.

Findings

Chow motive of the hyper-K"ahler manifold is a summand of the motive of X^5.

For a family of cubics, the motive is of abelian type.

Existence of a unique smooth compactification with irreducible fibers.

Abstract

Let $X$ be a smooth cubic fourfold over $\C$ and let $\pi : \sJ_U \to U$, with $ U \subset (\P^5)^*$, be the Lagrangian fibration whose fibres are the smooth hyperplane sections $Y_ H = X \cap H$, with $H \in U$. There always exists a (not unique) smooth compactification $\bar \sJ \to (\P^5)^*$ which is a hyper-K\"alher manifold of OG10 type. Since two different compactifications are birationally equivalent their Chow motives are isomorphic. For a general $X$ a geometrical construction of a smooth compactification $\sJ(X)$ with irreducible fibres has been described in [LSV]. In this note we prove that the Chow motive $h(\sJ(X)) $ is a direct summand of the (twisted) motive of $X^5$ and therefore is is of abelian type if $h(X)$ is of abelian type.We describe a 10 -dimensional family $\sF$ of cubics $X$ such that the compactification $\sJ(X)$ is unique, smooth, with irreducible fibres, and the Chow motive $h( \sJ(X) )$ is of abelian type.