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The Chow motive of LSV hyper-Kälher manifolds

Claudio Pedrini

Abstract

Let $X$ be a smooth cubic fourfold over $\C$ and let $π: \sJ_U \to U$, with $ U \subset (¶^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 $\bar \sJ \to (¶^5)^*$ which is a hyper-Kälher 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.

The Chow motive of LSV hyper-Kälher manifolds

Abstract

Let be a smooth cubic fourfold over and let , with , be the Lagrangian fibration whose fibres are the smooth hyperplane sections , with . There always exists a (not unique) smooth compactification which is a hyper-Kälher manifold of OG10 type. Since two different compactifications are birationally equivalent their Chow motives are isomorphic. For a general a geometrical construction of a smooth compactification with irreducible fibres has been described in [LSV]. In this note we prove that the Chow motive is a direct summand of the (twisted) motive of and therefore is is of abelian type if is of abelian type.We describe a 10 -dimensional family of cubics such that the compactification is unique, smooth, with irreducible fibres, and the Chow motive is of abelian type.
Paper Structure (4 sections, 7 theorems, 43 equations)

This paper contains 4 sections, 7 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. The motive of a general LSV
  3. The motives of $\bar{\mathcal{J}}$ and $\bar{\mathcal{J}}^t$
  4. Automorphisms of LSV manifolds

Key Result

Theorem 2.1

Let $X$ be general cubic fourfold and let $J(X)$ be the HK manifold of OG10 type constructed in [LSV]. Then $h(\mathcal{J}(X))$ is a direct summand of the motive in $\mathcal{M}_{rat}(\mathbf{C})$. Therefore $\mathcal{J}(X)$ has a motive of abelian type if $h(X)$ is of abelian type

Theorems & Definitions (15)

  • Conjecture 1.1
  • Theorem 2.1
  • Lemma 2.6
  • proof
  • Lemma 2.8
  • proof
  • Corollary 2.12
  • proof
  • Remark 2.13
  • Proposition 3.2
  • ...and 5 more