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On Chern classes of Lagrangian fibered hyper-Kähler manifolds

Claire Voisin

Abstract

We study the rank stratification for the differential of a Lagrangian fibration over a smooth basis. We also introduce and study the notion of Lagrangian morphism of vector bundles. As a consequence, we prove some of the vanishing, in the Chow groups of a Lagrangian fibered hyper-Kähler variety $X$, of certain polynomials in the Chern classes of $X$ and the Lagrangian divisor, predicted by the Beauville-Voisin conjecture. Under some natural assumptions on the dimensions of the rank strata, we also establish nonnegativity and positivity results for Chern classes.

On Chern classes of Lagrangian fibered hyper-Kähler manifolds

Abstract

We study the rank stratification for the differential of a Lagrangian fibration over a smooth basis. We also introduce and study the notion of Lagrangian morphism of vector bundles. As a consequence, we prove some of the vanishing, in the Chow groups of a Lagrangian fibered hyper-Kähler variety , of certain polynomials in the Chern classes of and the Lagrangian divisor, predicted by the Beauville-Voisin conjecture. Under some natural assumptions on the dimensions of the rank strata, we also establish nonnegativity and positivity results for Chern classes.
Paper Structure (11 sections, 30 theorems, 103 equations)

This paper contains 11 sections, 30 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theonew']}
  3. Lagrangian morphisms of vector bundles
  4. The case of Lagrangian fibrations
  5. Non-reduced fibers in codimension 1
  6. Higher rank loci for Lagrangian fibrations
  7. Support for Chern classes of Lagrangian fibered varieties
  8. Proof of Theorem \ref{['propnewauckland']}
  9. Proof of Theorems \ref{['theoLSV']} and \ref{['theoriess']}
  10. The cycle $c_2^2L^{n-1}$
  11. Riess' argument and proof of Theorem \ref{['theoriess']}

Key Result

Theorem 1.2

(Riess riess) If $X$ is a projective hyper-Kähler manifold which has an isotropic class in ${\rm NS}(X)$ and whose deformations satisfy the SYZ conjecture, $X$ satisfies Beauville's weak splitting conjecture.

Theorems & Definitions (75)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 1.10
  • ...and 65 more