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 $X$, 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.

arXiv commentFinal version, to appear in PAMQ, memorial volume for Jean-Pierre Demailly