Rationally connected manifolds and semipositivity of the Ricci curvature

TL;DR

This paper proves that compact Kähler manifolds with semipositive anticanonical bundles can be decomposed into Ricci flat and rationally connected parts, revealing their geometric structure.

Contribution

It provides a structure theorem for such manifolds, characterizing rational connectedness via tensor products and holonomy, and relates uniruledness to pseudoeffectiveness.

Findings

Manifolds split as products of Ricci flat and rationally connected varieties.

Rational connectedness characterized by non-existence of certain tensor products.

Uniruledness linked to the non-pseudoeffectiveness of the anticanonical bundle.

Abstract

This work establishes a structure theorem for compact K\"ahler manifolds with semipositive anticanonical bundle. Up to finite \'etale cover, it is proved that such manifolds split holomorphically and isometrically as a product of Ricci flat varieties and of rationally connected manifolds. The proof is based on a characterization of rationally connected manifolds through the non existence of certain twisted contravariant tensor products of the tangent bundle, along with a generalized holonomy principle for pseudoeffective line bundles. A crucial ingredient for this is the characterization of uniruledness by the property that the anticanonical bundle is not pseudoeffective.