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This work establishes a structure theorem for compact K\"ahler manifolds withsemipositive anticanonical bundle. Up to finite \'etale cover, it is provedthat such manifolds split holomorphically and isometrically as a product ofRicci flat varieties and of rationally connected manifolds. The proof is basedon a characterization of rationally connected manifolds through the nonexistence of certain twisted contravariant tensor products of the tangentbundle, along with a generalized holonomy principle for pseudoeffective linebundles. A crucial ingredient for this is the characterization of unirulednessby the property that the anticanonical bundle is not pseudoeffective.

Rationally connected manifolds and semipositivity of the Ricci curvature