Meromorphic maps of Kahler manifolds with trivial canonical bundles

TL;DR

This paper proves that meromorphic maps from complete Kähler manifolds with trivial canonical bundles to projective space cannot omit many hyperplanes in subgeneral position, generalizing Fujimoto's defect relation.

Contribution

It extends defect relations for meromorphic maps to broader classes of Kähler manifolds with trivial canonical bundles, beyond the case of the ball.

Findings

Meromorphic maps cannot omit a certain number of hyperplanes in subgeneral position.

Generalization of Fujimoto's defect relation to complete Kähler manifolds with trivial canonical bundle.

Provides a non-integrated defect relation for such maps.

Abstract

Let M be a (bounded or not) domain of C^n which is complete with respect to a K\"ahler metric, or more generally, a complete K\"ahler manifold with trivial canonical bundle. Let f be a linearly nondegenerate meromorphic map from M to the complex projective space P^m. Under an assumption on the positivity of the pull-back by f of the Fubini-Study form on P^m, we prove that f can not omit a certain number of hyperplanes in subgeneral position in P^m. This is deduced directly from a non-integrated defect relation for such f which generalizes that obtained by Fujimoto in the case where M is a ball.