Rational curves on compact Kähler manifolds (Q2283561)

For smooth projective varieties the relationship between the positivity of the canonical bundle and the existence of negative rational curves is well understood mainly thanks to the Cone Theorem and to [\textit{S. Boucksom} et al., J. Algebr. Geom. 22, No. 2, 201--248 (2013, Zbl 1267.32017)]. The existing proofs of these results are based on techniques that cannot be adapted to the more general analytic setting. Using deformation theory of curves on threefolds in the first case and rank one foliations in the second, these results can be generalised to compact Kähler threefolds, but only very particular cases are known for compact Kähler manifolds of higher dimension. The main result in this paper is the following: let \(X\) be a compact Kähler manifold of dimension \(n\). Assume that a compact Kähler manifold of dimension at most \(n-1\) is uniruled if and only if the canonical bundle is not pseudoeffective. If \(K_X\) is pseudo effective but is not nef, there exists a \(K_X\)-negative rational curve in \(X\). This result is a first step toward a generalization of some well-understood concepts of birational geometry to non projective varieties. The proof of this theorem is based on a weak subadjunction formula for log-canonical centers associated to certain cohomology classes. Most of the theory developed in this paper can be surely very useful for many related questions.