Answers to three problems in Dinh-Sibony's list on holomorphic foliation theory

Abstract

We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh--Sibony's list. For a saturated holomorphic curve foliation on a compact K\"ahler manifold with finitely many singularities, every positive directed $dd^c$-closed current has finite Poincar\'e mass. Thus linearizability is not needed for the geometric ergodic theorem in this setting. We construct a projective surface with dense hyperbolic leaves and a diffuse extremal current whose Nevanlinna averages converge at a uniform reciprocal logarithmic rate. This is an existence result for a chosen foliation, not a rate theorem for an arbitrary prescribed foliation. Under the hypotheses of the first result, every hyperbolic leaf has a logarithmic lower bound for its Nevanlinna characteristic, even if no diffuse directed current exists. Positive-dimensional singular sets are not treated.