A note on the transcendental basepoint-free conjecture for Calabi-Yau manifolds

In this note, we prove that the transcendental basepoint-free conjecture for Calabi-Yau manifolds holds if it holds for its hyperk{ä}hler factors in its Beauville-Bogomolov decomposition.

Based on a contraction theorem due to Bakker and Lehn, we show that the conjecture holds for a big and nef class $\alpha$ on a hyperk{ä}hler manifold under a mild condition on the dimension of the space generated by classes of rational curves on which $\alpha$ vanishes.