The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds

We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain Calabi–Yau complete intersections known as Wehler N N -folds. We find that the volume function exhibits a pathological behavior when N ≥ 3 N≥ 3 , we obtain examples of a pseudoeffective R R -divisor D D for which the volume of D + s A D+sA , with s s small and A A ample, oscillates between two powers of s s , and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that h 0 ( X , ⌊ m D ⌋ + A ) h^0(X, mD +A) displays a similar oscillatory behavior as m m increases, showing that several notions of numerical dimensions of D D do not agree and disproving a conjecture of Fujino. We accomplish this by relating the behavior of the volume function along a segment to the visits of a corresponding hyperbolic geodesics to the cusps of a hyperbolic manifold.