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

Abstract

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-folds. We find that the volume function exhibits a pathological behavior when N>=3, we obtain examples of a pseudoeffective R-divisor D for which the volume of D+sA, with s small and A ample, oscillates between two powers of s, and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that h^0(X,[mD]+A) displays a similar oscillatory behavior as m increases, showing that several notions of numerical dimensions of 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.

arXiv comment52 pages; final version to appear in J. Algebraic Geometry

References in corpus (8)

• Compactifications of moduli spaces inspired by mirror symmetry
• Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory
• Birational automorphism groups and the movable cone theorem for Calabi-Yau complete intersections of products of projective spaces
• Canonical currents and heights for K3 surfaces
• Remarks on Kawamata's effective non-vanishing conjecture for manifolds with trivial first Chern classes
• On effective non-vanishing and base-point-freeness
• Corrigendum: On subadditivity of the logarithmic Kodaira dimension
• On numerical dimensions of Calabi--Yau varieties