Work

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

Year: 2025

Type: article

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 N -folds. We find that the volume function exhibits a pathological behavior when N ≥ 3 N\geq 3 , we obtain examples of a pseudoeffective R \mathbb {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,\left \lfloor mD \right \rfloor +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.

Source: Journal of Algebraic Geometry

DOI: 10.1090/jag/851

Authors: Simion Filip, John Lesieutre, Valentino Tosatti

Institutions: University of Chicago, Pennsylvania State University, Courant Institute of Mathematical Sciences, New York University

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation