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

Simion Filip, John Lesieutre, Valentino Tosatti

TL;DR

This work analyzes the volume function on Wehler $N$-folds, revealing pathological boundary behavior that defies earlier expectations of polynomial-type growth and regularity. By translating divisor trajectories into hyperbolic-geometric cusp excursions via a pseudo-automorphism group $W$, the authors establish a divergent--recurrent decomposition of boundary points and prove that, for $N$, one can realize oscillations in $ ext{vol}(D+sA)$ and in $h^0(X,ig floor mDig floor+A)$ with prescribed liminf/limsup exponents. They show the volume along $D+sA$ is $C^1$-differentiable at $s=0$ but never $C^{1,eta}$ for any $eta>0$, thereby giving sharp regularity at the boundary. The results yield counterexamples to Fujino’s conjecture and to several proposed equalities among numerical dimensions, illustrating deep connections between birational geometry and hyperbolic dynamics. Overall, the paper broadens our understanding of the boundary behavior of numerical invariants and highlights the role of cusp dynamics in higher-dimensional algebraic geometry.

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.

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

TL;DR

This work analyzes the volume function on Wehler -folds, revealing pathological boundary behavior that defies earlier expectations of polynomial-type growth and regularity. By translating divisor trajectories into hyperbolic-geometric cusp excursions via a pseudo-automorphism group , the authors establish a divergent--recurrent decomposition of boundary points and prove that, for , one can realize oscillations in and in with prescribed liminf/limsup exponents. They show the volume along is -differentiable at but never for any , thereby giving sharp regularity at the boundary. The results yield counterexamples to Fujino’s conjecture and to several proposed equalities among numerical dimensions, illustrating deep connections between birational geometry and hyperbolic dynamics. Overall, the paper broadens our understanding of the boundary behavior of numerical invariants and highlights the role of cusp dynamics in higher-dimensional algebraic geometry.

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.
Paper Structure (86 sections, 23 theorems, 185 equations, 1 figure)

This paper contains 86 sections, 23 theorems, 185 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The volume function
  3. Volume of divisors
  4. Behavior at the pseudoeffective boundary
  5. Growth of sections
  6. Iitaka dimension
  7. $\mathbb{R}$-divisors
  8. The main result
  9. Regularity of the volume function
  10. $C^1$ differentiability
  11. Higher regularity
  12. Numerical dimensions
  13. Nakayama's $\kappa_\sigma$
  14. Real number extensions
  15. A conjecture of Fujino
  16. ...and 71 more sections

Key Result

Theorem 1.3.1

For every $N\geq 3$ there exists a smooth Calabi--Yau $N$-fold $X$ such that given any real number $\delta\in [1,\frac{N}{2}]$ there exists a pseudoeffective $\mathbb R$-divisor $D$ with $\mathop{\mathrm{vol}}\nolimits(D)=0$ so that given any sufficiently ample divisor $A$ we have

Figures (1)

  • Figure 1.3.1: Volume on the Wehler 3-fold

Theorems & Definitions (47)

  • Theorem 1.3.1: Oscillation of volume
  • Theorem 1.3.2: Limits for every divisor
  • Corollary 1
  • Corollary 2
  • Conjecture 1
  • Corollary 3
  • Proposition 1: The pseudoeffective and big cones
  • proof
  • Proposition 2: Convex core fundamental domain properties
  • proof
  • ...and 37 more