Holes in Calabi-Yau Effective Cones

TL;DR

This paper investigates the existence and properties of non-effective divisor classes, called 'holes', in the effective cones of Calabi-Yau threefolds, with implications for string theory and algebraic geometry.

Contribution

It introduces the concept of holes in Calabi-Yau effective cones, characterizes their behavior, and establishes conditions for their existence, advancing understanding of divisor classes in algebraic geometry.

Findings

Some divisor classes in effective cones are not effective, called holes.

Holes can be characterized by necessary and sufficient conditions.

Holes form semigroups and have moduli-dependent volume bounds.

Abstract

Motivated by their role in non-perturbative potentials in string theory, we study divisors in effective cones of Calabi-Yau threefolds. We give examples of geometries for which some divisor classes in the effective cone are not themselves effective: i.e., they have no global sections. We call these non-holomorphic divisor classes "holes," and characterize their behavior in an ensemble of toric hypersurface Calabi-Yau threefolds. We prove some necessary and sufficient conditions for the existence of holes, show consequences of holes that follow from the minimal model program, and demonstrate that a class of holes come in semigroups (with this class conjectured to constitute all holes). Furthermore, we provide moduli-dependent bounds on the volumes of four-cycles representing holes.