Iitaka dimension for cycles

TL;DR

This paper introduces the Iitaka dimension for numerical cycle classes, explores its properties, and provides evidence for its integrality, focusing on Schubert cycles and contracted cycles.

Contribution

It defines the Iitaka dimension for cycles, develops its theoretical framework, and investigates its properties in specific geometric contexts.

Findings

Conjecture that the Iitaka dimension is always an integer.

Evidence supporting the integrality of the Iitaka dimension.

Application to Schubert cycles and contracted cycles.

Abstract

We define the Iitaka dimension of a numerical cycle class and develop its theory. We conjecture that the Iitaka dimension is integer-valued, and give some evidence in this direction. We focus on two cases of geometric interest: Schubert cycles on Grassmannians and cycles contracted by morphisms.