Some Results about the Index Conjecture for log Calabi-Yau Pairs

TL;DR

This paper investigates the index conjecture for log Calabi-Yau pairs, establishing boundedness results in specific dimensions and conditions, advancing understanding of their algebraic structure.

Contribution

It provides new inductive methods and proves boundedness of the log canonical index for certain classes of log Calabi-Yau pairs in dimensions 3 and 4.

Findings

Boundedness of log canonical index in dimension 3 for pairs with rational DCC coefficients.

Boundedness of log canonical index in dimension 4 for klt log Calabi-Yau pairs with non-zero boundary.

Development of inductive techniques for the index conjecture in log Calabi-Yau pairs.

Abstract

We show some inductive statements for the index conjecture for log canonical Calabi-Yau pairs. Using it, we show that boundedness of log canonical index for log canonical Calabi Yau pairs with rational DCC coefficients in dimension 3. We also show boundedness of log canonical index for klt log Calabi-Yau pairs in dimension 4 with $B\neq 0$.