Angehrn-Siu-Helmke's method applied to abelian varieties

TL;DR

This paper applies Angehrn-Siu-Helmke's method to polarized abelian varieties, establishing new bounds on basepoint freeness thresholds and confirming a conjecture for very general cases.

Contribution

It extends Angehrn-Siu-Helmke's method to higher dimensions, improves bounds for specific abelian varieties, and verifies Caucci's conjecture in general settings.

Findings

Confirmed Caucci's conjecture for very general polarized abelian varieties.

Improved bounds on basepoint freeness thresholds for 4-folds and 5-folds.

Established finiteness of exceptions in polarization types for certain moduli spaces.

Abstract

We apply Angehrn-Siu-Helmke's method to estimate basepoint freeness thresholds of higher dimensional polarized abelian varieties. We showed that a conjecture of Caucci holds for very general polarized abelian varieties in the moduli spaces $\mathcal A_{g, l}$ with only finitely many possible exceptions of polarization types $l$ in each dimension $g$. We improved the bound of basepoint freeness thresholds of any polarized ableian $4$-folds and simple abelian $5$-folds.