  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.
