Projective normality and basepoint-freeness thresholds of general polarized abelian varieties

TL;DR

This paper establishes sharp bounds for projective normality of general polarized abelian varieties using the invariant eta, and explores its application to the Infinitesimal Torelli Theorem.

Contribution

It introduces the invariant eta to determine projective normality thresholds for polarized abelian varieties, providing sharp bounds and applications to Torelli-type results.

Findings

Projectively normal if hi(L)  2^{2g-1} and type not (2,4,...,4)

Sharp bound for projective normality established

Application to Infinitesimal Torelli Theorem for divisors in |L|

Abstract

For a polarized abelian variety $(X,L)$, Z. Jiang and G. Pareschi introduce an invariant $\beta(X,L)$, called the basepoint-freeness threshold. Using this invariant, we show that a general polarized abelian variety $(X,L)$ of dimension $g$ is projectively normal if $\chi(L) \geq 2^{2g-1}$ and the type of $L$ is not $(2,4,\dots,4)$. This bound is sharp since it is known that any polarized abelian variety of type $(2,4,\dots,4)$ is not projectively normal. We also give an application of $\beta(X,L)$ to the Infinitesimal Torelli Theorem for $Y \in |L|$.