# Some positivity properties of ample line bundles on abelian 3-folds via higher syzygies

For ample line bundles on projective varieties, $p$-jet ampleness or Property $(N_p)$ consists an increasing sequence of positivity properties respectively. For example, 0-jet ampleness is equivalent to basepoint freeness and $(N_0)$ is equivalent to projective normality.

On abelian varieties, it is known that $(p+2)L$ is $p$-jet ample and $(p+3)L$ satisfies $(N_p)$. In this talk, we discuss these positivity properties for ample line bundles which are not necessarily high multiples of other line bundles. Inspired by (a generalized version of) Fujita’s conjecture, we consider a question which asks if a line bundle satisfies these positivity properties when the degree of every abelian subvarieties are large. Using an invariant introduced by Z. Jiang-G. Pareschi recently and techniques in the study of Fujita’s conjecture, we answer this question affirmatively for abelian 3-folds.

(This is a talk of Tokyo-Kyoto AG seminar on Zoom, and will be given in Japanese.)