Buser-Sarnak invariant and projective normality of abelian varieties

Abstract

Abstract. We show that a general n-dimensional polarized abelian variety (A, L) of a given polarization type and satisfying h 0 (A, L) ≥ 8 n 2 · n n n! is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety in a geodesic tubular neighborhood of a subtorus of a compact complex torus.

Local positivity, multiplier ideals, and syzygies of abelian varieties

“…For p > 1 the condition is that the first p modules of syzygies among these quadrics are generated in minimal possible degree. The result of Hwang and To in [10] is essentially the case p = 0 of Theorem A.…”

“…In a recent paper [10], Hwang and To observed that there is a relation between local positivity on an abelian variety A and the projective normality of suitable embeddings of A. The purpose of this note is to extend their result to higher syzygies, and to show that the language of multiplier ideals renders the computations extremely quick and transparent.…”

Geometric aspects of Newton–Okounkov bodies

“…Motivated by Kollár's line of thought regarding Fujita-type conjectures, it is natural to ask whether it is feasible to study property (N p ) for a given line bundle with certain numerics instead. Such a new line of attack in the case of abelian varieties has been recently initiated by Hwang-To [HT11] where complex analytic techniques (more precisely upper bounds on volumes of tubular neighbourhoods of subtori of abelian varieties) were used to control projective normality of line bundles on abelian varieties in terms of Seshadri constants. The next step was taken by Lazarsfeld-Pareschi-Popa [LPP10], who used multiplier ideal methods to extend the results of [HT11] to higher syzygies on abelian varieties.…”

Basepoint-freeness thresholds and higher syzygies on abelian threefolds

“…Inspired by works [HT11], [LPP11], [KL19] on (N p ) of polarized abelian varieties and a generalization of Fujita's basepoint freeness conjecture [Kol97], the author [Ito18] asks the (N p )-part of the following question. After that, both the (N p )-part and the K p,1 -part (for d 2) are stated as a conjecture by V. Lozovanu: Question 1.3 ([Ito18, Question 4.2],[Loz18, Conjecture 7.1]).…”