Lower bounds for Seshadri constants via successive minima of line bundles

Abstract

Given a nef and big line bundle L L on a projective variety X X of dimension d ≥ 2 d \geq 2 , we prove that the Seshadri constant of L L at a very general point is larger than ( d + 1 ) 1 d − 1 (d+1)^{\frac {1}{d}-1} . This slightly improves the lower bound 1 / d 1/d established by Ein, Küchle and Lazarsfeld [J. Differential Geom. 42 (1995), pp. 193–219]. The proof relies on the concept of successive minima for line bundles recently introduced by Ambro and Ito [Adv. Math. 365 (2020), 38 pp.].