Seshadri Constants via Lelong Numbers

One of Demailly’s characterizations of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note sections of multiples of the line bundle are used to produce such metrics and then to deduce another formula for Seshadri constants. It is applied to compute Seshadri constants on blown up products of curves, to disprove a conjectured characterization of maximal rationally connected quotients and to introduce a new approach to Nagata’s conjecture. In 1990 Demailly introduced Seshadri constants ǫ(L, x) for nef line bundles L on projective complex manifolds X [Dem92]: ǫ(L, x) := inf C∋x L.X multxC where the infimum is taken over all irreducible curves passing through x. They refine constants ǫ(L) appearing in Seshadri’s ampleness criterion [Laz04, Thm.1.4.13] and quantify how much of the positivity of an ample line bundle can be localized at a given point. These constants gained immediately a lot of interest in algebraic geometry; for example lower bounds on Seshadri constants were used to produce sections in adjoint bundles [Laz97]. It also turned out that explicit calculations of Seshadri constants are difficult in almost every concrete situation (see for example the work of Garcia [Gar05] on ruled surfaces) and it is not easier to give (interesting) upper and lower bounds for them. From their very definition it seems easier to determine upper bounds (by showing that a curve with appropriate intersection number and multiplicity exists) than lower bounds (via the non-existence of such curves) [EKL95, Bau99]. On the other hand Demailly gave two more equivalent definitions of Seshadri constants ([Dem92, Thm.6.4] or Prop. 6): ǫ(L, x) = γ(L, x) := sup γ∈R+ { ∃ singular metric h on L : iΘh ≥ 0, x isolated pole of Θh, ν(Θh, x) = γ } , where ν(Θh, x) is the Lelong number of the curvature current iΘh in x, and ǫ(L, x) = σ(L, x) := sup k∈N 1 k s(kL, x)