Multipoint Okounkov bodies

Abstract: During the nineties, the field medallist Okounkov found a way to associate to an ample line bundle L over an n−complex dimensional projective manifold X a convex body in Rn, now called Okounkov body ∆(L). The construction depends on the choice of a valuation centered at one point p∈X and it works even if L is a big line bundle. In the last decade ∆(L) turned out to be an accurate simplied image of (L→X;p). Indeed it encodes important global invariants like the volume, Vol(L), and it can be a finer invariant of the Seshadri constant of L at p. Moreover it can be useful to approximates L→X through an n−complex dimensional torus-invariant domain equipped with the standard flat metric. In this thesis I propose a generalization of the Okounkov bodies. Namely, starting from a big line bundle L over an n−complex dimensional projective manifold X, and from the choice of N valuations centered at N different points p1,...,pN∈X, I construct N multipoint Okounkov bodies ∆1(L),...,∆N(L)⊂Rn. They are a simpler copy of (L→X;p1,...,pN) since they forms a finer invariant of the volume Vol(L) and of the multipoint Seshadri constant of L at p1,...,pN. The latter in particular is related to several important conjectures in Algebraic Geometry, like the Nagata's conjecture which concerns the projective plane P2. Related to this, in the thesis there are further small results for surfaces. Moreover the multipoint Okounkov bodies consent to define N torus-invariant domains in Cn which approximate simultaneously L→X, i.e. they produce a perfect Kähler packing (the holomorphic analogue of the symplectic packing), and this leads to an interpretation of the multipoint Seshadri constant in terms of packings. Finally in the toric case, in different situations, the multipoint Okounkov bodies can be recovered directly subdividing the polytope.