40 research outputs found

    Multipoint Okounkov bodies

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    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\u27s 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 K\ue4hler 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

    Multipoint Okounkov bodies, strong topology of ω-plurisubharmonic functions and K\ue4hler-Einstein metrics with prescribed singularities

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    The most classical topic in K\ue4hler Geometry is the study of K\ue4hler-Einstein metrics as solution of complex Monge-Amp\ue8re equations. This thesis principally regards the investigation of a strong topology for ω-plurisubharmonic functions on a fixed compact K\ue4hler manifold (X,ω), its connection with complex Monge-Amp\ue8re equations with prescribed singularities and the consequent study of singular K\ue4hler-Einstein metrics. However the first part of the thesis, Paper I, provides a generalization of Okounkov bodies starting from a big line bundle over a projective manifold and a bunch of distints points. These bodies encode renowned global and local invariants as the volume and the multipoint Seshadri constant.In Paper II the set of all ω-psh functions slightly more singular than a fixed singularity type are endowed with a complete metric topology whose distance represents the analog of the L1 Finsler distance on the space of K\ue4hler potentials. These spaces can be also glued together to form a bigger complete metric space when the singularity types are totally ordered. Then Paper III shows that the corresponding metric topology is actually a strong topology given as coarsest refinement of the usual topology for ω-psh functions such that the relative Monge-Amp\ue8re energy becomes continuous. Moreover the main result of Paper III proves that the extended Monge-Amp\ue8re operator produces homeomorphisms between these complete metric spaces and natural sets of singular volume forms endowed their strong topologies. Such homeomorphisms extend Yau\u27s famous solution to the Calabi\u27s conjecture and the strong topology becomes a significant tool to study the stability of solutions of complex Monge-Amp\ue8re equations with prescribed singularities. Indeed Paper IV introduces a new continuity method with movable singularities for classical families of complex Monge-Amp\ue8re equations typically attached to the search of log K\ue4hler-Einstein metrics. The idea is to perturb the prescribed singularities together with the Lebesgue densities and asking for the strong continuity of the solutions. The results heavily depend on the sign of the so-called cosmological constant and the most difficult and interesting case is related to the search of K\ue4hler-Einstein metrics on a Fano manifold. Thus Paper V contains a first analytic characterization of the existence of K\ue4hler-Einstein metrics with prescribed singularities on a Fano manifold in terms of the relative Ding and Mabuchi functionals. Then extending the Tian\u27s α-invariant into a function on the set of all singularity types, a first study of the relationships between the existence of singular K\ue4hler-Einstein metrics and genuine K\ue4hler-Einstein metrics is provided, giving a further motivation to study these singular special metrics since the existence of a genuine K\ue4hler-Einstein metric is equivalent to an algebrico-geometric stability notion called K-stability which in the last decade turned out to be very important in Algebraic Geometry

    Multipoint Okounkov bodies

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    Symplectic geometry of Cartan–Hartogs domains

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    This paper studies the geometry of Cartan–Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan–Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan–Hartogs domains and their dual. Further, we compute their symplectic capacity and show that a Cartan–Hartogs domain admits a symplectic duality if and only if it reduces to be a complex hyperbolic space
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