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Generalized Okounkov bodies, hyperbolicity-related and direct image problems

Abstract : In Part 1 of this thesis, we construct “Okounkov bodies” for an arbitrary pseudo-effective (1,1-class on a Kähler manifold. We prove the differentiability formula of volumes of big classes for Kähler manifolds on which modified nef cones and nef cones coincide. As a consequence we prove Demailly’s transcendental Morse inequality for these particular Kähler manifolds; this includes Kähler surfaces. Then we construct the generalized Okounkov body for any big (1,1)-class, and give a complete characterization of generalized Okounkov bodies on surfaces. We show that this relates the standard Euclidean volume of the body to the volume of the corresponding big class as defined by Boucksom; this solves a problem raised by Lazarsfeld and Mustaţă in the case of surfaces. We also study the behavior of the generalized Okounkov bodies on theboundary of the big cone.Part 2 deals with Kobayashi hyperbolicity-related problems. Chapter 2’s goal is to study the degeneracy of leaves of the one-dimensional foliations on higher dimensional manifolds. The first part of Chapter 2 generalizes McQuillan’s Diophantine approximations for one-dimensional foliations with absolutely isolated singularities, on higher dimensional manifolds. As an application, we give a new proof of Brunella’s hyperbolicity theorem, that is, all the leaves of a generic foliation of degree larger than 2 in CP 6n is hyperbolic. In the second part of Chapter 2 we introduce the so-called weakly reduced singularities for one-dimensional foliations on higher dimensional manifolds. The “weakly reduced singularities” assumption is less demanding than the one required for “reduced singularities”, but play the same role in studying the Green-Griffiths-Lang conjecture. Finally we discuss a strategy to prove the Green-Griffiths-Lang conjecture for complex surfaces.In Chapter 3, assuming that the canonical sheaf is big in the sense of Demailly, we prove theKobayashi volume-hyperbolicity for any (possibly singular) directed variety.In Chapter 4, our first goal is to deal with effective questions related to the Kobayashi and Debarre conjectures, relying on the work of Brotbek and his joint work with Darondeau. We then combine these techniques to study the conjecture on the ampleness of the Demailly-Semple bundles raised by Diverio and Trapani, and also obtain some effective estimates related to this problem. Our result integrates both the Kobayashi and Debarre conjectures, with some effective estimates.The purpose of Chapter 5 is twofold: on the one hand we study a Fujita-type conjecture by Popa and Schnell, and give an effective (linear) bound on the generic global generation of the direct image of the twisted pluricanonical bundle. We also point out the relation between the Seshadri constant and the optimal bound. On the other hand, we give an affirmative answer to a question by Demailly-Peternell-Schneider in a more general setting. As applications, we generalize the theorems by Fujino and Gongyo on images of weak Fano manifolds to the Kawamata log terminal cases, and refine a result by Broustet and Pacienza on the rational connectedness of the image.In Chapter 6, we give a concrete and constructive proof of the equivalence between the category of semistable Higgs bundles with vanishing Chern classes and the category of all representations of the fundamental groups on smooth Kähler manifolds. This chapter is written for the complex geometers who are not familiar with the language of differential graded category used by Simpson to prove the above equivalence on smooth projective manifolds, and for those who would like to see an elementary proof of Corlette-Simpson correspondence for semistable Higgs bundles.
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Submitted on : Friday, January 12, 2018 - 2:36:07 PM
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  • HAL Id : tel-01682739, version 1



Ya Deng. Generalized Okounkov bodies, hyperbolicity-related and direct image problems. Algebraic Geometry [math.AG]. University of science and technology of China, 2017. English. ⟨NNT : 2017GREAM026⟩. ⟨tel-01682739⟩



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