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Positivity of direct images and projective varieties with nonnegative curvature

Abstract : The birational classification of algebraic varieties is a central problem in algebraic geometry. Recently great progress has been made towards the establishment of the MMP and the Abundance and by these works, smooth (or mildly singular) projective varieties can be birationally divided into two categories: 1. varieties with pseudoeffective canonical divisor, which are shown to reach a minimal model under the MMP; 2. uniruled varieties, which are covered by rational curves. In this thesis refined studies of these two categories of varieties are carried out respectively, by following the philosophy of studying the canonical fibrations associated to them.For any variety X in the first category, the most important canonical fibration associated to X is the Iitaka-Kodaira fibration whose base variety is of dimension equal to the Kodaira dimension of X. This thesis tacles an important corollary of the Abundance conjecture, namely, the Iitaka conjecture C_{n,m}, which states the supadditivity of the Kodaira dimension with respect to algebraic fibre spaces. In this thesis the Kähler version of C_{n,m} is proved under the assumption that the base variety of the fibre space is a complex torus by further developping the positivity theorem of direct images and the pluricanonical version of the Green-Lazarsfeld-Simpson type theorem on cohomology jumping loci. This generalizes the main result of Cao-Păun (2017).As for varieties in the second category, one studies the Albanese map and the MRC fibration, instead of the Iitaka-Kodaira fibration. A philosophy in this investigation is that when the tangent bundle or the anticanonical divisor admits certain positivity, the aforementioned two fibrations of the variety should have a rigid structure. In this thesis I study in this thesis the structure of (mildly singular) projective varieties with nef anticanonical divisor. By again applying the positivity of direct images and by applying results from the foliation theory, I manage to prove that the Albanese map of such variety is a locally constant fibration and that if its smooth locus is simply connected then the MRC fibration induces a splitting into a product. These generalize the corresponding results for smooth projective varieties in Cao (2019) and Cao-Höring (2019)
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Submitted on : Thursday, October 29, 2020 - 10:37:29 AM
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Juanyong Wang. Positivity of direct images and projective varieties with nonnegative curvature. Algebraic Geometry [math.AG]. Institut Polytechnique de Paris, 2020. English. ⟨NNT : 2020IPPAX048⟩. ⟨tel-02982921⟩

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