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Register a new userA transcendental approach to Kollars injectivity theorem
النهج الطولي لمبرهنة كولار الحصول على الحصول
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Authors
Osamu Fujino
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نحن نعالج مبرهنة كولار للحقن من النظرة التحليلية (أو الهندسة التفاضلية). بشكل أكثر دقة، نقدم شرطا خطوط الإنحناء الذي يشير إلى أحكام كولار للحقن الشعبية. يتم تصريح أساسنا لهذا المبرهنة للمخلوق الكاهلي المضغوط، ولكن دليلنا يستخدم مساحة الأشكال الهارمونية على مجموعة زاريسكي مناسبة مع قياس كاهلي مكتمل. لا نحتاج إلى الحيل التغطية، ولا التجزئة، ولا سلسلة اسكترال ليراي.
We treat Kollars injectivity theorem from the analytic (or differential geometric) viewpoint. More precisely, we give a curvature condition which implies Kollar type cohomology injectivity theorems. Our main theorem is formulated for a compact Kahler manifold, but the proof uses the space of harmonic forms on a Zariski open set with a suitable complete Kahler metric. We need neither covering tricks, desingularizations, nor Lerays spectral sequence.
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We give a new proof of Kollars conjecture on the pushforward of the dualizing sheaf twisted by a variation of Hodge structure. This conjecture was settled by M. Saito via mixed Hodge modules and has applications in the investigation of Albanese maps. Our technique is the $L^2$-method and we give a concrete construction and proofs of the conjecture. The $L^2$ point of view allows us to generalize Kollars conjecture to the context of non-abelian Hodge theory.
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Transcendental Brauer elements are notoriously difficult to compute. Work of Wittenberg, and later, Ieronymou, gives a method for computing 2-torsion transcendental classes on surfaces that have a genus 1 fibration with rational 2-torsion in the Jacobian fibration. We use ideas from a descent paper of Poonen and Schaefer to remove this assumption on the rational 2-torsion.
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A mixture of an historical article, and of a survey of recent developments, containing also a couple of new results.
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