L2-Methods and Effective Results in Algebraic Geometry

Jean-Pierre Demailly
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Jean-Pierre Demailly: Université de Grenoble I, Institut Fourier, URA 188 du CNRS

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 817-827 from Springer

Abstract: Abstract One important problem arising in algebraic geometry is the computation of effective bounds for the degree of embeddings in a projective space of given algebraic varieties. This problem is intimately related to the following question: Given a positive (or ample) line bundle L on a projective manifold X, can one compute explicitly an integer m0 such that mL is very ample for m ≥ m0 ? It turns out that the answer is much easier to obtain in the case of adjoint line bundles 2(K X + mL), for which universal values ofm0 exist. We indicate here how such bounds can be derived by a combination of powerful analytic methods: theory of positive currents and plurisubharmonic functions (Lelong), L2 estimates for $$ \bar \partial$$ (Andreotti-Vesentini, Hörmander, Bombieri, Skoda), Nadel vanishing theorem, Aubin-Calabi-Yau theorem, and holomorphic Morse inequalities.

Date: 1995
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-9078-6_75

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DOI: 10.1007/978-3-0348-9078-6_75

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