Demailly, Jean-Pierre \(L^2\) methods and effective results in algebraic geometry. (Méthodes \(L^2\) et résultats effectifs en géométrie algébrique.) (French) Zbl 0962.14014 Séminaire Bourbaki. Volume 1998/99. Exposés 850-864. Paris: Société Mathématique de France, Astérisque. 266, 59-90, Exp. No. 852 (2000). The paper is a review of analytic methods (\(L^2\) Hodge theory) used in algebraic geometry for studying adjoint linear systems, vanishing theorem for algebraic vector bundles and invariance of plurigenera of general type families. Among the topics discussed in the paper are singular metrics, applications to Fujita’s conjecture [T. Fujita in: Algebraic Geometry, Proc. Symp., Sendai 1985, Adv. Stud. Pure Math. 10, 167–178 (1987; Zbl 0659.14002)] on global generation of adjoint linear systems, and analytic tools in Siu’s proof [Y.-T. Siu, Invent. Math. 134, No. 3, 661–673 (1998; Zbl 0955.32017)] of invariance of plurigenera for a family of general type.For the entire collection see [Zbl 0939.00019]. Reviewer: Taras E.Panov (Moskva) Cited in 1 Document MSC: 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14C20 Divisors, linear systems, invertible sheaves 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 14N30 Adjunction problems Keywords:adjoint linear systems; line bundle; Fujita conjecture; variety of general type; invariance of plurigenera; Hodge theory Citations:Zbl 0659.14002; Zbl 0955.32017 PDFBibTeX XMLCite \textit{J.-P. Demailly}, in: Séminaire Bourbaki. Volume 1998/99. Exposés 850--864. Paris: Société Mathé\-ma\-ti\-que de France. 59--90, Exp. No. 852 (2000; Zbl 0962.14014) Full Text: Numdam EuDML