This paper by Jean-Pierre Demailly discusses the use of $ L^2 $-methods in algebraic geometry to derive effective results, particularly concerning the very ampleness of line bundles. The key problem addressed is determining effective bounds for the degree of embeddings of algebraic varieties into projective space. The paper focuses on adjoint line bundles $ 2(K_X + mL) $, for which universal bounds $ m_0 $ exist, and explains how these bounds can be derived using analytic methods such as the theory of positive currents, plurisubharmonic functions, $ L^2 $ estimates for $ \overline{\partial} $, Nadel vanishing theorem, Aubin-Calabi-Yau theorem, and holomorphic Morse inequalities. The paper begins with an overview of basic concepts in hermitian differential geometry, including connections, curvature, and hermitian metrics. It then discusses the concepts of positivity and ampleness of holomorphic vector bundles, with examples of line bundles and their associated metrics. The paper then presents the Bochner technique and vanishing theorems, including the Bochner-Kodaira-Nakano formula and the Akizuki-Kodaira-Nakano vanishing theorem. Next, the paper discusses Hörmander's $ L^2 $ estimates and existence theorems, leading to the concept of multiplier ideal sheaves. It then presents the Nadel vanishing theorem, which is a key result in the study of very ample line bundles. The paper also discusses numerical criteria for very ample line bundles, including a theorem by Fujita and a numerical criterion by Demailly. Finally, the paper presents holomorphic Morse inequalities, which provide a powerful tool for studying the cohomology of line bundles. These inequalities are used to derive effective results on the very ampleness of line bundles, including a theorem by Demailly that improves upon Siu's result. The paper concludes with a discussion of effective bounds for very ample line bundles and references to various theorems and results in algebraic geometry.This paper by Jean-Pierre Demailly discusses the use of $ L^2 $-methods in algebraic geometry to derive effective results, particularly concerning the very ampleness of line bundles. The key problem addressed is determining effective bounds for the degree of embeddings of algebraic varieties into projective space. The paper focuses on adjoint line bundles $ 2(K_X + mL) $, for which universal bounds $ m_0 $ exist, and explains how these bounds can be derived using analytic methods such as the theory of positive currents, plurisubharmonic functions, $ L^2 $ estimates for $ \overline{\partial} $, Nadel vanishing theorem, Aubin-Calabi-Yau theorem, and holomorphic Morse inequalities. The paper begins with an overview of basic concepts in hermitian differential geometry, including connections, curvature, and hermitian metrics. It then discusses the concepts of positivity and ampleness of holomorphic vector bundles, with examples of line bundles and their associated metrics. The paper then presents the Bochner technique and vanishing theorems, including the Bochner-Kodaira-Nakano formula and the Akizuki-Kodaira-Nakano vanishing theorem. Next, the paper discusses Hörmander's $ L^2 $ estimates and existence theorems, leading to the concept of multiplier ideal sheaves. It then presents the Nadel vanishing theorem, which is a key result in the study of very ample line bundles. The paper also discusses numerical criteria for very ample line bundles, including a theorem by Fujita and a numerical criterion by Demailly. Finally, the paper presents holomorphic Morse inequalities, which provide a powerful tool for studying the cohomology of line bundles. These inequalities are used to derive effective results on the very ampleness of line bundles, including a theorem by Demailly that improves upon Siu's result. The paper concludes with a discussion of effective bounds for very ample line bundles and references to various theorems and results in algebraic geometry.