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This lecture explores the local dynamics of tangent to the identity biholomorphisms in C2, with a focus on the existence of invariant manifolds. Delve into the problem of existence of invariant curves for two-dimensional vector fields and examine classical results including Seidenberg's resolution of singularities, Briot-Bouquet theorem, and Camacho-Sad theorem. This is the first in a series of three lectures by Lorena López-Hernanz, recorded during the thematic meeting "Foliations, birational geometry and applications" on February 3, 2024, at the Centre International de Rencontres Mathématiques in Marseille, France. The subsequent lectures will cover the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms and their extensions, developed in collaboration with several mathematicians. Access this video with helpful features like chapter markers, keywords, abstracts, and bibliographies through CIRM's Audiovisual Mathematics Library.
