Appears in collection : Cours du M2R 2019 - Introduction to analytic geometry

Algebraic varieties over the eld of complex numbers can be studied through a transcen-dental point of view : such varieties, when they are non singular, are in fact complex analyticmanifolds. Their study is then intimately related to the study of holomorphic functions ofseveral complex variables. Just as in the case of one complex variable, holomorphic functionsof several variables enjoy many deep properties that pertain to the fact that complex numbersform an algebraically closed eld; also, holomorphic functions are in some sense a completionof the space of polynomials. Analytic properties can be linked to algebraic facts by somefundamental theorems such as Serre's GAGA theorem, which asserts that every holomor-phic object living in a complex projective manifold is in fact algebraic. Such properties arere ected by strong \rigidity properties" of holomorphic functions, and can also be detectedby computing relevant cohomology groups : these groups somehow describe the fundamen-tal analytic \invariants" of complex manifolds. The basic Dolbeault-Grothendieck lemmaasserts the local triviality of Dolbeault cohomology. The concepts of sheaf and locally freesheaf allow to establish certain canonical cohomology isomorphisms, and the fundamentalSerre duality theorem generalizes Poincaré duality to complex geometry.