Complex Manifolds

A variety over $\mathbb{C}$ in the Euclidean topology also has an analytic structure sheaf consisting of local holomorphic functions that makes it into a complex space, or a complex manifold if the variety is nonsingular. Many features of the geometry of algebraic varieties carry over to the complex analytic setting, such as the link between divisors and line bundles. The relation is especially close for complex manifolds that arise from complete varieties. Then global meromorphic functions are rational functions, analytically defined divisors and sheaves are algebraic, and two varieties are isomorphic as algebraic varieties if and only if the corresponding complex manifolds are isomorphic. However, in dimension ≥2 there are many complex manifolds not arising from algebraic varieties, for example, some having no nonconstant global meromorphic functions or no nontrivial submanifolds; there are also algebraic varieties (necessarily noncomplete) whose complex spaces are isomorphic, but that are not isomorphic as varieties. The first three sections discuss these ideas, with examples constructed as quotient manifolds by a discrete group action. The final section contains a discussion from first principles of Kähler metrics on complex manifolds. A Kähler metric on a complex manifold is a natural additional structure that makes it closer to a projective algebraic variety. The chapter concludes with a description of the Hodge structure on the cohomology of a Kähler manifold.