Compact Complex Manifolds

In this chapter we shall apply the differential equations and differential geometry of the previous two chapters to the study of compact complex manifolds. In Sec. 1 we shall present a discussion of the exterior algebra on a Hermitian vector space, introducing the fundamental 2-form and the Hodge *-operator associated with the Hermitian metric. In Sec. 2 we shall discuss and prove the principal results concerning harmonic forms on compact manifolds (real or complex), in particular, Hodge's harmonic representation for the de Rham groups, and special cases of Poincaré and Serre duality. In Sec. 3 we present the finite-dimensional representation theory for the Lie algebra sl(2, C), from which we derive the Lefschetz decomposition theorem for a Hermitian exterior algebra. In Sec. 4 we shall introduce the concept of a Kähler metric and give various examples of Kähler manifolds (manifolds equipped with a Kähler metric). In terms of a Hermitian metric we define the Laplacian operators associated with the operators d, ∂, and ∂ and show that when the metric is Kähler that the Laplacians are related in a simple way. We shall use this relationship in Sec. 5 to prove the Hodge decomposition theorem expressing the de Rham group as a direct sum of the Dolbeault groups (of the same total degree). In Sec. 6 we shall state and prove Hodge's generalization of the Riemann period relations for integrals of harmonic forms on a Kähler manifold. We shall then use the period relations and the Hodge decomposition to formulate the period mapping of Griffiths. In particular, we shall prove the Kodaira-Spencer upper semicontinuity theorem for the Hodge numbers on complex-analytic families of compact manifolds.