Differential Forms and Hermitian Geometry
In this chapter we collect the technical tools from the calculus of differential forms and from complex differential geometry which will be needed in the following chapters. Section 1 deals with differentiable manifolds; the principal goal here is a thorough understanding of Stokes’ Theorem in the language of differential forms. In §2 we discuss the additional structures which arise when the manifold under consideration is complex. The main topics here are the natural intrinsic complex structure on the (real) tangent space of a complex manifold M, the direct sum decomposition of the algebra of complex valued differential forms into forms of type (p, q), 0 ≤ p, q ≤ dimℂ M, and the Cauchy-Riemann complex with its associated ∂̅-cohomology groups. In §3 we discuss the elementary aspects of Riemannian geometry in ℂ n in complex form. Of major importance for our purposes are the inner product of differential forms defined by integration over regions in ℂ n , the Hodge *- operator, which allows us to freely go back and forth between the geometric inner product and the algebraic wedge product of forms, the various formulas for integration by parts, and the natural differential operators associated to the Cauchy-Riemann operator, i.e., the (formal) adjoint ϑof ∂̅ and the complex Laplacian \( \square = \vartheta \overline \partial + \overline \partial \vartheta . \). In this paragraph, which is more computational than the preceding ones, we consider only the case of ℂ n rather than general Hermitian manifolds; not only does this simplify matters quite a bit, but it allows us to state certain basic formulas in exact form without having to introduce the numerous error terms which occur in the general setting.
KeywordsCompact Support Differential Form Complex Manifold Complex Vector Space Exterior Derivative
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