The Hodge Theorem for Riemannian Manifolds

Abstract

Thus far, our approach has been primarily algebraic or topological.We are going to need a basic analytic result, namely the Hodge theorem. This says that on a compact oriented manifold equipped with a metric, every de Rham cohomology class has a unique “smallest” element, called its harmonic representative. When combined with the Kähler identities in later chapters, this will have strong consequences for complex algebraic geometry. The original proof is due to Hodge [64] with a correction by Weyl [121]. Different proofs were given by Bidal and de Rham [12] and Kodaira [72] soon there after. Standard accounts of these results, with the necessary details, can be found in the books of Griffiths and Harris [49], Warner [117], andWells [120]. These books follow a similar approach of first establishing a weak Hilbert space version of the Hodge theorem, and then applying some regularity results from elliptic PDE theory to deduce the stronger statement. We will depart slightly from these treatments by outlining the heat equation method of Milgram and Rosenbloom [84]. This is an elegant and comparatively elementary approach to the Hodge theorem. As a warmup, we will do a combinatorial version that requires nothing more than linear algebra.