Hodge Decomposition

On a given complex manifold X, there are two natural cohomologies to consider. One is the de Rham Cohomology which can be defined on a general, possibly non complex, manifold. The second one is the Dolbeault cohomology which uses the complex structure. We’ll quickly go over the definitions of these cohomologies in order to set notation but for a precise discussion I recommend Huybrechts or Wells. If X is a manifold, we can define the de Rahm Complex to be the chain complex 0→ E(Ω)(X) d −→ E(Ω)(X) d −→ . . . Where Ω = T∗ denotes the cotangent vector bundle and Ωk is the alternating product Ωk = ΛkΩ. In general, we’ll use the notation E(E) to denote the sheaf of sections associated to a vector bundle E. The boundary operator is the usual differentiation operator. The de Rham cohomology of the manifold X is defined to be the cohomology of the de Rham chain complex: