elements from Ω1,0 and q elements from Ω0,1. Symbolically, where there are p factors of Ω1,0 and q factors of Ω0,1. Just as with the two spaces of 1-forms, these are stable under holomorphic changes of coordinates, and so determine vector bundles. If Ek is the space of all complex differential forms of total degree k, then each element of Ek can be expressed in a unique way as a linear combination of elements from among the spaces Ωp,q with . More succinctly, there is a direct sum decomposition Because this direct sum decomposition is stable under holomorphic coordinate changes, it also determines a vector bundle decomposition. In particular, for each k and each p and q with , there is a canonical projection of vector bundles The Dolbeault operators The usual exterior derivative defines a mapping of sections via The exterior derivative does not in itself reflect the more rigid complex structure of the manifold. Using d and the projections defined in the previous subsection, it is possible to define the Dolbeault operators: To describe these operators in local coordinates, let where I and J are multi-indices. Then The following properties are seen to hold: These operators and their properties form the basis for Dolbeault cohomology and many aspects of Hodge theory. On a star-shaped domain of a complex manifold the Dolbeault operators have dual homotopy operators that result from splitting of the homotopy operator for . This is a content of the Poincare lemma on a complex manifold. The Poincaré lemma for and can be improved further to the local -lemma, which shows that every -exact complex differential form is actually -exact. On compact Kähler manifolds a global form of the local -lemma holds, known as the -lemma. It is a consequence of Hodge theory, and states that a complex differential form which is globally -exact (in other words, whose class in de Rham cohomology is zero) is globally -exact. Holomorphic forms For each p, a holomorphic p-form is a holomorphic section of the bundle Ωp,0. In local coordinates, then, a holomorphic p-form can be written in the form where the are holomorphic functions. Equivalently, and due to the independence of the complex conjugate, the (p, 0)-form α is holomorphic if and only if The sheaf of holomorphic p-forms is often written Ωp, although this can sometimes lead to confusion so many authors tend to adopt an alternative