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A compact, oriented Riemannian 4-manifold whose intersection form is not negative definite has a non-trivial closed, self-dual 2-form. Such a form defines a symplectic structure away from its zero set, and the metric can be used to defines a compatible almost complex structure. Although this almost complex structure is singular across the zero set of the given 2-form, one can none-the-less study the associated pseudoholomorphic subvarieties in the compliment of the zero set. In this regard, it is natural to restrict attention to those which have the following three properties: First, they are closed subsets of the compliment of the zero set of the given self-dual 2-form. Second, they are submanifolds except at a set which can be empty, but is at worst countable and non-accumulating. Third, the given self-dual 2-form has finite integral over the sub variety. With regard to the second point, remark that the condition of pseudoholomorphicity is no more nor less than the requirement that the almost complex structure preserve the tangent space at all manifold points. With regard to the third point, remember that a pseudoholomorphic variety is naturally oriented at its manifold points by the restriction of the symplectic form.

Moduli spaces and Fredholm theory for pseudoholomorphic subvarieties associated to self-dual, harmonic 2-forms