The Dolbeault Complex in Infinite Dimensions I

Infinite dimensional complex analysis was a popular subject in the sixties and seventies, but in the last fifteen years enjoyed much less attention. Oddly enough, this very same period began to see the emergence of examples of infinite dimensional complex manifolds, in mathematical physics, representation theory, and geometry. Thus it appears to be a worthwhile undertaking to revisit infinite dimensional complex analysis, and, in particular, to clarify fundamental properties of infinite dimensional complex manifolds. The most fundamental questions at this point seem to be related to the solvability of the inhomogeneous Cauchy-Riemann, or ∂̄, equations, and more generally to the study of the Dolbeault complex. (As an example, in [Le3] we use a result on the ∂̄ equation to describe the Virasoro group as an infinite dimensional complex manifold.) Up to now precious little has been known about the solvability of the infinite dimensional ∂̄ equation; the available results pertain to solving the equation in domains in or over locally convex topological vector spaces, and almost exclusively on the level of (0, 1)-forms. Postponing the precise definitions to sections 1, 2, below we discuss those results that are of immediate relevance to this paper; see [D2] for more. In [Li] Ligocka observes that Hörmander’s proof of Ehrenpreis’ theorem (see [E], [Hor]) can be extended to infinite dimensions to solve the equation