The Involutive Structure on the Blow-up of R in C Michael Eastwood and C. Robin Graham

Many interesting geometric structures can be defined by specifying a smooth subbundle V of the complexified tangent bundle of the underlying smooth manifold, subject to an integrability condition. Examples include foliations, complex structures, and CR structures. In general such structures are called involutive, or formally integrable, and their study has been the starting point of far-reaching general investigations (see, for example, [T], [CT], and [HJ]). However, most naturally occurring examples, for instance all those mentioned above, have the property that V ∩V has constant rank. In recent work on integral geometry ([BEGM], [E], and [BaE]), natural examples of involutive structures have arisen for which the rank of V ∩ V changes along a hypersurface. For these examples the underlying manifold is the real blow-up of RP in CP for various n. In this article we consider these new involutive structures from an analytic point of view. Locally we may as well consider the blow-up of R in C. This blowup B is a smooth real 2n-manifold with a distinguished hypersurface Σ, the inverse image of R under the blow-down map b : B → C. The blow-up is defined precisely so that the image under b of a neighborhood of a point of Σ is a localized wedge in C, i.e. the product of an open set in R with a localized cone in iR. Away from Σ, b is a diffeomorphism and the involutive structure is just the lift of the complex structure on C \ R; the bundle V of (0, 1) vectors extends smoothly across Σ but dim(V ∩ V) = n− 1 there. A solution of an involutive structure is a function or distribution annihilated by all sections of V. An important analytic problem is to understand the regularity of solutions of a given involutive structure. We show that the involutive structure on the blow-up of R in C