On the failure of the Poincaré lemma for $\overline\partial_M$∂−M. II

The purpose of this paper is to repair some inaccuracies in the formulation of the main result of [Part I, C. D. Hill and M. Nacinovich, Math. Ann. 324 (2002), no. 2, 213--224; MR1933856 (2003g:32056)]. But this is not simply an erratum. Instead this article contains now two different results. The first one has the same conclusion as the main result from above, namely the fact that certain cohomology groups are infinite-dimensional; however, this requires a slightly stronger assumption than before. Also, the proof is now much more complicated and uses the CR structure of the characteristic bundle. This is an interesting technique which might be of independent interest. The authors also give a couple of natural examples which show that the stronger assumption is actually satisfied in many important cases. The second result uses the same assumptions as in the main theorem of the other article but has a weaker conclusion, which is nevertheless quite interesting because it involves some invariant which measures the relative rate of shrinking. To make this a little bit clearer, assume that at some fixed point $x_0$x0 the local Poincaré lemma holds. After fixing a Riemannian metric and giving a $\overline\partial_M$∂−M-closed form in the ball $B(x_0,r)$B(x0,r), there must then be a solution $u$u for $\overline\partial_M u=f$∂−Mu=f in some smaller ball $B(x_0,r')$B(x0,r′). One of the consequences of this second main result is then that asymptotically (for $r\to0$r→0) there must be some relation of the form $r'\le Cr^{3/2}$r′≤Cr3/2, which gives some estimate for the relative rate of shrinking.