100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

482 R. Penroseexamples of certain widespread classes of Stein manifolds, so that we cansee how easy it is to ensure that we do have a Stein covering. In the firstplace, any region of C n that is delineated by the vanishing of a family of(holomorphic) polynomial equations is Stein (an “affine variety”). Secondly,any region of CP n obtainable by excluding the zero locus of a single homogeneouspolynomial will be Stein. Thirdly, any region of C n which has asmooth boundary satisfying a certain “convexity” condition — referred toas holomorphic pseudo-convexity (specified by a positive-definiteness criterionon a certain Hermitian form defined by the equation of the boundary,called the Levy form r ). This last property tells us that we can always findsmall Stein-manifold open neighbourhoods of any point of a complex manifold(e.g. a small spherical ball).An important additional property is that the intersection of Stein manifoldsis again always Stein. We can use this fact to compare 1st cohomologyelements defined with respect to two different Stein coverings {U i } and{V I }. All we need to do is find the common refinement of the two coverings,which is a covering {W iJ } each of whose members is the intersection of oneset from each of the two coverings:W iJ = U i ∩ V J(where we may ignore the empty intersections) and this will also be a Steincovering. Thus, if we have cochains {f ij } with respect to {U i } and {g IJ }with respect to {V I }, then we can form the sum “f +g”, with respect to thecommon refinement {W iJ } as the cochain {f ij | IJ + g IJ | ij }, where f ij | IJ isf ij on U i ∩ U j , restricted to its the intersection with V I ∩ V J , and g IJ | ij isg IJ on V I ∩ V J , restricted to its the intersection with U i ∩ U j . It is a directmatter to show that this leads to a sum of the corresponding cohomologyelements. Hence, the notion of a 1st cohomology element of PT + is intrinsicto PT + , and does not really care about how we have chosen to cover it witha collection of open (Stein) sets. s This now gets us out of our difficulty withthe twistor representation of massless fields. But it leads us to a greaterdegree of sophistication in the mathematics needed for the description ofphysical fields than we might have expected.One aspect of this sophistication is that our twistor description is alreadya non-local one, even for the description of certain things which, inordinary space-time, are perfectly local, like a physical field. But we shouldbear in mind that this twistor description is primarily for wavefunctions,and we recall from §1 that the wavefunction of a single particle is alreadysomething with puzzling non-local features (since the detection of the par-