Analytic Cohomology on Blown - Up Twistor Space

Abstract

A flat twistor space is a complex 3 - manifold having the property that every point of the manifold has a neighbourhood which is biholomorphic to a neighbourhood of a complex projective line in complex projective 3 - space. The Penrose transform provides an isomorphism between holomorphic structures on twistor spaces and certain field equations on (Riemannian or Lorentzian) space - times. The initial examples studied by Penrose were solutions to zero rest mass equations and, amongst these, the elementary states were of particular interest. These were elements of a sheaf cohomology group having a singularity on a particular complex projective line, with a codimension-2 structure similar, in some sense, to a Laurent series with a pole of finite order. In this work we extend this idea to the notion of codimension-2 poles for analytic cohomology classes on a punctured flat twistor space, by which we mean a general, compact, flat, twistor space with a finite number of non-intersecting complex, projective lines removed. We define a holomorphic line bundle on the blow-up of the compact flat twistor space along these lines and show that elements of the first cohomology group with coefficients in the line bundle, when restricted to the punctured twistor space, are cohomology classes with singularities on the removed lines which have precisely the kind of codimension - 2 structure which we define as codimension-2 poles. The dimension of this cohomology group on the blown-up manifold is then calculated for the twistor space of a compact, Riemannian, hyperbolic 4-manifold. The calculation uses the Hirzebruch - Riemann - Roch theorem to find the holomorphic Euler characteristic of the line bundle, (in chapter 3) together with vanishing theorems. In chapter 4 we show that it is sufficient to find vanishing theorems for the compact flat - twistor space. In chapter 5 we prove a number of vanishing theorems to be used. The technique uses the Penrose transform to convert the theorem to a vanishing theorem for spinor fields. These are then proved by using Penrose's Spinor calculus.