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.

Date of Award | 1994 |
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Original language | English |
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Awarding Institution | |
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Analytic Cohomology on Blown - Up Twistor Space

HORAN, R. E. (Author). 1994

Student thesis: PhD