Witt groups of complex varieties
University of Cambridge
Department of Pure Mathematics and Mathematical Statistics
Doctor of Philosophy (PhD)
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Zibrowius, M. (2011). Witt groups of complex varieties (Doctoral thesis). https://doi.org/10.17863/CAM.16217
The thesis Witt Groups of Complex Varieties studies and compares two related cohomology theories that arise in the areas of algebraic geometry and topology: the algebraic theory of Witt groups, and real topological K-theory. Specifically, we introduce comparison maps from the Grothendieck-Witt and Witt groups of a smooth complex variety to the KO-groups of the underlying topological space and analyse their behaviour. We focus on two particularly favourable situations. Firstly, we explicitly compute the Witt groups of smooth complex curves and surfaces. Using the theory of Stiefel-Whitney classes, we obtain a satisfactory description of the comparison maps in these low-dimensional cases. Secondly, we show that the comparison maps are isomorphisms for smooth cellular varieties. This result applies in particular to projective homogeneous spaces. By extending known computations in topology, we obtain an additive description of the Witt groups of all projective homogeneous varieties that fall within the class of hermitian symmetric spaces.
Witt groups, KO-theory
This work was supported by the Engineering and Physical Sciences Research Council [grant number LFAG/067] and by the Cambridge Philosophical Society.
This record's DOI: https://doi.org/10.17863/CAM.16217