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Title: Witt groups of complex varieties
Author: Zibrowius, Marcus
ISNI:       0000 0004 2708 382X
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2011
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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 resultapplies in particular to projective homogeneous spaces. By extending knowncomputations 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.
Supervisor: Totaro, Burt Sponsor: ; Cambridge Philosophical Society
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
Keywords: Witt groups ; KO-theory