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Hodge-theoretic obstruction to existence of quaternion algebras

Andrew Kresch
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Abstract

The class in the Brauer group of a quaternion algebra over a field is 2-torsion. We study the following question: Which 2-torsion elements of the Brauer group of a complex function field are representable by quaternion algebras? Using intersection theory to show that a certain cohomology class (on a smooth projective model) is the class of an algebraic cycle, we arrive at an obstruction, defined on a subgroup of the 2-torsion of the Brauer group, to representability by quaternion algebras. For the function fields of some complex threefolds, the obstruction map is computed and found to be nontrivial.

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Andrew Kresch. 2000-09-12. Hodge-theoretic obstruction to existence of quaternion algebras. https://arxiv.org/abs/math/0009115

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