  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.
