Hodge-theoretic Obstruction to Existence of Quaternion Algebras
نویسنده
چکیده
The subject of this paper is the Brauer group of a nonsingular complex projective variety. More specifically, we study the question of whether a 2-torsion element of the cohomological Brauer group is representable by a quaternion algebra over the generic point. Using intersection theory – on schemes and on algebraic stacks – we are able to describe an obstruction to such a representation, and therefore, to give examples of varieties with 2-torsion classes that are not representable by quaternion algebras. Let X be a nonsingular complex algebraic variety. It is a well-known consequence of the Grauert–Remmert theorem, plus GAGA, that topological 2sheeted covers, classified byH(X,Z/2), correspond exactly to algebraic degree 2 unramified covers. There is similarly an algebraic object that we can associate to an element α ∈ H(X,Z/2) – a kind of algebraic stack overX known as a gerbe. The element α determines a 2-torsion element of Br(X) = H(X,Gm), and if this element is represented by a quaternion algebra then the quaternion algebra can be identified with the (descent to X of the) endomorphism algebra of a rank 2 vector bundle on the gerbe. The second Chern class of this bundle (or rather, a specific multiple of this) is an algebraic class on X, and if X is projective this leads to a Hodge-theoretic obstruction to representing the element of Br(X) by a quaternion algebra. We turn now to the statement of our main result.
منابع مشابه
Hodge-theoretic obstruction to the existence of quaternion algebras
This paper gives a necessary criterion in terms of Hodge theory for representability by quaternion algebras of certain 2-torsion classes in the unramified Brauer group of a complex function field. This criterion is used to give examples of threefolds with unramified Brauer group elements which are the classes of biquaternion division algebras. DOI: https://doi.org/10.1112/S0024609302001546 Post...
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