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Hodge-theoretic obstruction to the existence of quaternion algebras. (English) Zbl 1028.16010

Let \(X\) be a nonsingular projective complex variety, and \(\alpha\) an element of the cohomology group \(H^2(X,\mathbb{Z}/2)\) with the following properties: (i) the image of \(\alpha\) in the Brauer group \(\text{Br}(\mathbb{C}(X))\) of the function field \(\mathbb{C}(X)\) is represented by a quaternion \(\mathbb{C}(X)\)-algebra; (ii) there exists a preimage \(\alpha_0\in H^2(X,\mathbb{Z})\) under the natural map \(H^2(X,\mathbb{Z})\to H^2(X,\mathbb{Z}/2)\). The paper under review shows that then there exists an algebraic cycle on \(X\) of codimension \(2\), whose class in cohomology is equal to \(4(\alpha_0^2+2\varepsilon)\), for some \(\varepsilon\in H^4(X,\mathbb{Z})\). The result obtained gives a general Hodge-theoretic obstruction (stated as Corollary 1) to the presentability of certain \(2\)-torsion elements of the unramified Brauer group of \(\mathbb{C}(X)\) by similarity classes of quaternion algebras. This, applied to the special case where \(X\) is a surface with \(h^{2,0}(X)\neq\{0\}\), implies the existence of a smooth conic \(V\to X\), whose function field \(\mathbb{C}(V)\) admits a biquaternion division algebra \(B\) that is a restriction of a globally defined sheaf of Azumaya algebras on \(V\). The proof uses a cycle map to equivariant cohomology [see D. Edidin and W. Graham, Invent. Math. 131, No. 3, 595-634 (1998; Zbl 0940.14003)].

MSC:

16K50 Brauer groups (algebraic aspects)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F22 Brauer groups of schemes

Citations:

Zbl 0940.14003
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