Kresch, Andrew Hodge-theoretic obstruction to the existence of quaternion algebras. (English) Zbl 1028.16010 Bull. Lond. Math. Soc. 35, No. 1, 109-116 (2003). 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)]. Reviewer: Ivan D.Chipchakov (Sofia) Cited in 1 ReviewCited in 3 Documents MSC: 16K50 Brauer groups (algebraic aspects) 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14F22 Brauer groups of schemes Keywords:projective complex varieties; cohomology groups; Brauer groups; function fields; quaternion algebras; biquaternion division algebras; Azumaya algebras; equivariant cohomology Citations:Zbl 0940.14003 PDFBibTeX XMLCite \textit{A. Kresch}, Bull. Lond. Math. Soc. 35, No. 1, 109--116 (2003; Zbl 1028.16010) Full Text: DOI arXiv