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Topology of conic bundles. (English) Zbl 0668.14013

Let \(P\to X\) be a bundle of conics on a smooth algebraic variety X which degenerates into a pair of distinct lines over a smooth irreducible divisor Y. The 2-sheeted covering of Y thus obtained defines an element \(\alpha \in H^ 1(Y,{\mathbb{Z}}/2)\). On the other hand we have a \({\mathbb{P}}^ 1\)-bundle on X-Y, and the topological obstruction to this \({\mathbb{P}}^ 1\)- bundle to be SL(2)-banal, that is, to be the projective bundle of a rank 2-topological vector bundle with trivial determinant, is an element \(\beta \in H^ 2(X-Y,{\mathbb{Z}}/2)\) (see § 1.1). Consider the Gysin map \(H^ 2(X-Y,{\mathbb{Z}}/2)\to H^ 1(Y,{\mathbb{Z}}/2)\), the composite of the coboundary map \(H^ 2(X-Y,{\mathbb{Z}}/2)\to H^ 3(X,X-Y,{\mathbb{Z}}/2)\) with the Thom isomorphism \(H^ 3(X,X-Y,{\mathbb{Z}}/2)\to H^ 1(Y,{\mathbb{Z}}/2)\), by definition.
Theorem 1. If the total space P of the conic bundle is a smooth algebraic variety then under the Gysin map, the image of the obstruction class \(\beta \in H^ 2(X-Y,{\mathbb{Z}}/2)\) is the cohomology class \(\alpha \in H^ 1(Y,{\mathbb{Z}}/2)\) defined by the 2-sheeted covering. In particular, if the 2-sheeted covering is not split, then the \({\mathbb{P}}\)-bundle on X-Y is not topologically SL(2)-banal.
Corollary 1. Under the hypothesis of theorem 1, the topological Brauer class \(\beta '\in H^ 3(X-Y,{\mathbb{Z}})\) of the \({\mathbb{P}}^ 1\)-bundle (see § 1.1) maps under the Gysin homomorphism to the Chern class \(\alpha '\in H^ 2(Y,{\mathbb{Z}})\) of the line bundle determined by the 2-sheeted cover of Y. In particular if the Chern class is not zero, then the \({\mathbb{P}}^ 1\)-bundle on X-Y is not topologically banal, that is, it is not associated to any rank-2 topological vector bundle.

MSC:

14F45 Topological properties in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
57R20 Characteristic classes and numbers in differential topology
55S35 Obstruction theory in algebraic topology
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