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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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