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2007 Even-relative-dimensional vanishing cycles in bivariant intersection theory
Hiroshi Saito
Nagoya Math. J. 187: 49-73 (2007).

Abstract

For a smooth variety proper over a curve having a fibre with isolated ordinary quadratic singularities, it is well-known that we have the vanishing cycles associated to the singularities in the étale cohomology of the geometric generic fibre. The base-change by a double cover of the base curve ramified at the image of the singular fibre has singularities corresponding to the singularities in the fibre. In this note, we show that in the even relative-dimensional case, there exist elements of the bivariant Chow group of the base-change with supports in the singularities and hence their images in the bivariant Chow group with supports in the special fibre and that the usual cohomological vanishing cycles are obtained as their images by a natural map, a kind of "cycle map" so that the elements in the bivariant Chow groups can be regarded as the vanishing cycles. The bivariant Chow group with supports in the special fibre has a ring structure and the natural map is a ring homomorphism to the cohomology ring of the geometric generic fibre. Also discussed is the relation of the bivariant Chow group with supports in the special fibre to the specialization map of Chow groups.

Citation

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Hiroshi Saito. "Even-relative-dimensional vanishing cycles in bivariant intersection theory." Nagoya Math. J. 187 49 - 73, 2007.

Information

Published: 2007
First available in Project Euclid: 4 September 2007

zbMATH: 1129.14012
MathSciNet: MR2354555

Subjects:
Primary: 14C17 , 14F17

Rights: Copyright © 2007 Editorial Board, Nagoya Mathematical Journal

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