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The characteristic cycle and the singular support of a constructible sheaf

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A Correction to this article was published on 03 April 2019

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Abstract

We define the characteristic cycle of an étale sheaf as a cycle on the cotangent bundle of a smooth variety in positive characteristic using the singular support recently defined by Beilinson. We prove a formula à la Milnor for the total dimension of the space of vanishing cycles and an index formula computing the Euler–Poincaré characteristic, generalizing the Grothendieck–Ogg–Shafarevich formula to higher dimension. An essential ingredient of the construction and the proof is a partial generalization to higher dimension of the semi-continuity of the Swan conductor due to Deligne–Laumon. We prove the index formula by establishing certain functorial properties of characteristic cycles.

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Change history

  • 03 April 2019

    The first part of Proposition 7.4 and its proof in p. 670-671 should be corrected as follows. The author apologizes for the mistake.

  • 03 April 2019

    The first part of Proposition 7.4 and its proof in p. 670-671 should be corrected as follows. The author apologizes for the mistake.

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Authors and Affiliations

  1. Department of mathematical sciences, University of Tokyo, Tokyo, 153-8914, Japan

    Takeshi Saito

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  1. Takeshi Saito
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Saito, T. The characteristic cycle and the singular support of a constructible sheaf. Invent. math. 207, 597–695 (2017). https://doi.org/10.1007/s00222-016-0675-3

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