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Perverse sheaves

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Sheaves on Manifolds

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 292))

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Summary

Although perverse sheaves have a short history, they play an important role in various branches of mathematics, such as algebraic geometry or group representation. This theory is now well understood, but it is difficult to find in the literature a systematic treatment of it in the analytical case.

In this chapter, we define perverse sheaves on a real analytic manifold X and show that they form an abelian subcategory of D ℝ−c b(X). Then we study the case where X is a complex manifold. We prove that perversity is a microlocal property, that is, an object of D ℂ−c b(X) is perverse if and only if it is pure of shift — dim X at generic points of its micro-support. Because Morse theory is well-adapted to the microlocal characterization, we can prove a vanishing theorm for perverse sheaves on Stein manifolds. Then we prove that perversity is preserved by various operations and in particular by specialization, microlocalization and Fourier-Sato transformation. To achieve this program we need some complements of homological algebra. In section 1, we explain the notion of t-structures which permits to construct abelian subcategories in triangulated categories.

References are made to Beilinson-Bernstein-Deligne [1] and Goresky-MacPherson [2, 3], (cf. also Borel et al. [1] and Gelfand-Manin [1]).

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© 1990 Springer-Verlag Berlin Heidelberg

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Kashiwara, M., Schapira, P. (1990). Perverse sheaves. In: Sheaves on Manifolds. Grundlehren der mathematischen Wissenschaften, vol 292. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02661-8_12

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  • DOI: https://doi.org/10.1007/978-3-662-02661-8_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08082-1

  • Online ISBN: 978-3-662-02661-8

  • eBook Packages: Springer Book Archive

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