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The Decomposition Package and Applications

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Intersection Homology & Perverse Sheaves

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 281))

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Abstract

This chapter is devoted to the “decomposition package”, consisting of Lefschetz-type results for perverse sheaves and intersection homology (Sections 9.1 and 9.2), as well as the BBDG decomposition theorem (Section 9.3). A sample of the numerous applications of the decomposition package is presented in Section 9.4.

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Notes

  1. 1.

    f : X → Y  is projective if it can be factored as \(X \overset {i}{\hookrightarrow }Y \times {\mathbb C} P^N \overset {p}{\to } Y\) for some N, with i a closed embedding, and p a projection.

  2. 2.

    A very ample line bundle is one with enough global sections to set up an embedding of its base variety into projective space. An ample line bundle is one such that some positive power is very ample. For a morphism f : X → Y , an f-ample line bundle on X is a line bundle that is ample on every fiber of f.

  3. 3.

    An example of such an f-ample line bundle on X can be obtained as follows: pull back the hyperplane bundle from \({\mathbb C} P^N\) to \(Y \times {\mathbb C} P^N\), and then restrict to X.

  4. 4.

    A more general decomposition theorem (for semi-simple coefficients) has been obtained by Mochizuki [183, 182] (with substantial contributions of Sabbah [204]), in relation to a conjecture of Kashiwara [121].

  5. 5.

    A local system \({\mathcal L}\) on Y  is semi-simple if every local subsystem \({\mathcal L}'\) of \({\mathcal L}\) admits a complement, i.e., a local subsystem \({\mathcal L}''\) of \({\mathcal L}\) such that \({\mathcal L} \simeq {\mathcal L}' \oplus {\mathcal L}''\).

  6. 6.

    The original formulations of the Dowling–Wilson and Rota conjectures concern matroids. The proof by Huh–Wang is applicable only for matroids realizable over some field.

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  1. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA

    Laurenţiu G. Maxim

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Maxim, L.G. (2019). The Decomposition Package and Applications. In: Intersection Homology & Perverse Sheaves. Graduate Texts in Mathematics, vol 281. Springer, Cham. https://doi.org/10.1007/978-3-030-27644-7_9

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