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Correction to: The characteristic cycle and the singular support of a constructible sheaf

  • Takeshi SaitoEmail author
Correction
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1 Correction to: Invent. math. (2017) 207:597–695  https://doi.org/10.1007/s00222-016-0675-3

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

Proposition 7.4

(Beilinson) Let \({{\mathbf {P}}}={{\mathbf {P}}}^n\) be a projective space, and let \({{\mathbf {P}}}^\vee \) be the dual projective space. Let \({{{\mathcal {G}}}}\) be a constructible complex of \(\Lambda \)-modules on \({{\mathbf {P}}}^\vee \), and let \({{{\mathcal {F}}}}\) denote the naive inverse Radon transform \(R{{{\varvec{p}}}}_*{{{\varvec{p}}}}^{\vee *}{{\mathcal {G}}}\). Let \(C^\vee \subset T^*{{\mathbf {P}}}^\vee \) be a closed conical subset such that every irreducible component is of dimension n, and let \(C={{{\varvec{p}}}}_{\circ } {{{\varvec{p}}}}^{\vee \circ }C^\vee \subset T^*{{\mathbf {P}}}\). Assume that \({{\mathcal {G}}}\) is micro-supported on \(C^\vee \subset T^*{\mathbf {P}}^\vee \).

Let X be a smooth subscheme of \({{\mathbf {P}}}\), and assume that the immersion \(h:X\rightarrow {{\mathbf {P}}}\) is properly C-transversal. Using the notation in (3.10), let \(p:X\times _{{\mathbf {P}}}Q \rightarrow X\) be the projection and \(p^\vee :X\times _{{\mathbf {P}}}Q \rightarrow {{\mathbf {P}}}^\vee \) be the restriction of \({{{\varvec{p}}}}^\vee \).

1. We have
$$\begin{aligned} {{\mathbf {P}}}(CC Rp_*p^{\vee *}{{\mathcal {G}}})= {{\mathbf {P}}}(p_!CC p^{\vee *}{{\mathcal {G}}}) = {{\mathbf {P}}}(p_!p^{\vee !}CC {{\mathcal {G}}}). \end{aligned}$$
(7.4)
In particular, for \(X={{\mathbf {P}}}\), we have
$$\begin{aligned} {{\mathbf {P}}}(CC {{\mathcal {F}}})= {{\mathbf {P}}}({{{\varvec{p}}}}_!CC {{{\varvec{p}}}}^{\vee *}{{\mathcal {G}}}) = {{\mathbf {P}}}({{{\varvec{p}}}}_!{{{\varvec{p}}}}^{\vee !}CC \mathcal{G}). \end{aligned}$$
(7.2)
2. We have
$$\begin{aligned} CC h^*{{\mathcal {F}}}= h^!CC{{\mathcal {F}}}. \end{aligned}$$
(7.3)

Proof

1. First, we prove the second equality in (7.4) for properly C-transversal immersion \(h:X\rightarrow {{\mathbf {P}}}\). By Corollary 3.13.2, \(p^\vee :X\times _{{\mathbf {P}}}Q\rightarrow {\mathbf {P}}^\vee \) is \(C^\vee \)-transversal and hence \(p^*{{\mathcal {G}}}\) is micro-supported on \(p^{\vee \circ }C^\vee \). Since \(p^\vee :X\times _{{\mathbf {P}}}Q\rightarrow {{\mathbf {P}}}^\vee \) is smooth outside \(\Delta _X\subset X\times _{{\mathbf {P}}}Q\) (3.11), we have \(CC p^{\vee *}{{\mathcal {G}}} = p^{\vee !}CC {{\mathcal {G}}}\) outside \(\Delta _X\subset X\times _{{\mathbf {P}}}Q\) by Proposition 5.17. By the assumption that \(h:X\rightarrow {{\mathbf {P}}}\) is C-transversal, the pair \((p^\vee , p)\) is \(C^\vee \)-transversal on a neighborhood of \(\Delta _X\subset X\times _{{\mathbf {P}}}Q\) by Corollary 3.13.1 (1)\(\Rightarrow \)(2). Hence, we have the second equality in (7.4).

We prove the first equality in (7.4). We may assume that \(k \, \cdots \)

(We keep from the 3rd line of p. 671 to the displayed formula as it is.)
$$\begin{aligned} \phi _u(Rp_*p^{\vee *}{{\mathcal {G}}},f) \rightarrow R\Gamma (Q\times _Xu, \phi (p^{\vee *}{{\mathcal {G}}},fp)) \rightarrow \bigoplus _v \phi _v(p^{\vee *}\mathcal{G},fp). \end{aligned}$$
For equalities (7.2), it suffices to take \(X={\mathbf {P}}\).

(From the beginning of the proof of 2. on, no change is necessary.)\(\square \)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of TokyoTokyoJapan

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