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

The Original Article was published on 15 July 2016

## 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.)$$\square$$

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Correspondence to Takeshi Saito.

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