# Comparison of the two notions of characteristic cycles

Ankit Rai

###### Abstract

Given a constructible sheaf $F$ on a complex manifold, Kashiwara-Schapira
defined the notion of singular support and characteristic cycle of $F$. On the
other hand for a Zariski constructible étale sheaf $F$ on an algebraic variety
$X$, Beilinson defined the notion of singular support of $F$ and Saito defined
the notion of characteristic cycle of $F$. In this article we compare these
notions and prove that they agree in a suitable sense. In the appendix we
discuss extension of the notions of singular support and characteristic cycles
developed by Umezaki-Yang-Zhao and Barrett.

###### Contents

  1. 1 Introduction
  2. 2 Basics
    1. 2.1 Assumptions in this article
    2. 2.2 Constructible sheaves
    3. 2.3 Six functor formalism
    4. 2.4 Comparison of analytic and étale topos
    5. 2.5 Vanishing cycles
    6. 2.6 Cycle class
    7. 2.7 Closed conical subsets of cotangent bundle
  3. 3 Singular supports and characteristic cycles
    1. 3.1 Analytic
    2. 3.2 Algebraic
    3. 3.3 Three lemmas
    4. 3.4 Summary of properties of singular supports and characteristic cycles
  4. 4 Main theorem and proof
  5. A Coefficients in $\mathrm{CC^{KS}}$ and $\mathrm{CC^{S}}$
  6. B Characteristic cycles for $\mathbb{Q}_{\ell}$-sheaves

##  1 Introduction

The comparison theorems between the classical and the modern geometry are of
interest for multiple reasons. On one hand it allows the use of powerful
algebro-geometric methods to prove theorems in classical geometry, and on the
other hand it provides a much needed testing ground for the intuitions behind
developing new notions in modern geometry.

In the next paragraph we discuss the necessary notation to state the main
theorem of this article. Let $X$ be a separated smooth scheme of finite type
over the field of complex numbers $\mathbb{C}$ and $X^{an}$ denotes the
associated complex analytic space $X(\mathbb{C})$. Let $\Lambda$ be a finite
local ring and $\mathrm{D}^{b}_{ctf}(X^{an},\Lambda)$ (respectively
$\mathrm{D}^{b}_{ctf}(X_{\text{\'{e}t}},\Lambda)$) denote the bounded derived
category of tor-finite complexes $F$ of sheaves (respectively the étale
sheaves) with coefficients in finite $\Lambda$-modules, such that the
cohomology sheaves $\mathrm{H}^{i}(F)$ are Zariski constructible and are of
finite tor-dimension. Kashiwara-Schapira refined111Let $\pi:T^{*}X\rightarrow
X$ be the cotangent bundle then $\pi(\mathrm{SS^{KS}}(F))=$ the support of
$F\in\mathrm{D}^{b}_{ctf}(X,\Lambda)$ and this justifies the claim that
singular support refines the notion of the support of a sheaf. the notion of
the support of $F\in\mathrm{D}^{b}_{ctf}(X^{an},\Lambda)$ to that of the
singular support. The singular support of
$F\in\mathrm{D}^{b}_{ctf}(X^{an},\Lambda)$ is a closed complex analytic
conical subset of $(T^{*}X)(\mathbb{C})$ which we denote by
$\mathrm{SS^{KS}}(F)$. In [Bei17], Beilinson defines the notion of the
singular support of $F\in\mathrm{D}^{b}_{ctf}(X_{\text{\'{e}t}},\Lambda)$ as a
Zariski closed conical subset of $T^{*}X$ denoted in this article by
$\mathrm{SS^{B}}(F)$. The notion of the singular support in the sense of
Beilinson has been extended to any
$F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda)$ for
$\Lambda\in\\{\mathbb{Z}_{\ell},\mathbb{Q}_{\ell}\\}$ (See [UYZ20, §5.15, Thm.
5.17]). We denote this by $\mathrm{SS^{BUYZ}}(F)$. In this article, we prove
the following

######  Theorem 1.1.

Let $F\in\mathrm{D}^{b}_{c}(X_{\textup{\'{e}t}},\Lambda)$, where
$\Lambda\in\\{\mathbb{Z}/\ell^{n}\mathbb{Z},\mathbb{Z}_{\ell},\mathbb{Q}_{\ell}\\}$
and let $\overline{\mathrm{SS^{KS}}(F)}^{Zar}$ denote the Zariski closure of
$\mathrm{SS^{KS}}(F)$ with the reduced induced subscheme structure. Then
$\overline{\mathrm{SS^{KS}}(F)}^{Zar}=\mathrm{SS^{BUYZ}}(F)$.

The notion of the singular support can be upgraded to a cycle supported on the
singular support. This has been achieved by Kashiwara-Schapira in the analytic
setting, and by Saito in the algebraic setting. The characteristic cycle
defined by Kashiwara-Schapira is defined only for
$F\in\mathrm{D}^{b}_{c}(X^{an},\Lambda)$, where $\Lambda$ is a field of
characteristic 0. We denote this cycle by $\mathrm{CC^{KS}}(F)$. The
characteristic cycle defined by Saito, a priori, only makes sense for
$F\in\mathrm{D}^{b}_{ctf}(X_{\text{\'{e}t}},\Lambda)$, where $\Lambda$ is a
finite local ring. This notion has also been extended to any
$F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda)$ for
$\Lambda\in\\{\mathbb{Z}_{\ell},\mathbb{Q}_{\ell}\\}$ (See [UYZ20, §5.15]). We
denote it by $\mathrm{CC^{SUYZ}}(F)$. The two notions are compared in the
following

######  Theorem 1.2.

For any $F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})$, suppose
$\mathrm{CC^{KS}}(F)=\underset{i}{\sum}m_{i}[X_{i}]$ as a Lagrangian cycle
(see equation (2)), where $X_{i}$ are irreducible components of
$\mathrm{SS^{KS}}(F)$ then
$\mathrm{CC^{S}}(F)=\underset{i}{\sum}m_{i}[\overline{X_{i}}^{Zar}]$ as
cycles.

### Organization of the paper and the strategy of the proof

In §2 we lay down the assumptions made in this article and discuss some basic
terminology which will be needed in the later sections. In §3.1 we recall the
definition of the singular support and the characteristic cycle of a complex
of sheaves on a complex algebraic variety. We have taken the liberty of
stating a theorem of Kashiwara-Schapira as the definition of the singular
support. Section 3.2 recalls the definition of the weak singular support given
by Beilinson and the definition of the characteristic cycle given by Saito.
The extension of these notions to complexes of sheaves with coefficients in
$\mathbb{Z}_{\ell}$ or $\mathbb{Q}_{\ell}$ is also discussed there.

A few (perhaps well known) statements whose proofs we were unable to find in
the literature are needed in the sequel. The statements along with their
proofs are stated as Lemma 3.5, 3.6 and 3.8 in §3.3. In §3.4 we summarise
various properties of the singular supports and of the characteristic cycles
that are to be used in the proofs of Theorems 1.1 and 1.2. The proofs of
Theorem 1.1 and Theorem 1.2 are completed in §4. The strategy of the proof of
these theorems is to recursively use properties 1 and 2 from §3.4 in order to
reduce to the case of irreducible perverse sheaves. Further Hironaka’s
resolution of singularities and the decomposition theorem with respect to
support for perverse sheaves (Step 3 of proof of Theorem 4.1) are used to
reduce to the case of irreducible perverse sheaves of the form $j_{!}L$, where
$L$ is a simple local system on $U$, and $j:U\hookrightarrow X$ is an open
subset whose complement is a strict normal crossing divisor. The theorem is
achieved thereafter by an explicit computation (See §4, Step 4).

Finally Appendix A we discusses the dependence of the characteristic cycle
$\mathrm{CC^{KS}}$ on the coefficient system and Appendix B compares the two
ways of extending of the notion of singular support developed by Umezaki-Yang-
Zhao and Barrett.

### Acknowledgements

The author thanks K. V. Shuddhodan for suggesting the question and for
numerous discussions. Thanks are due to Najmuddin Fakhruddin for directing the
author to relevant reference in Fulton and for the accompanied explanation
which forms the proof of Lemma 2.3.

##  2 Basics

###  2.1 Assumptions in this article

In this article an algebraic variety will mean a separated smooth scheme of
finite type over the field $\mathbb{C}$ of complex numbers. Any morphism of
algebraic varieties that appear will automatically be of finite type. By
$X^{an}$ we will mean the complex points of an algebraic variety, and any
morphism will be assumed to be one induced by a morphism of algebraic
varieties. To keep matters simple and the exposition lucid we restrict
ourselves to the case of
$\Lambda\in\\{\mathbb{Z}/\ell^{n}\mathbb{Z},\mathbb{Z}_{\ell},\mathbb{Q}_{\ell}\\}$.

###  2.2 Constructible sheaves

Notion of a constructible sheaf depends on the stratification of a space and
the fundamental groups of these strata. Since the latter two notions differ in
the settings of algebraic varieties and complex analytic varieties, the notion
of a constructible sheaf differs as well. We recall the basics of
constructible sheaves which also serves the purpose of itroducing the relevant
notation.

Let $\Lambda$ be a noetherian ring. A sheaf $F$ valued in finitely generated
$\Lambda$-modules on $X^{an}$ is called constructible if there exists finitely
many Zariski locally closed subsets $X^{an}_{i}\subset X^{an}$ such that, (1)
$X^{an}=\underset{i}{\sqcup}X^{an}_{i}$, (2) the Zariski closure
$\overline{X^{an}_{i}}$ of each $X^{an}_{i}$ is a union of finitely many of
the $X^{an}_{j}$, and (3) $F|_{X^{an}_{i}}$ is a local system valued in finite
$\Lambda$-modules. We can furthermore choose $X^{an}_{i}$ to be complex
submanifolds of $X^{an}$(See [KS, Prop. 8.5.4]). Let
$\mathrm{D}^{b}(X^{an},\Lambda)$ be the bounded derived category of sheaves on
$X^{an}$ valued in $\Lambda$-modules. The full subcategory
$\mathrm{D}^{b}_{ctf}(X^{an},\Lambda)\subset\mathrm{D}^{b}(X^{an},\Lambda)$ is
then defined by declaring that
$F\in\mathrm{Ob}(\mathrm{D}^{b}(X^{an},\Lambda))$ belongs to
$\mathrm{D}^{b}_{ctf}(X^{an},\Lambda)$ if $\mathrm{H}^{i}(F)$ is a
constructible sheaf, and $\mathrm{H}^{q}(F\otimes^{L}Q)\neq 0$ for finitely
many $q$ and any finite $\Lambda$-module $Q$. The assumptions on the ring
$\Lambda$ ensure that the conditions on $F$ in order to belong to the
subcategory $\mathrm{D}^{b}_{ctf}(X^{an},\Lambda)$ is equivalent to the
condition that $F_{m}$, the stalk of $F$ at any $m\in M$, is a perfect complex
of $\Lambda$-modules (thanks to the standard results [Sta, Lemma 066E, Lemma
0658]). We will denote $\mathrm{D}^{b}_{ctf}(X^{an},\Lambda)$ by
$\mathrm{D}^{b}_{c}(X^{an},\Lambda)$ henceforth.

Let $\Lambda$ be a finite local ring. Let $F$ be a Zariski constructible étale
sheaf valued in finite $\Lambda$-modules on the algebraic variety $X$ that is
$F$ is an étale sheaf valued in finitely generated $\Lambda$-modules and there
exists finitely many locally closed subvarieties $X_{i}\subset X$ such that
(1) $X=\underset{i}{\sqcup}X_{i}$, (2) the Zariski closure $\overline{X_{i}}$
of each $X_{i}$ is a union of finitely many of the $X_{j}$, and (3)
$F|_{X_{i}}$ is a étale-locally constant sheaf valued in finite
$\Lambda$-modules. The Zariski locally closed subvarieties $X_{i}$ can always
be chosen to be smooth. Let $\mathrm{D}^{b}(X_{\text{\'{e}t}},\Lambda)$ denote
the bounded derived category of étale sheaves valued in finite
$\Lambda$-modules. The full subcategory
$\mathrm{D}^{b}_{ctf}(X_{\text{\'{e}t}},\Lambda)\subset\mathrm{D}^{b}(X_{\text{\'{e}t}},\Lambda)$
is defined by declaring that
$F\in\mathrm{Ob}(\mathrm{D}^{b}(X_{\text{\'{e}t}},\Lambda))$ belongs to
$\mathrm{D}^{b}_{ctf}(X_{\text{\'{e}t}},\Lambda)$ if $\mathrm{H}^{i}(F)$ is a
Zariski constructible étale $\Lambda$-sheaf which is nonzero for atmost
finitely many $i$, and
$F\otimes^{L}_{\Lambda}Q\in\mathrm{D}^{b}(X_{\text{\'{e}t}},\Lambda)$ for any
finitely generated module $Q$ over $\Lambda$. In the sequel we will denote
$\mathrm{D}^{b}_{ctf}(X_{\text{\'{e}t}},\Lambda)$ by
$\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda)$.

For $\Lambda=\mathbb{Z}_{\ell}$, the derived category
$\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Z}_{\ell})$ is defined to be the
category $2$-$\lim\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda/\ell^{n})$
whose objects are projective systems $F=\\{F_{n}\\}_{n\geq 1}$ with
$F_{n}\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Z}/\ell^{n}\mathbb{Z})$
such the induced map $F_{n+1}\otimes^{L}\mathbb{Z}/\ell^{n}\mathbb{Z}\cong
F_{n}$ is an isomorphism, and the morphisms
$f\in\hom_{\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Z}_{\ell})}(F,G)$ is a
collection $\\{f_{n}\\}$ of morphisms, where
$f_{n}\in\hom_{\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Z}/\ell^{n}\mathbb{Z})}(F_{n},G_{n})$
which renders the following diagram

${F_{n+1}\otimes^{L}\mathbb{Z}/\ell^{n}\mathbb{Z}}$${G_{n+1}\otimes^{L}\mathbb{Z}/\ell^{n}\mathbb{Z}}$${F_{n}}$${G_{n}}$$\scriptstyle{\sim}$$\scriptstyle{f_{n+1}\otimes\mathbb{Z}/\ell^{n}\mathbb{Z}}$$\scriptstyle{\sim}$$\scriptstyle{f_{n}}$

to be commutative. The $\ell$-adic derived category
$\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})$ is obtained by
inverting the multiplication by $\ell$ map on
$\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Z}_{\ell})$. The definition of
$\mathrm{D}^{b}_{c}(X^{an},\Lambda)$ makes sense without any further
qualifications. Over the algebraically closed field $\mathbb{C}$, any étale-
locally constant sheaf on an algebraic variety $X$ over $\mathbb{C}$ valued in
a finite ring $\Lambda$ gives a local system on $X^{an}$. This allows us to
associate to $F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Z}_{\ell})$
$\left(\mathrm{resp.\,}F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Z}_{\ell})[\ell^{-1}]\right)$
an element in $\mathrm{D}^{b}_{c}(X^{an},\mathbb{Z}_{\ell})$ (in
$\mathrm{D}^{b}_{c}(X^{an},\mathbb{Q}_{\ell})$) in a canonical manner. The
associated object is denoted by $\varepsilon^{*}F$ in §2.4.

###  2.3 Six functor formalism

Let $X$ and $Y$ be algebraic varieties and $f:X\rightarrow Y$ be a morphism of
algebraic varieties. Let
$\Lambda\in\\{\mathbb{Z}/\ell^{n}\mathbb{Z},\mathbb{Z}_{\ell},\mathbb{Q}_{\ell}\\}$
we may associate to a morphism $f:X\rightarrow Y$ two pairs of adjoint
functors

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The adjointness of the pair $(f_{!},f^{!})$ is called as the Verdier duality
which is a vast generalization of the Poincaré duality. The existence of the
pair of adjoint functors maybe found in [Sch, Cor. 2.2.2, Cor. 2.2.5] and
[SGA4$\frac{1}{2}$, Cor. 1.5].

Consider the triple ${U}$${X}$${Z}$$\scriptstyle{j}$$\scriptstyle{i}$, where
$j$ is an open immersion and $i$ is a closed immersion. It is easy to check
from the definitions that $i_{*}=i_{!}$, $j^{*}=j^{!}$ and $j^{*}i_{*}=0$.
Moreover, $i_{*},j_{*}$, and $j_{!}$ are fully faithful. For a sheaf
$F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda)$ (resp.
$\mathrm{D}^{b}_{c}(X^{an},\Lambda)$) we have the following distinguished
triangles

$\rightarrow j_{!}j^{*}F\rightarrow F\rightarrow
i_{*}i^{*}F\xrightarrow{+1}\qquad\text{and,}\qquad\rightarrow
i_{*}i^{!}F\rightarrow F\rightarrow j_{*}j^{*}F\xrightarrow{+1}$

in $\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda)$ (resp.
$\mathrm{D}^{b}_{c}(X^{an},\Lambda)$). The data described in this paragraph
form a recollement of $\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda)$ (resp.
$\mathrm{D}^{b}_{c}(X^{an},\Lambda)$) in the sense of [BBD82, §1.4.3].

###  2.4 Comparison of analytic and étale topos

Recall from [BBD82, §6.1, 6.2] that there is a morphism of topos
$X^{an}\rightarrow X_{\text{\'{e}t}}$ which induces a fully faithful functor

${\varepsilon^{*}:\\{\text{Zariski constructible \'{e}tale
}\Lambda\text{-sheaf}\\}}$${\\{\text{Zariski constructible
}\Lambda\text{-sheaf}\\}}$${\varepsilon^{*}:\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})}$${\mathrm{D}^{b}_{c}(X^{an},\mathbb{Q}_{\ell}).}$

The essential image of the functor $\varepsilon^{*}$ consists of objects $F$
such that $\mathrm{H}^{i}F$ is the image of a Zariski constructible étale
$\Lambda$-sheaf under $\varepsilon^{*}$. In fact, as noted in [BBD82] the
functor is not an equivalence of categories.

###  2.5 Vanishing cycles

Let $X^{an}$ be a complex algebraic variety,
$F\in\mathrm{D}^{b}_{c}(X^{an},\Lambda)$ with
$\Lambda\in\\{\mathbb{Z}/\ell^{n}\mathbb{Z},\mathbb{Z}_{\ell},\mathbb{Q}_{\ell}\\}$,
and $f:X^{an}\rightarrow\mathbb{C}$ be a morphism. Set $X^{an}_{0}=f^{-1}(0)$
and denote by $i$ the closed embedding $i:X^{an}_{0}\hookrightarrow X^{an}$.
Let $\widetilde{p}:\widetilde{X}\rightarrow X^{an}$ be the pullback of $f$
along the composite
$\widetilde{\mathbb{C}^{\times}}\rightarrow\mathbb{C}^{\times}\hookrightarrow\mathbb{C}$,
where $\widetilde{\mathbb{C}^{\times}}$ is the universal cover of
$\mathbb{C}^{\times}$. The vanishing cycle $\phi^{an}_{f}(F)$ of $F$ with
respect to the map $f$ is defined to be the cone
$\mathrm{Cone}(i^{*}F\rightarrow i^{*}\widetilde{p}_{*}\widetilde{p}^{*}F)$.
The superscript an has been added to the standard notation to remind us that
the objects $f,X^{an}$, and $F$ are analytic in nature. The functor
$\phi^{an}_{f}:\mathrm{D}^{b}(X^{an},\Lambda)\rightarrow\mathrm{D}^{b}(X^{an}_{0},\Lambda)$
preserves constructibility ([Sch, Thm. 4.0.2]).

Let $X$ be an algebraic variety and $f:X\rightarrow\mathbb{A}^{1}$ be a
morphism of algebraic varieties. Let
$\mathcal{O}^{sh}_{\mathbb{A}^{1},\\{0\\}}$ denote the strict henselization of
the local ring of $\mathbb{A}^{1}$ at the point $\\{0\\}$. We continue to
denote by $f$ the base change of $f$ via the map
$\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{\mathbb{A}^{1},\\{0\\}})\rightarrow\mathbb{A}^{1}$.
Let $\bar{s}$ and $\bar{\eta}$ respectively be a closed geometric and a
generic geometric point of
$\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{\mathbb{A}^{1},\\{0\\}})$.
Let $i:X_{\bar{s}}\rightarrow X$ and $j:X_{\bar{\eta}}\rightarrow X$
respectively be the pullback of
$\bar{s}\rightarrow\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{\mathbb{A}^{1},\\{0\\}})$
and
$\bar{\eta}\to\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{\mathbb{A}^{1},\\{0\\}})$
along the morphism $f$. The natural map $F\rightarrow j_{*}j^{*}F$ induces the
map $i^{*}F\rightarrow i^{*}j_{*}j^{*}F$. The vanishing cycle
$\phi^{alg}_{f}(F)$ of $F$ with respect to the map $f$ is defined to be the
cone $\mathrm{Cone}(i^{*}F\rightarrow i^{*}j_{*}j^{*}F)$. The superscript alg
has been added to the standard notation to remind ourselves that the objects
$f,X,$ and $F$ are algebraic in nature. The functor
$\phi^{alg}_{f}:\mathrm{D}^{b}(X,\Lambda)\rightarrow\mathrm{D}^{b}(X_{0},\Lambda)$
preserves constructibility. This is given by [SGA4$\frac{1}{2}$, Thm. 3.2]
when $\Lambda=\mathbb{Z}/\ell^{n}\mathbb{Z}$ for some $n$. The case of
$\Lambda=\mathbb{Z}_{\ell}$ follows from applying the aforementioned theorem
for $\Lambda=\mathbb{Z}/\ell^{n}\mathbb{Z}$ for all $n\geq 1$ by the standard
arguments, and finally the case of $\Lambda=\mathbb{Q}_{\ell}$ is immediate.

When $X$ is an algebraic variety and $f:X\rightarrow\mathbb{A}^{1}$ is a
morphism of algebraic varieties, we have two notions of vanishing cycles
$\phi^{alg}_{f}(F)$ and $\phi^{an}_{f}(\varepsilon^{*}F)$. The morphism of
topos $\varepsilon$ gives the canonical map

$\mathrm{comp}_{\text{\'{e}t,
Betti}}:\varepsilon^{*}(\phi^{alg}_{f}(F))\rightarrow\phi^{an}_{f(\mathbb{C})}(\varepsilon^{*}(F)).$

The following comparison result for vanishing cycles is due to Deligne.

######  Theorem 2.1 ([SGA$7_{\textup{II}}$], Expose XIV, Theorem 2.8).

The map $\mathrm{comp}_{\textup{\'{e}t, Betti}}$ is an isomorphism.

Let $X$ and $S$ be algebraic varieties, $f:X\rightarrow S$ be a morphism, and
$F$ be in $\mathrm{D}^{b}_{ctf}(X_{\text{\'{e}t}},\Lambda)$ with $\Lambda$ a
finite local ring. Let $\bar{x}\in X(\mathbb{C})$ be a geometric point of $X$
and $f(\bar{x})\in S(\mathbb{C})$ be the image of $\bar{x}$ under $f$. The map
$f$ induces a local homomorphism of local rings
$\mathcal{O}_{S,f(\bar{x})}\rightarrow\mathcal{O}_{X,\bar{x}}$, and hence also
a homomorphism on their strict henselizations
$\mathcal{O}^{sh}_{S,f(\bar{x})}\rightarrow\mathcal{O}^{sh}_{X,\bar{x}}$. Let
$f^{sh}:\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{X,\bar{x}})\rightarrow\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{S,f(\bar{x})})$
be the induced map. Let $M_{\bar{x},\bar{s}}$ denote the fiber product
$\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{X,\bar{x}})\times_{\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{S,f(\bar{x})})}\bar{s}$
for a geometric point
$\bar{s}\in\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{S,f(\bar{x})})$.
The canonical map
$M_{\bar{x},\bar{s}}\rightarrow\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{X,\bar{x}})$
induces a pullback map

$\alpha_{\bar{x},\bar{s}}:\Gamma(\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{X,\bar{x}}),F)\rightarrow\Gamma(M_{\bar{x},\bar{s}},F).$

We say that $F$ is locally acyclic with respect to $f$ if
$\alpha_{\bar{x},\bar{s}}$ are isomorphisms for all geometric points
$\bar{x}\in X$ and a generization $\bar{s}$ of $f(\bar{x})\in S$. In the
situation when the algebraic variety $S$ is of dimension 1, we have the
following well known lemma which relates the local acyclicity with the
vanishing cycles.

######  Lemma 2.2.

Suppose that $X,S,f:X\rightarrow S$, and $F$ are as above. Then $F$ is locally
acyclic with respect to the morphism $f\Leftrightarrow\phi_{f}(F)=0$.

###### Proof.

$\phi_{f}(F)=0\Leftrightarrow i^{*}F\rightarrow i^{*}j_{*}j^{*}F$ is an
isomorphism $\Leftrightarrow F\rightarrow j_{*}j^{*}F$ is an isomorphism.
Taking stalks at a geometric point $\bar{x}\in X_{\bar{s}}$ we get

$\Gamma(\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{X,\bar{x}},F)\cong(j_{*}j^{*}F)_{\bar{x}}\cong\Gamma(\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{X,\bar{x}}),j_{*}j^{*}F)\cong\Gamma(\operatorname*{\mathrm{Spec}}(\mathcal{O}^{sh}_{X,\bar{x}})_{\bar{\eta}},F).$

The isomorphism in the last step follows due to the property that nearby
cycles commutes with a finite base change (See [Sai17, Prop. 2.7]). The
isomorphism of the two extreme terms precisely means that $F$ is locally
acyclic with respect to the morphism $f$. The lemma follows. ∎

In a more general setup than the one considered above, the given definition of
local acyclicity has several issues such as, the sheaf of vanishing cycles may
not be constructible222But it is constructible after a modification of the
base (See [Org06, Thm. 6.1]), the vanishing cycles functor may not commute
with the base change, and the data of the local acyclicity cannot be captured
by a lemma as simple as above. The appearance of non-finite type schemes such
as the Milnor tubes can be circumvented by using the characterization of
universally acyclic complex of sheaves developed by Lu-Zheng [LZ19] and
Hansen-Scholze [HS23].

###  2.6 Cycle class

Let $X$ be an equidimensional algebraic variety. For a natural number $n$, let
$\mathrm{CH}^{n}(X)$ denote the formal sum of irreducible algebraic varieties
of $X$ of codimension $n$ upto rational equivalence. There is the following
cycle class map

$\mathrm{CH}^{n}(X)\xrightarrow{\text{cl}}\mathrm{H}^{2n}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell}),$

assigning to a closed subvariety $Y$ of $X$ of codimension $n$, a refined
cycle class $[Y]\in\mathrm{H}^{2n}_{Y}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})$
(See [El78, Ch. III, Def. 2, Remark 2]) which under the canonical map
$\mathrm{H}^{2n}_{Y}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})\rightarrow\mathrm{H}^{2n}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})$
maps to $\text{cl}(Y)$. We will write $\mathrm{cl}$ for the refined cycle
class map as well. Let $c$ be a correspondence as below

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where $C$ is an algebraic variety, $f$ is a locally complete intersection
morphism and $g$ is proper. Then, $c_{*}$ the pushforward map along $c$ is
defined to be the composite $g_{*}f^{*}$,

$c_{*}:\mathrm{CH}^{n}(X)\rightarrow\mathrm{CH}^{n+\dim Z-\dim C}(Z).$

Similarly333Here we do not need the assumption of $f$ beiung lci., the maps

$c_{*}:\mathrm{H}^{2n}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})\rightarrow\mathrm{H}^{2n+2\dim
Z-2\dim C}(Z_{\text{\'{e}t}},\mathbb{Q}_{\ell})\qquad\text{ and }\qquad
c_{*}:\mathrm{H}^{2n}_{Y}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})\rightarrow\mathrm{H}^{2n+2\dim
Z-2\dim C}_{g\circ f^{-1}(Y)}(Z_{\text{\'{e}t}},\mathbb{Q}_{\ell})$

is defined to be the composite $g_{*}f^{*}$.

Recall that by our assumption the varieties $X,Z$, and $C$ are smooth. This in
particular implies that the morphisms are locally complete
intersections444abrreviated as lci in the rest of the document.. The cycle
class map is functorial with respect to the pushforward maps under
correspondences, that is, the diagram below commutes.

${\mathrm{CH}^{n}(X)}$${\mathrm{H}^{2n}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})}$${\mathrm{CH}^{n+\dim
Z-\dim C}(Z)}$${\mathrm{H}^{2n+2\dim Z-2\dim
C}(Z_{\text{\'{e}t}},\mathbb{Q}_{\ell}).}$cl$\scriptstyle{c_{*}}$$\scriptstyle{c_{*}}$cl

######  Lemma 2.3.

Let $c$ be a correspondence as above, $Y\subset X$ be an equidimensional
Zariski closed subset with the reduced induced subscheme structure, let
$c^{0}(Y)$ denote the Zariski closed subset $g\circ f^{-1}(Y)$, and let $n$ be
the codimension of $Y$ in $X$. Then the refined cycle class map is also
functorial. That is, the diagram below commutes.

${\mathrm{CH}^{0}(Y)}$${\mathrm{H}^{2n}_{Y}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})}$${\mathrm{CH}^{\dim
Z-\dim C}(c^{0}(Y))}$${\mathrm{H}^{2n+2\dim Z-2\dim
C}_{c^{0}(Y)}(Z_{\text{\'{e}t}},\mathbb{Q}_{\ell})}$$\scriptstyle{\mathrm{cl}}$$\scriptstyle{c_{*}}$$\scriptstyle{c_{*}}$$\scriptstyle{\mathrm{cl}}$

###### Proof.

The commutativity of the above diagram can be broken into functoriality of the
refined cycle class map for 1. pullback along locally complete intersection
morphism555Note that any morphism among smooth varieties is a locally complete
intersection. and 2. pushforward along proper morphism. Note that any lci
morphism can be factored as a composition of regular embedding followed by a
projection map which is flat. The Functoriality of the cycle class map for
pullbacks under the projection map is clear. See [Ful, Chapter 19, Lemma
19.2(a)] for the functoriality of the cycle class maps for pullback via a
regular embedding. See [Ful, Lemma 19.1.2] for functoriality of the cycle
class map under pushforward along proper morphisms. ∎

###  2.7 Closed conical subsets of cotangent bundle

Let $X$ be an algebraic variety, then it’s cotangent bundle $T^{*}X$ is again
an algebraic variety equipped with a canonical 1-form $\omega$. With this
choice $d\omega$ is a closed 2-form on $T^{*}X$ making it into a symplectic
manifold. A Zariski closed subset $C\subset T^{*}X$ is called a conical subset
if it is stable under the obvious action of $\mathbb{G}_{m}$ on $T^{*}X$. The
first and perhaps the example that is most pertinent to us is
$\overline{T^{*}_{S}X}$, the closure in the Zariski topology of the conormal
bundle of $X$ along a smooth locally closed subset in the Zariski topology
$S\subset X$(See [KS, Prop. 8.4.1]). Let $C^{\mathrm{sm}}\subset C$ be the
smooth locus of $C$ which is in fact dense in the Zariski topology. We call
$C$ to be isotropic if $d\omega|_{C^{\text{sm}}}\equiv 0$, and $C$ is said to
be involutive if for all $p\in C^{\mathrm{sm}}$, $T_{p}C$ is an involutive
subset of $T_{p}T^{*}X$ for the symplectic pairing on $T_{p}T^{*}X$ induced by
$\omega$. A subset $C$ is called lagrangian if $C$ is an isotropic and
involutive subset.

In the case of complex analytic variety $X^{an}$, the above paragraph must be
read replacing Zariski topology, and $\mathbb{G}_{m}$ respectively by
Euclidean topology and, $\mathbb{C}^{\times}$.

##  3 Singular supports and characteristic cycles

###  3.1 Analytic

Let
$\Lambda\in\\{\mathbb{Z}/\ell^{n}\mathbb{Z},\mathbb{Z}_{\ell},\mathbb{Q}_{\ell}\\}$
except in the last paragraph where we assume $\Lambda=\mathbb{Q}_{\ell}$.

To any sheaf $F\in\mathrm{D}^{b}_{c}(X^{an},\Lambda)$, Kashiwara-Schapira
associates a closed conical isotropic involutive subset of $T^{*}X^{an}$
denoted in this article by $\mathrm{SS^{KS}}(F)$ (See [KS, Thm. 8.5.5]). The
definition in the book perhaps cannot be seen to be immediately related to the
definition of the singular supports due to Beilinson. We quote here a result
from the book which resembles Beilinson’s definition

######  Theorem-Definition 3.1 ([KS], Prop. 8.6.4).

Let $\pi:T^{*}X^{an}\rightarrow X^{an}$ be the cotangent bundle of $X^{an}$.
The following are equivalent

  1. 1.

$p\in T^{*}X^{an}$ does not belong to the singular support of $F$.

  2. 2.

There exists an open neighbourhood $U$666$U$ is open subset in the Euclidean
topology. of $p$ and a holomorphic function $f$ defined on some open
neighbourhood $V$ of $\pi(p)$ satisfying $f(\pi(p))=0$ and $df(\pi(p))\in U$,
such that $\phi^{an}_{f}(F)_{\pi(p)}=0$.

Kashiwara-Schapira further define a cycle which is a formal sum of closed
conical Lagrangian subsets with certain integer coefficients as explained
below. Let $\pi:T^{*}X^{an}\rightarrow X^{an}$ be the cotangent bundle of
$X^{an}$, $p_{1},p_{2}$ respectively be the first and second projections of
$X^{an}\times X^{an}\rightarrow X^{an}$, and $\delta:X^{an}\rightarrow
X^{an}\times X^{an}$ be the diagonal embedding. The characteristic class
$\mathrm{C}(F)\in\mathrm{H}^{0}_{\mathrm{supp}(F)}(X^{an},\omega_{X^{an}})$ is
defined to be the image of the identity map of $F$, denoted by
$\mathrm{id}_{F}\in\hom(F,F)$, under the following composite

$\mathrm{R}\hom(F,F)\xrightarrow{\sim}\delta^{!}(F\boxtimes\mathrm{D}_{X^{an}}F)\rightarrow\delta^{*}(F\boxtimes
D_{X^{an}}F)\xrightarrow{\sim}F\otimes
D_{X^{an}}F\xrightarrow{tr}\omega_{X^{an}}.$

Here $\omega_{X}(=\Lambda_{X}[2\dim X])$ denote the dualizing sheaf of $X$.
The above morphisms can be lifted to a map of sheaves on the cotangent bundle
of $X^{an}$ using the technique of microlocalization. We refer the reader to
[KS, §9.4] for the definition of the characteristic cycle, and to [KS, Ch. IV]
for the definition of microlocalization. This allows us to write a morphism
(See [KS, pp. 352])

$\mathrm{R}\hom(F,F)\rightarrow
R\pi_{*}R\Gamma_{\mathrm{SS}^{\mathrm{KS}}(F)}(\pi^{-1}\omega_{X^{an}}).$ (1)

The image of $\mathrm{id}_{F}\in\mathrm{R}\hom(F,F)$ under the above map is an
equivalence class
$\mathrm{H}^{0}_{\mathrm{SS^{KS}(F)}}(T^{*}X^{an},\pi^{-1}\omega_{X^{an}})$,
to be denoted by $\mathrm{CC^{KS}}(F)$. We now make the assumption that
$\Lambda$ is a field of characteristic 0. Since $X^{an}$ is a complex
manifold, we have a canonical isomorphism
$\omega_{X^{an}}\simeq\Lambda_{X^{an}}[2\dim X^{an}]$. Denote by
$\mathscr{CS}^{\bullet}(T^{*}X^{an})$ the sheaf of subanalytic chains in
$T^{*}X^{an}$ (See [KS, §9.2]). Using [KS, Prop. 9.2.6(iv)] we get the
following series of isomorphisms

$\begin{split}\mathrm{H}^{0}_{\mathrm{SS^{KS}(F)}}(T^{*}X^{an},\Lambda_{T^{*}X^{an}}[2\dim
X^{an}])&=\mathrm{H}^{-2\dim
X^{an}}_{\mathrm{SS^{KS}}(F)}(T^{*}X^{an},\mathscr{CS}^{\bullet}(T^{*}X^{an}))=\mathscr{CS}^{T^{*}X^{an}}_{2\dim
X^{an}}(\mathrm{SS^{KS}}(F))\\\ &\subset\Bigg{\\{}\sum
a_{i}X^{an}_{i}\;\Bigg{|}\;\begin{aligned}
&X^{an}_{i}\subset\mathrm{SS^{KS}}(F)\text{ locally closed subanalytic},\\\
&\dim(X^{an}_{i})=2\dim X^{an}\text{ and
}a_{i}\in\Lambda\end{aligned}\Bigg{\\}}.\end{split}$ (2)

The image of the cohomology class $\mathrm{CC^{KS}}(F)$ under the
identification in (2) will again be denoted by $\mathrm{CC^{KS}}(F)$. We will
also need the following

######  Lemma 3.2.

For a perverse sheaf $F$, $\mathrm{CC^{KS}}(F)\geq 0$ and is supported on
$\mathrm{SS^{K}S}(F)$, the singular support of $F$.

###### Proof.

The singular support commutes with the Riemann-Hilbert correspondence functor
$\mathrm{DR}_{X^{an}}$ (See [KS, Theorem 11.3.3]). The characteristic cycle
commutes with $\mathrm{DR}_{X^{an}}$ as well (See [SV00, pp.1115-1116]). But
for holonomic $D$-modules the assertion that the characteristic cycle is
supported on the characteristic variety is clear from the definition. ∎

###  3.2 Algebraic

We assume $\Lambda=\mathbb{Z}/\ell^{n}\mathbb{Z}$ in this section unless
otherwise mentioned. In [Bei17] Beilinson defines the notion of the weak
singular support of a constructible sheaf
$F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda)$ to be the smallest closed
conical subsets $C$ of $T^{*}X$ satisfying the following : for every
$C$-transversal777Recall that a pair $(h,f)$ is said to be $C$-transversal if
$df_{x}^{-1}(C_{x})\setminus\\{0\\}=\emptyset$. Here $C_{x}$ denotes the set
$C\cap T^{*}_{x}X$, and $df_{x}$ denotes the stalk at $x\in X$ of the morphism
$df:T^{*}\mathbb{A}^{1}\rightarrow T^{*}X$. test pair $(j,f)$ with
$j:U\hookrightarrow X$ an open embedding, and a morphism
$f:X\rightarrow\mathbb{A}^{1}$, $F|_{U}$ is locally acyclic with respect to
the map $f$. Explicitly the weak singular support can also be described as the
Zariski closure in $T^{*}X$ of the set

$\\{(x,df(x))\;|\;x\in X(\mathbb{C})\text{ and }f\text{ is locally acyclic
relative to $F$ at $x\in X(\mathbb{C})$}\\}.$

It is proved that $\mathrm{SS^{B}}(F)$ is a closed conical isotropic subset
and each of it’s irreducible components are of dimension $n=\dim(X)$ (See
[Sai20, Prop. 2.2.7]).

######  Remark 3.3.

  1. 1.

Let $f:U\rightarrow\mathbb{A}^{1}$ be a
$\overline{\mathrm{SS^{KS}}(F)}^{Zar}$-transversal pair, then
$df(x)\notin\overline{\mathrm{SS^{KS}}(F)}^{Zar}$ for any $x\in U$. Hence
$\phi^{an}_{f}(F|_{U})=0$. Using Theorem 2.1 we get that
$\phi^{alg}_{f}(F|_{U})=0$. Thus, $F$ is microsupported on the Zariski closed
subset $\overline{\mathrm{SS^{KS}}(F)}^{Zar}$. Hence the inclusion
$\mathrm{SS^{B}}(F)\subset\overline{\mathrm{SS^{KS}}(F)}^{Zar}$ holds.

  2. 2.

For the sake of clarity we mention here that the notion of the weak singular
support coincides with the notion of singular support as has been proved by
Beilinson [Bei17, §1.5, Theorem].

To any $F\in\mathrm{D}^{b}_{ctf}(X,\Lambda)$, Saito [Sai17] associates a cycle
(not just a class!) supported on $\mathrm{SS^{B}}(F)$ with integer
coefficients (See [Sai17, Prop. 5.18]). We denote this cycle by
$\mathrm{CC^{S}}(F)$. More precisely if
$\mathrm{SS^{B}}(F)=\underset{i}{\cup}C_{i}$ then
$\mathrm{CC^{S}}(F)\colonequals\underset{i}{\sum}m_{i}[C_{i}]$ is such that
for any triple $(j,f,u)$ in the set

$\\{(j:U\rightarrow X,f:U\rightarrow\mathbb{A}^{1},u\in
U(\mathbb{C}))\;|\;(j,f)\text{ is a test pair and},(j|_{U\setminus
u},f|_{U\setminus u})\text{ is a }C\text{-transversal test pair}.\\},$

the equality

$-\mathrm{tot}\dim(\phi^{alg}_{u}(j^{*}F,f))=(\mathrm{CC^{S}}(F),df)_{T^{*}U,u}$
(Milnor formula)

holds. Here $\phi^{alg}_{u}(j^{*}F,f)$ denotes the vanishing cycle of $j^{*}F$
with respect to the morphism $f$ and $(\mathrm{CC^{S}}(F),df)_{T^{*}U,u}$
denotes the intersection of the cycle $\mathrm{CC^{S}}(F)$ with the graph of
the map $df$ induced by the morphism $f$. The intersection number is well
defined since $f$ is assumed to be transversal to
$\mathrm{SS^{B}}(F)\setminus\\{u\\}$. For a perverse sheaf $F$, the
coefficients of the cycle $\mathrm{CC^{S}}(F)$ are nonnegative integers
([Sai17, Prop. 5.14]).

For the remaining part of this subsection we assume
$\Lambda=\mathbb{Z}_{\ell}$. For any torsion free Zariski constructible étale
sheaf $\mathcal{F}$ with coefficients in $\Lambda$, there exists étale sheaves
$\mathcal{F}_{n}$ with coefficients in $\Lambda/\lambda^{n}$ such that
$\mathcal{F}_{n}$ is flat over $\Lambda/\lambda^{n}$, and
$\mathcal{F}_{n+1}\otimes_{\Lambda/\lambda^{n+1}}\Lambda/\lambda^{n}\cong\mathcal{F}_{n}$.
It follows from the definitions of the singular support and of the
characteristic cycle that
$\mathrm{SS^{B}}(\mathcal{F}_{n+1})=\mathrm{SS^{B}}(\mathcal{F}_{n})$ and
$\mathrm{CC^{S}}(\mathcal{F}_{n+1})=\mathrm{CC^{S}}(\mathcal{F}_{n})$. Hence,
it is meaningful to define

$\mathrm{SS^{B}}(\\{\mathcal{F}_{n}\\}_{n})\colonequals\mathrm{SS^{B}}(\mathcal{F}_{0}),\quad\mathrm{CC^{S}}(\\{\mathcal{F}_{n}\\}_{n})\colonequals\mathrm{CC^{S}}(\mathcal{F}_{0}).$

It is proved in [UYZ20, Prop. 5.9] that $\mathcal{F}$ is in fact
mircosupported on $\mathrm{SS^{B}}(\mathcal{F})$ and that
$\mathrm{CC^{S}}(\mathcal{F})$ satisfies the (Milnor formula). For any
$F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda)[\lambda^{-1}]$, there exist
torsion free sheaves $\mathcal{F}^{i}$ with coefficients in $\Lambda$ such
that
$\mathcal{F}^{i}\otimes_{\Lambda}\mathrm{Frac}(\Lambda)\cong\mathrm{H}^{i}(F)$.
Following a suggestion of Saito, Umezaki-Yang-Zhao [UYZ20] defines

$\mathrm{CC^{SUYZ}}(F)\colonequals\sum_{i}(-1)^{i}\mathrm{CC^{S}}(\mathcal{F}^{i}),\text{
and}\quad\mathrm{SS^{BUYZ}}(F)\colonequals\bigcup_{i}\mathrm{Supp}\left(\mathrm{CC^{SUYZ}}({}^{p}\mathrm{H}^{i}(F))\right).$

In the following proposition we list and indicate a quick proof of some
expected properties of $\mathrm{SS^{BUYZ}}$ and $\mathrm{CC^{SUYZ}}$.

######  Proposition 3.4.

With the notation as above, the following holds

  1. 1.

For any $F\in\mathrm{D}^{b}_{c}(X_{\textup{\'{e}t}},\mathrm{Frac}(\Lambda))$,
$\mathrm{SS^{BUYZ}}(F)=\bigcup_{i}\mathrm{SS^{BUYZ}}({}^{p}\mathrm{H}^{i}(F))$
and
$\mathrm{CC^{SUYZ}}(F)=\sum_{i}\mathrm{CC^{SUYZ}}({}^{p}\mathrm{H}^{i}(F))$.

  2. 2.

For a perverse sheaf
$F\in\mathrm{D}^{b}_{c}(X_{\textup{\'{e}t}},\Lambda)[\lambda^{-1}]$, the
coefficients of the cycle $\mathrm{CC^{SUYZ}}(F)$ are all nonnegative.

  3. 3.

Let $0\rightarrow F_{1}\rightarrow F_{2}\rightarrow F_{3}\rightarrow 0$ be an
exact sequence of perverse sheaves in
$\mathrm{D}^{b}_{c}(X_{\textup{\'{e}t}},\Lambda)[\lambda^{-1}]$. Then
$\mathrm{SS^{BUYZ}}(F_{2})=\mathrm{SS^{BUYZ}}(F_{1})\cup\mathrm{SS^{BUYZ}}(F_{3})$
and
$\mathrm{CC^{SUYZ}}(F_{2})=\mathrm{CC^{SUYZ}}(F_{1})+\mathrm{CC^{SUYZ}}(F_{3})$.

  4. 4.

Suppose that $U\overset{j}{\hookrightarrow}X$ is an open subset of $X$ such
that $Z=X\setminus U=\underset{i=1}{\overset{n}{\cup}}D_{i}$ is a normal
crossing divisor. Set $D_{I}\colonequals\cap_{\\{i\in I\\}}D_{i}$ and
$D_{\emptyset}\colonequals X$. Let $F$ be a locally constant constructible
sheaf on $U$ valued in $\mathrm{Frac}(\Lambda)$ whose complement is $D$. Then
$\mathrm{SS^{BUYZ}}(j_{!}F[\dim
X])=\underset{I\subset\\{1,2,\dots,n\\}}{\cup}T^{*}_{D_{I}}X$ and
$\mathrm{CC^{SUYZ}}(j_{!}F[\dim
X])=\operatorname*{\mathrm{rk}}(F)\underset{I\subset\\{1,2,\dots,n\\}}{\sum}T^{*}_{D_{I}}X$.

###### Proof.

  1. 1.

The equality on the singular supports and the characteristic cycles is clear
from the definition.

  2. 2.

We know from [UYZ20, Thm. 5.17(3)] that $\mathrm{CC^{SUYZ}}$ satisfies the
Milnor number formula. We may use [Bei17, §4.9(i)], more precisely it’s
refinement [Sai17, Lemma 4.10] to conclude that the coefficients are
nonnegative.

  3. 3.

The equality on the characteristic cycles follows from the definition. The
equality on the singular supports follows from the definition and part (2).

  4. 4.

Note that $j_{!}$ is $t$-exact and hence, $j_{!}F[\dim X]$ is also perverse.
In this case the equality
$\mathrm{SS^{BUYZ}}(j_{!}F)=\mathrm{Supp}(\mathrm{CC^{SUYZ}}(j_{!}F))$ follows
from the definitions of $\mathrm{SS^{BUYZ}}$ and $\mathrm{CC^{BUYZ}}$. Thus it
is enough to prove that $\mathrm{CC^{SUYZ}}(j_{!}F[\dim
X])=\operatorname*{\mathrm{rk}}(F)\underset{I\subset\\{1,2,\dots,n\\}}{\sum}T^{*}_{D_{I}}X$.
This follows by putting together [UYZ20, Thm. 5.17(1)] and [Sai17, Prop.
4.11].

∎

###  3.3 Three lemmas

In this subsection we assume that $\Lambda=\mathbb{Q}_{\ell}$.

Consider the following diagram
${U}$${X}$${D}$$\scriptstyle{j}$$\scriptstyle{i}$ where $j$ is an open
immersion and $i$ is a closed immersion such that
$D=\underset{i=1}{\overset{r}{\cup}}D_{i}$ is a strict normal crossing
divisor. Let $F$ be a local system on the open set $U^{an}$, then $j_{!}F$ is
a perverse sheaf on $X^{an}$ (See [BBD82, Cor. 4.1.10]).

######  Lemma 3.5.

Let $X,D$, and $U$ be as above. Define $D_{I}\colonequals\cap_{\\{i\in
I\\}}D_{i}$ and $D_{\emptyset}\colonequals X$. Then, for any
$F\in\mathrm{D}^{b}_{c}(X^{an},\Lambda)$, we have

$\mathrm{CC^{KS}}(j_{!}F)=(-1)^{\dim_{\mathbb{C}}X^{an}}\operatorname*{\mathrm{rk}}(F|_{U})\underset{I\subset\\{1,2,\dots,r\\}}{\sum}T^{*}_{D^{an}_{I}}X^{an}.$

###### Proof.

Since the question is local (in fact microlocal) we may assume that
$X^{an}=\mathbb{C}^{n}$, $D=\\{z_{1}z_{2}\dots z_{r}=0\\}$, and $F=j_{!}L$,
where $L$ is a $\mathbb{Q}_{\ell}$-local system on
$(\mathbb{C}^{\times})^{r}\times\mathbb{C}^{n-r}$. It is clear from the
definition of $\mathrm{CC^{KS}}$ that
$\mathrm{CC^{KS}}(F)=\mathrm{CC^{KS}}(F\otimes_{\Lambda}\mathbb{C})$. Thus we
may further assume
$F=j_{!}(L\otimes\mathbb{C})\in\mathrm{D}^{b}_{c}(X^{an},\mathbb{C})$. Since
$\mathrm{CC^{KS}}$ is additive under triangles (See §3.4(2)), in addition we
may assume that $L$ is an irreducible local system with complex coefficients.
So we may write $L=L_{1}\boxtimes\dots\boxtimes L_{r}$888Note that all the
irreducible representations of
$\pi_{1}((\mathbb{C}^{\times})^{r})=\mathbb{Z}^{r}$ over an algebraically
closed field is one dimensional and is product of one dimensional
representation. for certain irreducible local system $L_{i}$ of
$\mathbb{C}^{\times}$. Let
$j_{r}:\mathbb{C}^{\times}\hookrightarrow\mathbb{C}$ be the restriction of $j$
to the $r$-th coordinate, then $j^{*}=j^{*}_{1}\times\dots\times j^{*}_{r}$.
Since $j_{!}$ is left adjoint to $j^{*}=j^{*}_{1}\times\dots\times j^{*}_{r}$,
$j_{!}L=j_{1!}L_{1}\boxtimes j_{2!}L_{2}\boxtimes\cdots\boxtimes j_{r!}L_{r}$.
We know from [KS, pp. 378] that
$\mathrm{CC^{KS}}(j_{!}L)=\mathrm{CC^{KS}}(j_{1!}L_{1})\boxtimes\cdots\boxtimes\mathrm{CC^{KS}}(j_{r!}L_{r})$.
Supposing we also know that
$\mathrm{CC^{KS}}(j_{t!}L_{t})=-\operatorname*{\mathrm{rk}}(L_{t})([T^{*}_{\mathbb{C}}\mathbb{C}]+[T^{*}_{\\{0\\}}\mathbb{C}])$,
we get

$\mathrm{CC^{KS}}(j_{!}L)=(-1)^{\dim
X}\operatorname*{\mathrm{rk}}(L)([T^{*}_{\mathbb{C}}\mathbb{C}]+[T^{*}_{\\{0\\}}\mathbb{C}])\boxtimes\cdots\boxtimes([T^{*}_{\mathbb{C}}\mathbb{C}]+[T^{*}_{\\{0\\}}\mathbb{C}])=(-1)^{\dim
X}\operatorname*{\mathrm{rk}}(L)\underset{I\subset\\{1,\dots,r\\}}{\sum}T^{*}_{D_{I}}X.$

It remains to prove the equality
$\mathrm{CC^{KS}}(j_{!}L)=-\operatorname*{\mathrm{rk}}(L)([T^{*}_{\mathbb{C}}\mathbb{C}]+[T^{*}_{\\{0\\}}\mathbb{C}])$
where $L$ is a one dimensional local system with coefficients in $\mathbb{C}$.
We first prove this when $L$ is a trivial local system. Applying 1 of §3.4 to
the triangle $j_{!}j^{*}\mathbb{C}\rightarrow\mathbb{C}\rightarrow
i_{*}i^{*}\mathbb{C}$ we get

$\mathrm{CC}^{KS}(j_{!}\mathbb{C})=\mathrm{CC^{KS}}(\mathbb{C})-\mathrm{CC^{KS}}(i_{*}\mathbb{C})=-[T^{*}_{\mathbb{C}}\mathbb{C}]-[T^{*}_{\\{0\\}}\mathbb{C}].$

Thus proving the claim in this case. If $L$ is a nontrivial local system then
the canonical map $j_{!}L\rightarrow j_{*}L$ is an isomorphism. In this case
we have the following commutative diagram

${j_{!}(L\otimes L^{\vee})}$${j_{!}L\otimes j_{*}L^{\vee}}$${j_{!}L\otimes
D(j_{!}L)}$${j_{!}\mathbb{C}}$${\omega_{X^{an}}}$${j_{!}(\mathbb{C}\otimes\mathbb{C})}$${j_{!}(\mathbb{C})\otimes
j_{*}(\mathbb{C})}$${j_{!}(\mathbb{C})\otimes
D(j_{!}\mathbb{C})}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

Chasing the image of the identity morphism in $\mathrm{R}\hom(j_{!}L,j_{!}L)$
and $\mathrm{R}\hom(j_{!}\mathbb{C},j_{!}\mathbb{C})$ in the sequence of
arrows in the definition of characteristic cycle (See [KS, §9.4]) we conclude
using the diagram above that their images in
$R\pi_{*}R\Gamma_{\mathrm{SS}}(\pi^{-1}\omega_{X^{an}})$ coincides. Hence
$\mathrm{CC^{KS}}(j_{!}L)=\mathrm{CC^{KS}}(j_{!}\mathbb{C})=-[T^{*}_{\mathbb{C}}\mathbb{C}]-[T^{*}_{\\{0\\}}\mathbb{C}]$.
This finishes the proof of the lemma. ∎

######  Lemma 3.6.

Let $p:Y\rightarrow X$ be a projective morphism of algebraic varieties and
$F\in\mathrm{D}^{b}_{c}(Y_{\textup{\'{e}t}},\Lambda)$. Assume
$\Lambda\in\\{\mathbb{Z}/\ell^{n}\mathbb{Z},\mathbb{Z}_{\ell}\\}$ and
$\mathrm{CC^{KS}}(F)=\mathrm{CC^{S}}(F)$, or $\Lambda=\mathbb{Q}_{\ell}$ and
$\mathrm{CC^{KS}}(F)=\mathrm{CC^{SUYZ}}(F)$. Then
$\mathrm{CC^{KS}}(p_{*}F)=\mathrm{CC^{SUYZ}}(p_{*}F)$.

###### Proof.

Let the pushforward in cycles (resp. cohomology) along the correspondence

${T^{*}X\times_{X}Y}$${T^{*}Y}$${T^{*}X.}$$\scriptstyle{dp}$

be denoted by $p_{!}$ (resp. $p_{*}$). We have the following equality of
cycles

$\mathrm{CC^{SUYZ}}(p_{*}F)\overset{(1)}{=}p_{!}\mathrm{CC^{SUYZ}}(F)\overset{(2)}{=}p_{*}\mathrm{CC^{KS}}(F)\overset{(3)}{=}\mathrm{CC^{KS}}(p_{*}F).$

Using [UYZ20, Thm. 5.17(1)] we are reduced to proving (1) for a Zariski
constructible étale sheaf $F$ with coefficients in
$\mathcal{O}_{\Lambda}/\lambda^{n}$. The equality (1) for
$F\in\mathrm{D}^{b}_{c}(Y_{\text{\'{e}t}},\mathcal{O}_{\Lambda}/\lambda^{n})$
is the content of [Sai20, Prop. 2.2.7(2)]. Saito first proves that the
equality holds under certain addtional assumption on the dimension of
$f_{0}(\mathrm{SS}(F))$ (See [Sai20, Thm. 2.2.5]). He then proves that this
additional assumption is automatically satisfied when $X$ is defined over a
field of characteristic 0. This finishes the proof of equality (1) above. The
equality (3) is the content of [KS, Prop. 9.4.2]. The equality (2) is
explained in Lemma 2.3 of §2.6. ∎

######  Remark 3.7.

The assumption in the above lemma can be relaxed to - $f$ a quasi-projective
map and proper on the support of $F$. This assumption is forced on us since we
rely crucially on Saito’s result [Sai20, Prop. 2.2.7(2)].

######  Lemma 3.8.

Let $F$ be a perverse sheaf on $X$ such that $F|_{U}$ is isomorphic to
$L|_{U}$ where $L|_{U}$ is a simple local system. Then
$\mathrm{IC}_{X}(L|_{U})$ is the only simple subquotient of $F$ with support
containing the open subset $U$.

###### Proof.

This statement is about composition series and hence we may assume that $F$ is
semisimple. Moreover $F$ may be assumed to be simple since the local system of
interest is simple. So it suffices observe that for a simple perverse sheaf
$F$, $F|_{U}$ is isomorphic to a simple local system $L|_{U}$, then
$F\cong\mathrm{IC}_{X}(L|_{U})$. ∎

###  3.4 Summary of properties of singular supports and characteristic cycles

The properties of the singular supports and the characteristic cycles are
summarised below. In this subsection the notation $\mathrm{SS}(F)$ and
$\mathrm{CC}(F)$ are used to signify that the properties enumerated below
continues to hold true for both the notions of the singular support and of the
characteristic cycle discussed in §3.1 and §3.2.

  1. 1.

$\mathrm{SS}(F)=\cup_{i}\mathrm{SS}({}^{p}\mathrm{H}^{i}(F))$,
$\mathrm{CC}(F)=\underset{i}{\sum}\mathrm{CC}({}^{p}\mathrm{H}^{i}(F))$. See
[KS, Prop. 5.1.3(iii), Prop. 9.4.5], [Sai17, Lemma 5.13(1)], and Propisition
3.4(1) above.

  2. 2.

For an exact sequence of perverse sheaves $0\rightarrow F_{1}\rightarrow
F_{2}\rightarrow F_{3}\rightarrow 0$, the equalities
$\mathrm{SS}(F_{2})=\mathrm{SS}(F_{1})\cup\mathrm{SS}(F_{2})$ and
$\mathrm{CC}(F_{2})=\mathrm{CC}(F_{1})+\mathrm{CC}(F_{3})$ holds. See [KS,
Prop. 9.4.5(ii)], [Sai17, Lemma 5.13(1)], and Proposition 3.4(3).

  3. 3.

Suppose that $U\overset{j}{\hookrightarrow}X$ is an open subset of $X$ such
that $Z=X\setminus U=\underset{i=1}{\overset{n}{\cup}}D_{i}$ is a normal
crossing divisor. Let $F$ be a locally constant constructible sheaf on $U$,
whose complement is $D$. Then
$\mathrm{SS}(j_{!}F)=\underset{I\subset\\{1,2,\dots,n\\}}{\cup}T^{*}_{D_{I}}X$,
and $\mathrm{CC}(j_{!}F)=(-1)^{\dim
X}\operatorname*{\mathrm{rk}}(F)\underset{I\subset\\{1,2,\dots,n\\}}{\sum}T^{*}_{D_{I}}X$.
See Proposition 3.4(4) and Lemma 3.5 above.

  4. 4.

Let $p:X\rightarrow Y$ be a projective morphism between smooth varieties. Then
$\mathrm{CC^{KS}}(F)=\mathrm{CC^{SUYZ}}(F)\Rightarrow\mathrm{CC^{KS}}(p_{*}F)=\mathrm{CC^{SUYZ}}(p_{*}F)$.
See Lemma 3.6 above.

##  4 Main theorem and proof

######  Theorem 4.1.

Let $X$ be a smooth algebraic variety, and
$F\in\mathrm{D}^{b}_{c}(X_{\textup{\'{e}t}},\mathbb{Q}_{\ell})$. Then
$\mathrm{CC^{SUYZ}}(F)=\mathrm{CC^{KS}}(F)$.

###### Proof.

We may assume $F\neq 0$ since the theorem is clear when $F=0$. The proof of
the theorem proceeds via an induction argument on the dimension of the
support. The case of objects
$F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})$ supported of
dimension 0 is obvious. We may thus begin the induction process.

  1. Step 1.

Reduce to $F$ a simple perverse sheaf. An application of §3.4(1) implies that
it is enough to prove the theorem for a perverse sheaf $F$. Using §3.4(2) we
are reduced to considering only simple perverse sheaves.

  2. Step 2.

Behaviour under taking subobjects and quotients. Let $0\rightarrow
F_{1}\rightarrow F_{2}\rightarrow F_{3}\rightarrow 0$ be an exact sequence of
perverse sheaves. If the theorem holds for any two out of the three perverse
sheaves $F_{1},F_{2}$, and $F_{3}$, then it holds for the remaining one as
well. This is again clear by using §3.4(2).

  3. Step 3.

Reduction to $F=j_{!*}L_{U}$ where $j:U\hookrightarrow X$ is an open embedding
whose complement is strict normal crossing divisor. Let $F$ be a simple
perverse sheaf. We may assume that $F=j_{!*}L_{U}$, where $L_{U}$ is a simple
local system on $U$. After possibly choosing a smaller open subset $U$ and
using Hironaka’s theorem on resolution of singularities, we can ensure that
there exists a smooth variety $\widetilde{X}$ and a projective birational map
$r:\widetilde{X}\rightarrow X$, such that $r|_{r^{-1}(U)}:r^{-1}(U)\rightarrow
U$ is an isomorphism and
${r^{-1}(U)}$${\widetilde{X}}$${Z}$$\scriptstyle{\tilde{j}}$$\scriptstyle{\tilde{i}}$,
where $Z\subset\widetilde{X}$ is a strict normal crossing divisor. Let
$\widetilde{F}\colonequals j_{!*}L_{r^{-1}(U)}$. Then using the decomposition
theorem with respect to supports for perverse sheaf $\widetilde{F}$, we get

$r_{*}\widetilde{F}={}^{p}\mathrm{H}^{0}(r_{*}F)\oplus\\{\text{shifted direct
sum of perverse sheaves supported on smaller dimensional subvarieties}\\}.$

Assuming that the theorem holds for $\widetilde{F}$, we use §3.4(4) to
conclude that the theorem holds for $r_{*}\widetilde{F}$. An application of
Step 2 above, and the hypothesis that the theorem holds for all perverse
sheaves supported on closed subsets of $X$ of dimension $<\dim X$ implies that
the theorem holds for ${}^{p}\mathrm{H}^{0}(r_{*}F)$. Using Lemma 2.5 we get
that $F\hookrightarrow{}^{p}\mathrm{H}^{0}(r_{*}\widetilde{F})$ with the
quotient being supported on smaller dimensional smooth varieties. An
application of step 2 and the hypothesis that the theorem holds for all
perverse sheaves supported on closed subsets of $X$ of dimension $<\dim X$ we
get that the theorem is true for $F$. Thus completing the proof of this step.

  4. Step 4.

Proof of the theorem for $F$ as in Step 3. Let $F$ be a perverse sheaf on a
smooth variety such that $F=j_{!*}(L_{U})$ and $X\setminus U$ is a strict
normal crossing divisor. The triangle $\cdots\rightarrow
j_{!}j^{*}F\rightarrow F\rightarrow i_{*}i^{*}F\rightarrow\cdots$ gives the
following exact sequence of perverse sheaves (See [BBD82, Cor. 4.1.10(ii)])

$0\rightarrow i_{*}{}^{p}\mathrm{H}^{-1}i^{*}F\rightarrow
j_{!}j^{*}F\rightarrow F\rightarrow i_{*}{}^{p}\mathrm{H}^{0}i^{*}F\rightarrow
0.$

To prove the theorem for $F$, it is enough to prove the theorem for all
objects in the the above exact sequence other than $F$. By inductive
hypothesis we may asusme that the theorem holds for the extreme terms. Thus it
is enough to prove the theorem for $j_{!}j^{*}F$ which follows form §3.4(3).

∎

######  Corollary 4.2.

With notation as in the previous theorem,
$\mathrm{SS^{BUYZ}}(F)=\mathrm{SS^{KS}}(F)$

###### Proof.

Let $F$ be a perverse sheaf. We know from [Sai20, Prop. 5.14(2)] that support
of the cycle $\mathrm{CC^{S}}(F)$ is $\mathrm{SS^{B}}(F)$. It is clear from
the definitions of the extended notions of characteristic cycles to
$\mathbb{Q}_{\ell}$-sheaves that the support of the cycle
$\mathrm{CC^{SUYZ}}(F)$ is $\mathrm{SS^{BUYZ}}(F)$. On the other hand, the
arguments of Kashiwara-Schapira and Schmid-Vilonen as summarized in Lemma 3.2
implies that support of the Lagrangian cycle $\mathrm{CC^{KS}}(F)$ is
$\mathrm{SS^{KS}}(F)$. Now taking supports of both sides of the equality
established in Theorem 4.1 we get that for a perverse sheaf $F$, the equality
$\mathrm{SS^{BUYZ}}(F)=\mathrm{SS^{KS}}(F)$ holds. Now using §3.4(1) the
equality holds for all
$F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})$. ∎

##  Appendix A Coefficients in $\mathrm{CC^{KS}}$ and $\mathrm{CC^{S}}$

The characteristic cycle $\mathrm{CC^{S}}(F)$ of a constructible sheaf
$F\in\mathrm{D}^{b}(X,\Lambda)$ defined by Saito have coefficients in the ring
of integers irrespective of the ring of coefficients $\Lambda$. On the other
hand characteristic cycle as constructed999Note that in [KS, §9.4], it is
assumed that $\Lambda=k$ is a field of characteristic 0, but clearly the maps
make sense for any ring $\Lambda$ of finite global dimension. by Kashwara-
Schapira have coefficients in the ring $\Lambda$. Under the assumption that
$\Lambda$ is a field of characteristic 0, it is proved ([KS, Prop. 9.4.5])
that characteristic cycle $\mathrm{CC^{KS}}(-)$ has coefficients in the ring
$\mathbb{Z}$. In this section we wish to understand the dependence of the
characteristic cycle on the coefficient system.

Let $\Lambda\in\\{\mathbb{Z}/\ell^{n}\mathbb{Z},\mathbb{Z}_{\ell}\\}$ and
$F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\Lambda)$. For convenience we denote
the sheaf $\varepsilon^{*}F$ again by $F$ in this section. Recall that
$\mathrm{CC^{S}}(F)$ is an element of $\mathrm{CH}^{2\dim
X}(\mathrm{SS^{B}}(F))$ and $\mathrm{CC^{KS}}(F)$ is an element of
$\mathrm{H}^{0}_{\mathrm{SS}(F)}(T^{*}X^{an},\omega_{X^{an}})$. Since
$\Lambda$ is a noetherian ring we may associate a well-defined integer
$\dim\mathrm{tot}(K)$ to a perfect complex
$K\in\mathrm{D}^{b}_{perf}(\Lambda\text{-mod})$. We proceed as in [KS, pp.
382]; for $F\in\mathrm{D}^{b}_{c}(X^{an},\Lambda)$ and $p$ in an irreducible
component $V$ of $\mathrm{SS}(F)$ there exists
$K\in\mathrm{D}^{b}(\Lambda\text{-mod})$ such that $F\xrightarrow{\sim}A$
([KS, Prop. 6.6.2]) in $\mathrm{D}^{b}(X^{an};p)$ ([KS, §6.1]) and we define
$m_{V}\colonequals\dim\mathrm{tot}(A)$. Then the image of
$id\in\mathrm{R}\underline{\hom}(F,F)_{p}=\mathrm{H}^{0}(\mu\hom(F,F))_{p}$
([KS, Thm. 6.1.2]) in $\Lambda$ is given by the image of $\dim\mathrm{tot}(A)$
in the ring $\Lambda$ under the canonical map $\mathbb{Z}\rightarrow\Lambda$.
Now using Theorem 4.1 we get

######  Lemma A.1.

The image of $\mathrm{CC^{S}}(F)$ under the canonical map
$\mathbb{Z}\rightarrow\Lambda$ is the cycle $\mathrm{CC^{KS}}(F)$.

##  Appendix B Characteristic cycles for $\mathbb{Q}_{\ell}$-sheaves

We have already seen that the notion of $\mathrm{CC^{S}}$ has been extended
for $F\in\mathrm{D}^{b}_{c}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})$ by Umezaki-
Yang-Zhao. In a recent article [Bar23], Barrett has extended the definition of
singular support $\mathrm{SS^{B}}$ by utilizing the interpretation of the
condition of local acylicity ($\equiv$ universal local acyclicity) in terms of
dualizable objects in a certain 2-monoidal category developed by Lu-Zheng and
Hansen-Scholze. This allows him to bypass nonfinite type schemes such as
Milnor fibers (denoted $M_{\bar{x},\bar{s}}$ in §2.5) over which six functors
maynot preserve the derived category of Zariski constructible étale
$\mathbb{Q}_{\ell}$-sheaves. He also uses the proétale topology to bypass the
2-limit construction of derived category of $\mathbb{Q}_{\ell}$-sheaves which
allows for cleaner arguments once certain technical results are proved (See
[Bar23, §2, §3]).

In this article Barrett constructs a torsion free $\mathbb{Z}_{\ell}$-model of
a $\mathbb{Q}_{\ell}$-perverse sheaves. More precisely it is shown that given
a perverse sheaf $F\in\mathrm{Perv}(X,\mathbb{Q}_{\ell})$, there exists
$\mathcal{F}\in\mathrm{Perv}(X,\mathbb{Z}_{\ell})$ which is torsion free and
$\mathcal{F}\otimes\mathbb{Q}_{\ell}\cong F$. The singular support for $F$ can
now be defined as

$\mathrm{SS^{BB}}(F)\colonequals\mathrm{SS^{B}}(\mathcal{F}\otimes\mathbb{Z}_{\ell}/\ell),\text{
where $\mathcal{F}$ is torsion free and
}\mathcal{F}\otimes\mathbb{Q}_{\ell}\cong F.$

It is natural to extend the definition of characteristic cycles along the same
lines and define

$\mathrm{CC^{SB}}(F)\colonequals\mathrm{CC^{S}}(\mathcal{F}\otimes\mathbb{Z}_{\ell}/\ell).$

It can be proved as in [UYZ20, Prop. 5.9], that $\mathrm{CC^{SB}}$ satisfies
the Milnor number formula. Thus we may prove the following

######  Lemma B.1.

For a perverse sheaf
$F\in\mathrm{Perv}(X_{\textup{\'{e}t}},\mathbb{Q}_{\ell})$, we have
$\mathrm{CC^{SUYZ}}(F)=\mathrm{CC^{SB}}(F)$.

###### Proof.

This follows immediately from [Sai17, Lemma 4.10] and the fact that both the
notions satisfy Milnor number formula. ∎

We know from [Sai17, Prop. 5.14(2)] that
$\mathrm{CC^{SB}}(F)=\mathrm{SS^{BB}}(F)$ for any perverse sheaf
$F\in\mathrm{Perv}(X_{\text{\'{e}t}},\mathbb{Q}_{\ell})$. In light of the
above lemma, we get $\mathrm{SS^{BB}}=\mathrm{SS^{BUYZ}}$. Thus the definition
of singular support $\mathrm{SS^{BUYZ}}$ coincides with the definition of
singular supports developed by Barrett.

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