# Constructible sheaf complexes in complex geometry
and Applications

Laurenţiu G. Maxim  L. Maxim: Department of Mathematics, University of
Wisconsin-Madison, USA<EMAIL_ADDRESS>and  Jörg Schürmann  J. Schürmann
: Mathematische Institut, Universität Münster, Einsteinstr. 62, 48149 Münster,
Germany<EMAIL_ADDRESS>
###### Abstract.

We present a detailed introduction of the theory of constructible sheaf
complexes in the complex algebraic and analytic setting. All concepts are
illustrated by many interesting examples and relevant applications, while some
important results are presented with complete proofs. This paper is intended
as a broadly accessible user’s guide to these topics, providing the readers
with a taste of the subject, reflected by concrete examples and applications
that motivate the general theory. We discuss the stability of constructible
sheaf complexes under the standard functors, and explain the relation of these
functors to perverse sheaves and the perverse $t$-structure. We introduce the
main results of stratified Morse theory in the framework of constructible
sheaves, for proving the basic vanishing and finiteness results. Applications
are given to various index theorems, the functorial calculus of characteristic
cycles of constructible functions, and to weak Lefschetz and Artin-
Grothendieck type theorems. We recall the construction of Deligne’s nearby and
vanishing cycle functors, prove that they preserve constructible complexes,
and discuss their relation with the perverse $t$-structure. We finish this
paper with a description and applications of the Kähler package for
intersection cohomology of complex algebraic varieties, and the recent study
of perverse sheaves on semi-abelian varieties.

###### Key words and phrases:

constructible sheaves, perverse sheaves, stratified Morse theory,
characteristic cycles, vanishing cycles, nearby cycles, intersection
cohomology, Kähler package, jump loci

###### 2010 Mathematics Subject Classification:

32S60, 32S20, 32S05, 32S25, 32S40, 32S50, 58K05, 58K30, 14B05

L. Maxim is partially supported by the Simons Foundation Collaboration Grant
#567077. J. Schürmann is funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) Project-ID 427320536 – SFB 1442, as well as under
Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster:
Dynamics – Geometry – Structure.

###### Contents

  1. 1 Introduction
  2. 2 Constructible and perverse sheaf complexes
    1. 2.1 Constructibility
    2. 2.2 Perverse sheaves
    3. 2.3 Strongly perverse sheaves. dual t-structure and rectified homological depth
  3. 3 Stratified Morse theory for constructible sheaves
    1. 3.1 Morse functions, local and normal Morse data
    2. 3.2 Perverse sheaf description via normal Morse data
    3. 3.3 Characteristic cycles and index theorems
    4. 3.4 Functorial calculus of characteristic cycles
    5. 3.5 Vanishing results
  4. 4 Nearby and vanishing cycles, applications
    1. 4.1 Construction
    2. 4.2 Relation with perverse sheaves and duality
    3. 4.3 Thom-Sebastiani for vanishing cycles
    4. 4.4 Gluing perverse sheaves via vanishing cycles
    5. 4.5 Other applications
  5. 5 Intersection cohomology, the decomposition theorem, applications
    1. 5.1 Lefschetz type results for intersection cohomology
    2. 5.2 The decomposition theorem and immediate applications
    3. 5.3 A recent application of the Kähler package for intersection cohomology
  6. 6 Perverse sheaves on semi-abelian varieties
    1. 6.1 Cohomology jump loci
    2. 6.2 Jump loci via constructible complexes
    3. 6.3 Mellin transformation and applications
    4. 6.4 Characterization of perverse sheaves on semi-abelian varieties
    5. 6.5 Application: cohomology jump loci of quasi-projective manifolds
    6. 6.6 Application: topological characterization of semi-abelian varieties

##  1\. Introduction

The main goal of this paper is to provide a user’s guide, both for the novice
and the expert, for the theory of (weakly) constructible sheaf complexes in
complex geometry and their many applications. Our guiding principle for
writing these notes was to provide an explicit and geometric introduction of
the mathematical concepts, while also explaining some of the most important
examples of the general theory. For this reason, we aim to present as many of
the basic results as possible, sometimes even with complete proofs in some
special important cases. Moreover, these results and definitions are then
always explained and illustrated by many examples.

Constructible (complexes of) sheaves are the algebraic counterpart of the
decomposition of a variety into manifolds pieces (strata), and they are,
roughly speaking, obtained by gluing local systems defined along strata of a
Whitney stratification ${\mathscr{S}}$. Perverse sheaves are an important
class of constructible complexes, introduced in [6] as a formalization of the
celebrated Riemann-Hilbert correspondence of Kashiwara [65], which relates the
topology of algebraic, resp., analytic varieties (intersection homology) and
the algebraic, resp., analytic theory of differential equations (holonomic
$D$-modules).

In recent years, constructible sheaf complexes and especially perverse sheaves
have become indispensable tools for studying complex algebraic and analytic
varieties. They have seen spectacular applications in geometry and topology
(e.g., the decomposition theorem [6] and the topology of complex algebraic
maps), but also in fields such as representation theory (e.g., proof of the
Kazhdan-Lusztig conjecture, proof of the geometrization of the Satake
isomorphism, and proof of the fundamental lemma in the Langlands program) or
combinatorics (e.g., Stanley’s proof of McMullen’s conjecture, or the
resolution of the Dowling-Wilson top-heavy conjecture); see, e.g., [25, 92]
for more recent surveys of such applications. Furthermore, perverse sheaves
and the nearby and vanishing cycle functors of Deligne [52] are the backbone
of Saito’s mixed Hodge module theory [109, 110], a far-reaching generalization
of Deligne’s mixed Hodge theory.

However, despite their fundamental importance, perverse sheaves as special
constructible complexes of sheaves remain rather mysterious objects. It is our
hope that the present paper will help readers become better acquainted with
various aspects of the general theory. Those looking to delve further into
more specialized topics or wishing to explore problems of current research
will find ample references to facilitate navigation of both classic and recent
literature.

Let us next give a brief summary of the content of the paper.

In Section 2, we define the notion of (weak) constructibility, and discuss the
stability of (weakly) constructible sheaf complexes under the standard
functors. As far as possible, we allow also weakly constructible sheaf
complexes, where one does not impose any finiteness conditions for the stalks
(and which is sometimes even more natural or simpler to work with, especially
as long as no duality is used). Similarly, we try to work in such a way that
it applies to the complex algebraic as well as complex analytic context
(sometimes only under suitable compactness assumptions in the complex analytic
context). The presented calculus includes external products, Künneth
isomorphisms, Verdier duality and the relation to the Euler characteristic
calculus of constructible functios.

We also introduce here the perverse t-structure and perverse sheaves (with
respect to middle perversity), and explain their relation with the standard
functors (following their counterparts in $l$-adic cohomology as presented in
[6, Chapter 4]). Several aspects of the general theory are worked out in
detail for intersection cohomology complexes, which provide some of the main
examples of perverse sheaves. Furthermore, for constructible sheaf complexes
of $R$-modules with $R$ a Dedekind domain, we also consider the dual perverse
t-strucure and its relation to the rectified homological depth of
Grothendieck, as studied by Hamm and Lê [56] (also for the corresponding
homotopical notion).

In Section 3, we explain the basic results from [113] about stratified Morse
theory in the framework of (weakly) constructible sheaves in the complex
context, continuing and extending the recent survey of Goresky [49] in this
handbook series, as well as Massey’s survey [88]. We follow here the notions
of the geometric stratified Morse theory of Goresky-MacPherson [47], so that
one can easily compare the results of our paper with those of loc.cit.. We
introduce, for example, the sheaf theoretic counterparts of the local and
normal Morse data, as well as their relations for a stratified Morse critical
point of a $C^{\infty}$-function on a complex algebraic (or analytic) variety.
The normal Morse data of (weakly) constructible sheaf complexes are studied
via the complex link of a stratum of a Whitney stratification, which allows to
prove the basic vanishing and finiteness theorems by induction on the
dimension of the underlying complex analytic variety. In particular, we get a
description of the (dual) perverse t-structure in terms of properties of the
normal Morse data. Moreover, we also explain some relations to the general
micro-local sheaf theory of Kashiwara-Schapira [67], e.g. like a description
of the micro-support of a (weakly) constructible sheaf complex in terms of the
normal Morse data.

We use the language of stratified Morse theory for constructible functions and
sheaves to also give in this section an introduction to the functorial theory
of Lagrangian cycles in the complex analytic and algebraic context. We discuss
from this viewpoint the Euler isomorphism between constructible functions and
Lagrangian cycles given by the characteristic cycle of a constructible
function, together with some index theorems. Examples are given to Poincaré-
Hopf index theorems for singular spaces, effective characteristic cycles on
abelian varieties [4] and the global Euler obstruction for affine varieties
[120], as well as the famous local Euler obstruction of MacPherson [83]. We
also explain (using this language of stratified Morse theory for constructible
functions) the translation into the context of Lagrangian cycles of the
following operations for constructible functions and sheaves: external
product, proper direct image, non-characteristic pullback and specialization
(i.e., nearby cycles), together with an intersection formula for vanishing
cycles.

Finally, the last part of Section 3 deals with applications of the stratified
Morse theory for constructible sheaves to vanishing and weak Lefschetz
theorems in the complex algebraic and analytic context. This includes
different versions of the Artin vanishing theorem for complex algebraically
(weakly) constructible complexes on an affine variety, and vanishing theorems
for (weakly) constructible complexes on complex analytic Stein and
$q$-complete varieties, as well as relative counterparts for morphisms given
by Artin-Grothendieck types theorems in the complex algebraic and analytic
context.

In Section 4, we recall the construction of Deligne’s nearby and vanishing
cycle functors [52], and prove that they preserve (weakly) constructible
complexes. For the constructible context we also need and explain the
calculation of their (co)stalks in terms of the Milnor fibers for the
corresponding local Milnor fibrations, [72] based on the existence of an
adopted Whitney stratifications satisfying the $a_{f}$-condition of Thom [14,
57, 76]. Then we state the relation of nearby and vanishing cycle functors
with duality [89], and prove their relation with the (dual) perverse
$t$-structure and perverse sheaves (from the point of view of stratified Morse
theory, as developed in the previous section), i.e.,

$\psi_{f}[-1]\quad\text{and}\quad\varphi_{f}[-1]\quad\text{are {\it t-exact}
for the (dual) perverse t-structure.}$

We also include here a discussion on the Thom-Sebastiani theorem for vanishing
cycles [87, 113], and give a brief description of Beilinson’s and Deligne-
Verdier’s procedure for gluing perverse sheaves via vanishing cycles [7, 128].
As a final application we explain the equality $Rf_{!}=Rf_{*}$ on the level of
Grothendieck groups of algebraically constructible complexes [70, 129], as
well as analytically constructible functions in a compactifiable complex
analytic context [113].

In Section 5, we overview properties of the intersection cohomology groups of
complex algebraic varieties, which generalize the corresponding features of
the cohomology groups of smooth varieties. These properties, consisting of
Poincaré duality, (weak and hard) Lefschetz theorems and the decomposition
theorem, are collectively termed the Kähler package for intersection
cohomology [6, 24, 25, 109, 110].

We also mention briefly a recent combinatorial application of the Kähler
package for intersection cohomology, namely a proof (for realizable matroids)
by Huh-Wang [59] of the Dowling-Wilson top-heavy conjecture [32, 33].

Finally, in Section 6, we survey recent developments in the study of perverse
sheaves on semi-abelian varieties. We also include several concrete
applications of this theory, e.g., to the study of homotopy types of complex
algebraic manifolds (formulated in terms of their cohomology jump loci), as
well as new topological characterizations of semi-abelian varieties [77, 80,
79].

We assume reader’s familiarity with derived categories and the derived
calculus; for a quick refresher on these topics the interested reader may
consult [31], [92] or [67, Chapter I-III].

We work in the complex algebraic or analytic setting with reduced Hausdorff
spaces, e.g., in the complex algebraic context we are working with the complex
analytic space associated to a reduced separated scheme of finite type over
$\operatorname{Spec}({\mathbb{C}})$. Unless otherwise specified, all
dimensions are taken to be complex dimensions. In the complex analytic setting
we always assume that our spaces have bounded dimension and a countable
topology, e.g., the disjoint union of all ${\mathbb{C}}^{n}$’s
($n\in{\mathbb{N}}_{0}$) is not allowed.

##  2\. Constructible and perverse sheaf complexes

###  2.1. Constructibility

Let $X$ be a complex algebraic (or analytic) variety (here a variety does not
need to be irreducible). It is well known that such a variety can be endowed
with a Whitney stratification, i.e., a (locally) finite partition
$\mathscr{S}$ into non-empty, connected, locally closed nonsingular
subvarieties $S$ of $X$ (called “strata”) which satisfy the following
properties:

  1. (i)

Frontier condition: for any stratum $S\in\mathscr{S}$, the frontier $\partial
S:={\bar{S}}\setminus S$ is a union of strata of $\mathscr{S}$, where
${\bar{S}}$ denotes the closure of $S$.

  2. (ii)

Constructibility: the closure ${\bar{S}}$ and the frontier $\partial S$ of any
stratum $S\in\mathscr{S}$ are closed complex algebraic (respectively,
analytic) subspaces in $X$.

These conditions already imply that $X$ gets an induced dimension filtration

$X_{\bullet}:\emptyset:=X_{-1}\subset X_{0}\subset\cdots\subset X_{n}=X$

by the closed algebraic (or analytic) subsets $X_{i}:=\bigcup_{\dim S\leq i}S$
($0\leq i\leq n=\dim X$), with the strata $S$ of dimension $i$ given by the
connected components of $X_{i}\backslash X_{i-1}$.

In addition, whenever two strata $S_{1}$ and $S_{2}$ are such that
$S_{2}\subseteq{\partial S_{1}}$, the pair $(S_{2},S_{1})$ is required to
satisfy the following Whitney b-regularity conditions that guarantee that the
variety $X$ is topologically or cohomologically equisingular along each
stratum (as in [113, Section 4.2]):

  1. (iii)

Whitney b-condition: If $x_{n}\in S_{1}$ and $y_{n}\in S_{2}$ are sequences
converging to $x\in S_{2}$ such that the tangent planes $T_{x_{n}}S_{1}$
converge to some limiting plane $\tau$ and the secant lines
$l_{n}=\overline{x_{n},y_{n}}$ converge to some limiting line $l$ (in some
local coordinates), then $l\subset\tau$.

Note that the Whitney b-condition implies the following.

  1. (iv)

Whitney a-condition: If $x_{n}\in S_{1}$ is a sequence converging to $x\in
S_{2}$ such that the tangent planes $T_{x_{n}}S_{1}$ converge to some limiting
plane $\tau$, then $T_{x}S_{2}\subset\tau$.

These conditions are independent of the choice of local coordinates, and any
algebraic (or analytic) partition $\mathscr{S}$ as above has a refinement to
an algebraic (or analytic) Whitney stratification (see, e.g., [125, Theorem
1.2 and Proposition 2.1], [126], as well as the references given in [47,
Section 1.7]). Moreover, if the Whitney b-conditions hold for all connected
components $S$ of the $X_{i}\backslash X_{i-1}$ and a filtration $X_{\bullet}$
by closed algebraic (or analytic) subsets as above, then the partition
$\mathscr{S}$ with these strata $S$ is (locally) finite and satisfies the
frontier condition (see, e.g., [113, Section 4.2.1]) as well as the
constructibility condition.

######  Example 2.1 (Whitney umbrella).

The singular locus of the Whitney umbrella

$X=\\{z^{2}=xy^{2}\\}\subset{\mathbb{C}}^{3}$

is the $x$-axis, but the origin is “more singular” than any other point on the
$x$-axis. A Whitney stratification of $X$ is given by the strata

$S_{1}=X\setminus\\{x-\text{axis}\\},\quad S_{2}=\\{(x,0,0)\mid x\neq
0\\},\quad S_{3}=\\{(0,0,0)\\}.$

Another example is given by $X$ a complex manifold, with strata $S$ given by
its connected components.

In the following, let $R$ be a noetherian and commutative ring of finite
global dimension. Let $X$ be a complex algebraic (or analytic) variety, and
denote by $D^{b}(X;R)$ the derived category of bounded complexes of sheaves of
$R$-modules. By our assumptions on $X$ and $R$, these bounded derived
categories $D^{b}(-;R)$ are closed under Grothendieck’s six operations:
$Rf_{*}$, $Rf_{!}$, $f^{*}$, $f^{!}$, $R{\mathscr{H}}om^{\bullet}$ and
$\overset{L}{\otimes}$, with $f$ an algebraic (or analytic) morphism.

######  Definition 2.2.

A sheaf ${\mathscr{F}}$ of $R$-modules on $X$ is said to be weakly
constructible if there is a Whitney stratification ${\mathscr{S}}$ of $X$ so
that the restriction ${\mathscr{F}}|_{S}$ of ${\mathscr{F}}$ to every stratum
$S\in{\mathscr{S}}$ is an $R$-local system (i.e., a locally constant sheaf).
In this case we also say that ${\mathscr{F}}$ is ${\mathscr{S}}$-weakly
constructible. A (${\mathscr{S}}$-)weakly constructible sheaf ${\mathscr{F}}$
on $X$ is called (${\mathscr{S}}$-)constructible, if all stalks
${\mathscr{F}}_{x}$ for $x\in X$ are finitely generated $R$-modules. A bounded
complex ${\mathscr{F}}^{\bullet}\in D^{b}(X;R)$ is called (weakly)
constructible if all its cohomology sheaves
${\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})$ are (weakly) constructible.
Similarly for ${\mathscr{S}}$-(weakly) constructible complexes in case one
works with a fixed Whitney stratification ${\mathscr{S}}$.

Note that the category $Sh_{({\mathscr{S}}-)wc}(X)$ of
(${\mathscr{S}}$-)weakly constructible sheaves on $X$ is an abelian
subcategory of the category of all sheaves of $R$-modules on $X$, which is
stable under extensions (like the subcategory of $R$-local systems). Similarly
for the category $Sh_{({\mathscr{S}}-)c}(X)$ of
(${\mathscr{S}}$-)constructible sheaves (like the subcategory of $R$-local
systems with finitely generated stalks), since we assume $R$ to be noetherian.

######  Example 2.3.

On a point space $X=\\{pt\\}$ any sheaf is weakly constructible. For general
$X$, the constant sheaf ${R}_{X}$ (resp., an $R$-local system ${\mathscr{L}}$
on $X$) is (weakly) constructible on $X$ with respect to any Whitney
stratification. On the other hand, if $i\colon
X\hookrightarrow{\mathbb{C}}^{*}$ denotes the inclusion of the closed analytic
subset $X:=\\{\frac{1}{n}\mid n\in{\mathbb{N}}\\}\subset{\mathbb{C}}^{*}$,
then the direct image sheaf $i_{*}{R}_{X}$ is constructible on
${\mathbb{C}}^{*}$. But if
$k\colon{\mathbb{C}}^{*}\hookrightarrow{\mathbb{C}}$ is the open inclusion,
then the closure $\bar{X}=X\cup\\{0\\}$ in ${\mathbb{C}}$ is not analytic and
the direct image sheaf $k_{*}i_{*}{R}_{X}=(k\circ i)_{*}R_{X}$ is not weakly
constructible on ${\mathbb{C}}$.

We let $D^{b}_{(w)c}(X;R)$ be the full triangulated subcategory of
$D^{b}(X;R)$ consisting of (weakly) constructible complexes (that is,
complexes which are (weakly) constructible with respect to some Whitney
stratification). We will also use the simpler notation $D^{b}_{(w)c}(X)$ if
the coefficient ring $R$ is understood from the context. Similarly for
$D^{b}_{{\mathscr{S}}-(w)c}(X;R)$ in case we work with (weakly) constructible
complexes with respect to a fixed Whitney stratification ${\mathscr{S}}$. By
viewing a sheaf as a complex concentrated in degree zero, one gets an
isomorphism of Grothendieck groups (and similarly for weakly constructible
sheaves)

(1)
$\begin{CD}K_{0}\left(Sh_{({\mathscr{S}}-)c}(X)\right)@>{\sim}>{}>K_{0}\left(D^{b}_{({\mathscr{S}}-)c}(X;R)\right)\>,\end{CD}$

whose inverse is given by taking the alternating sum of (classes of)
cohomology sheaves (see [113, Lemma 3.3.1]).

######  Remark 2.4.

Even in the complex analytic setting a sheaf ${\mathscr{F}}$ (resp., a bounded
sheaf complex ${\mathscr{F}}^{\bullet}$) on $X$ is (weakly) constructible iff
this is locally the case, in the sense that there is an open covering
$(U_{i})$ of $X$ such that ${\mathscr{F}}|_{U_{i}}$ (resp.,
${\mathscr{F}}^{\bullet}|_{U_{i}}$) is (weakly) constructible for all $i$ (see
[31, Proposition 4.1.13] and compare also with [53, Corollary 3.4]). In fact,
the complement of the largest open $U$ of $X$ such ${\mathscr{F}}|_{U}$ is
locally constant (resp., ${\mathscr{F}}^{\bullet}|_{U}$ has locally constant
cohomology sheaves) is then an analytic subset of $X$ (see [31, Proposition
4.1.12]).

The derived category $D^{b}_{(w)c}(X;R)$ of bounded (weakly) constructible
complexes is closed under Grothendieck’s six operations: $Rf_{*}$, $Rf_{!}$,
$f^{*}$, $f^{!}$, $R{\mathscr{H}}om^{\bullet}$ and $\overset{L}{\otimes}$.
More precisely, one has the following (e.g., see [11] or the unified treatment
of [113, Theorem 4.0.2 and Proposition 4.0.2]):

######  Theorem 2.5.

Let $f\colon X\to Y$ be a morphism of complex algebraic (or analytic)
varieties, with $j\colon U\hookrightarrow Y$ the inclusion of the open
complement of a closed algebraic (or analytic) subset.

  1. (a)

If ${\mathscr{G}}^{\bullet}\in D^{b}_{c}(Y;R)$, then
$f^{*}{\mathscr{G}}^{\bullet},f^{!}{\mathscr{G}}^{\bullet}\in D^{b}_{c}(X;R)$.

  2. (b)

If ${\mathscr{F}}^{\bullet}\in D^{b}_{c}(X;R)$ and $f$ is an algebraic map,
then $Rf_{*}{\mathscr{F}}^{\bullet},Rf_{!}{\mathscr{F}}^{\bullet}\in
D^{b}_{c}(Y;R)$.
If ${\mathscr{F}}^{\bullet}\in D^{b}_{c}(X;R)$ and $f$ is an analytic map so
that the restriction of $f$ to ${\rm supp}({\mathscr{F}}^{\bullet})$ is proper
(e.g., $f$ is proper), then $Rf_{*}{\mathscr{F}}^{\bullet}\simeq
Rf_{!}{\mathscr{F}}^{\bullet}\in D^{b}_{c}(Y;R)$.

  3. (c)

If ${\mathscr{G}}^{\bullet}\in D^{b}_{c}(Y;R)$, then
$Rj_{*}j^{*}{\mathscr{G}}^{\bullet},Rj_{!}j^{*}{\mathscr{G}}^{\bullet}\in
D^{b}_{c}(Y;R)$.

  4. (d)

If ${\mathscr{F}}^{\bullet},{\mathscr{G}}^{\bullet}\in D^{b}_{c}(X;R)$, then
${\mathscr{F}}^{\bullet}\overset{L}{\otimes}{\mathscr{G}}^{\bullet},R{\mathscr{H}}om^{\bullet}({\mathscr{F}}^{\bullet},{\mathscr{G}}^{\bullet})\in
D^{b}_{c}(X;R)$.

Similarly for weakly constructible instead of constructible complexes.

In fact, the above theorem in an application of the following more precise
version, where one considers a stratified morphism with respect to given
Whitney stratifications (e.g., see [11] or [113, Proposition 4.0.2 and
Corollary 4.2.1]):

######  Theorem 2.6.

Let $f\colon X\to Y$ be a stratified morphism of complex algebraic (or
analytic) varieties mapping all strata $S\in{\mathscr{S}}$ of a Whitney
stratification of $X$ into strata $T\in{\mathscr{T}}$ of a Whitney
stratification of $Y$, with $j\colon U\hookrightarrow Y$ the inclusion of the
open complement of a closed union of strata of ${\mathscr{T}}$ (with its
induced Whitney stratification ${\mathscr{T}}|_{U}$).

  1. (a)

If ${\mathscr{G}}^{\bullet}\in D^{b}_{{\mathscr{T}}-c}(Y;R)$, then
$f^{*}{\mathscr{G}}^{\bullet},f^{!}{\mathscr{G}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-c}(X;R)$.

  2. (b)

If ${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-c}(X;R)$ and $f$ is a
stratified submersion (i.e., it maps all strata $S\in{\mathscr{S}}$
submersively to a stratum $T\in{\mathscr{T}}$) so that the restriction of $f$
to ${\rm supp}({\mathscr{F}}^{\bullet})$ is proper (e.g., $f$ is proper), then
$Rf_{*}{\mathscr{F}}^{\bullet}\simeq Rf_{!}{\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{T}}-c}(Y;R)$.

  3. (c)

If ${\mathscr{G}}^{\bullet}\in D^{b}_{{\mathscr{T}}|_{U}-c}(U;R)$, then
$Rj_{*}{\mathscr{G}}^{\bullet},Rj_{!}{\mathscr{G}}^{\bullet}\in
D^{b}_{{\mathscr{T}}-c}(Y;R)$.

  4. (d)

If ${\mathscr{F}}^{\bullet},{\mathscr{G}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-c}(X;R)$, then
${\mathscr{F}}^{\bullet}\overset{L}{\otimes}{\mathscr{G}}^{\bullet},R{\mathscr{H}}om^{\bullet}({\mathscr{F}}^{\bullet},{\mathscr{G}}^{\bullet})\in
D^{b}_{{\mathscr{S}}-c}(X;R)$.

Similarly for weakly constructible instead of constructible complexes.

By considering the constant map $c\colon X\to pt$ to a point, one gets the
following.

######  Example 2.7.

Let $X$ be a complex algebraic (or analytic) variety with a Whitney
stratification ${\mathscr{S}}$, and let $i\colon Z\hookrightarrow X$ be the
inclusion of a closed algebraic (or analytic) subset given as a union of
strata of ${\mathscr{S}}$ with its induced Whitney stratification
${\mathscr{S}}|_{Z}$.

  1. (1)

The constant sheaf $R_{X}=c^{*}R$ is constructible with respect to
${\mathscr{S}}$, with

$H^{k}_{c}(X;R)=H^{k}(Rc_{!}c^{*}R)\quad\text{resp.,}\quad
H^{k}(X;R)=H^{k}(Rc_{*}c^{*}R)\quad(k\in{\mathbb{Z}})$

the corresponding cohomology (with compact support) of $X$.

  2. (2)

The sheaf complex $i^{!}R_{X}=i^{!}c^{*}R$ is constructible with respect to
${\mathscr{S}}|_{Z}$, with

$H^{k}_{Z}(X;R)=H^{k}(Rc_{*}i^{!}c^{*}R)=H^{k}(X,X\backslash
Z;R)\quad(k\in{\mathbb{Z}})$

the corresponding cohomology with support in $Z$ of $X$ (or relative
cohomology of the pair $(X,X\backslash Z)$).

  3. (3)

The dualizing complex ${\mathbb{D}}_{X}^{\bullet}:=c^{!}R$ of $X$ is
constructible with respect to ${\mathscr{S}}$, with

$H_{k}(X;R)=H^{-k}_{c}(X;{\mathbb{D}}_{X}^{\bullet})=H^{-k}(Rc_{!}c^{!}R)\quad(k\in{\mathbb{Z}})$

resp.,

$H_{k}^{BM}(X;R)=H^{-k}(X;{\mathbb{D}}_{X}^{\bullet})=H^{-k}(Rc_{*}c^{!}R)\quad(k\in{\mathbb{Z}})$

the corresponding (Borel-Moore) homology of $X$. If $X$ is smooth (or more
generally an $R$-homology manifold) of pure dimension $d$, then
${\mathbb{D}}_{X}^{\bullet}=c^{!}R\simeq c^{*}R[2d]$, which implies the
Poincaré duality:

$H_{2d-k}(X;R)\simeq H^{k}_{c}(X;R)\quad\text{and}\quad
H_{2d-k}^{BM}(X;R)\simeq H^{k}(X;R)\quad(k\in{\mathbb{Z}})\>.$

######  Corollary 2.8 (External tensor product).

Let $X_{i}$ be complex algebraic (or analytic) varieties, with
${\mathscr{F}}_{i}^{\bullet}\in D^{b}_{(w)c}(X_{i};R)$ for $i=1,2$. Consider
the external tensor product

${\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}:=p_{1}^{*}\left({\mathscr{F}}_{1}^{\bullet}\right)\overset{L}{\otimes}p_{2}^{*}\left({\mathscr{F}}_{2}^{\bullet}\right)\in
D^{b}_{(w)c}(X_{1}\times X_{2};R)\>,$

with $p_{i}\colon X_{1}\times X_{2}\to X_{i}$ the projection on the
corresponding factor for $i=1,2$. If ${\mathscr{F}}_{i}^{\bullet}$ is (weakly)
constructible with respect to a Whitney stratification ${\mathscr{S}}_{i}$ of
$X_{i}$ for $i=1,2$, then
${\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}$
is (weakly) constructible with respect to the product Whitney stratification
${\mathscr{S}}_{1}\times{\mathscr{S}}_{2}$ of $X_{1}\times X_{2}$ (with strata
$S_{1}\times S_{2}$ for $S_{1}\in{\mathscr{S}}_{1}$ and
$S_{2}\in{\mathscr{S}}_{2}$).

The external tensor product of (weakly) constructible complexes behaves nicely
with respect to products of two morphisms, as the following result shows (see,
e.g., [113, Eq. (1.16) on p.78, Proposition 2.0.1 and Corollary 2.0.4] for
more general versions).

######  Proposition 2.9 (Künneth isomorphisms).

Let $f_{k}\colon X_{k}\to Y_{k}$ be two morphisms of complex algebraic (or
analytic) varieties, with $i_{k}\colon Z_{k}\hookrightarrow X_{k}$ the
inclusion of a closed complex algebraic (or analytic) subset, and $j_{k}\colon
U_{k}:=X_{k}\backslash Z_{k}\hookrightarrow X_{k}$ the inclusion of the open
complement ($k=1,2$).

  1. (1)

For any ${\mathscr{F}}_{k}^{\bullet}\in D^{b}(Y_{k};R)$ for $k=1,2$ one has:

$(f_{1}\times
f_{2})^{*}\left({\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}\right)\simeq
f_{1}^{*}\left({\mathscr{F}}_{1}^{\bullet}\right)\overset{L}{\boxtimes}f_{2}^{*}\left({\mathscr{F}}_{2}^{\bullet}\right)\>.$

  2. (2)

For any ${\mathscr{F}}_{k}^{\bullet}\in D^{b}(X_{k};R)$ for $k=1,2$ one has:

$R(f_{1}\times
f_{2})_{!}\left({\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}\right)\simeq
Rf_{1!}\left({\mathscr{F}}_{1}^{\bullet}\right)\overset{L}{\boxtimes}Rf_{2!}\left({\mathscr{F}}_{2}^{\bullet}\right)\>.$

  3. (3)

For any ${\mathscr{F}}_{k}^{\bullet}\in D^{b}_{wc}(Y_{k};R)$ for $k=1,2$ one
has:

$(i_{1}\times
i_{2})^{!}\left({\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}\right)\simeq
i_{1}^{!}\left({\mathscr{F}}_{1}^{\bullet})\right)\overset{L}{\boxtimes}i_{2}^{!}\left({\mathscr{F}}_{2}^{\bullet}\right)\>.$

  4. (4)

For any ${\mathscr{F}}_{k}^{\bullet}\in D^{b}_{wc}(X_{k};R)$ for $k=1,2$ one
has in the complex algebraic context:

$R(f_{1}\times
f_{2})_{*}\left({\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}\right)\simeq
Rf_{1*}\left({\mathscr{F}}_{1}^{\bullet}\right)\overset{L}{\boxtimes}Rf_{2*}\left({\mathscr{F}}_{2}^{\bullet}\right)\>.$

  5. (5)

For any ${\mathscr{F}}_{k}^{\bullet}\in D^{b}_{wc}(X_{k};R)$ for $k=1,2$ one
has:

$R(j_{1}\times j_{2})_{*}(j_{1}\times
j_{2})^{*}\left({\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}\right)\simeq
Rj_{1*}j_{1}^{*}\left({\mathscr{F}}_{1}^{\bullet}\right)\overset{L}{\boxtimes}Rj_{2*}j_{2}^{*}\left({\mathscr{F}}_{2}^{\bullet}\right)\>.$

By taking for both morphisms $f_{k}$ the constant map $f_{k}=c:X_{k}\to pt$ to
a point space ($k=1,2$) one gets the following.

######  Example 2.10 (Classical Künneth formulae).

For any ${\mathscr{F}}_{k}^{\bullet}\in D^{b}_{wc}(X_{k};R)$ for $k=1,2$ one
has:

$R\Gamma_{c}\left(X_{1}\times
X_{2};{\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}\right)\simeq
R\Gamma_{c}\left(X_{1};{\mathscr{F}}_{1}^{\bullet}\right)\overset{L}{\otimes}R\Gamma_{c}\left(X_{2};{\mathscr{F}}_{2}^{\bullet}\right)\>,$
$R\Gamma_{Z_{1}\times Z_{2}}\left(X_{1}\times
X_{2};{\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}\right)\simeq
R\Gamma_{Z_{1}}\left(X_{1};{\mathscr{F}}_{1}^{\bullet}\right)\overset{L}{\otimes}R\Gamma_{Z_{2}}\left(X_{2};{\mathscr{F}}_{2}^{\bullet}\right)\>,$

and in the complex algebraic context:

$R\Gamma\left(X_{1}\times
X_{2};{\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}\right)\simeq
R\Gamma\left(X_{1};{\mathscr{F}}_{1}^{\bullet}\right)\overset{L}{\otimes}R\Gamma\left(X_{2};{\mathscr{F}}_{2}^{\bullet}\right)\>.$

In the special case when $R$ is a field, one further gets

$H^{*}_{c}\left(X_{1}\times
X_{2};{\mathscr{F}}_{1}^{\bullet}\boxtimes{\mathscr{F}}_{2}^{\bullet}\right)\simeq
H^{*}_{c}\left(X_{1};{\mathscr{F}}_{1}^{\bullet}\right)\otimes
H^{*}_{c}\left(X_{2};{\mathscr{F}}_{2}^{\bullet}\right)\>,$

and in the complex algebraic context also

$H^{*}\left(X_{1}\times
X_{2};{\mathscr{F}}_{1}^{\bullet}\boxtimes{\mathscr{F}}_{2}^{\bullet}\right)\simeq
H^{*}\left(X_{1};{\mathscr{F}}_{1}^{\bullet}\right)\otimes
H^{*}\left(X_{2};{\mathscr{F}}_{2}^{\bullet}\right)\>.$

Another important application of the general calculus of (weakly)
constructible complexes deals with Verdier duality (see, e.g., [113, Corollary
4.2.2]):

######  Corollary 2.11.

Let $X$ be a complex algebraic (or analytic) variety with a Whitney
stratification ${\mathscr{S}}$, with morphisms $f\colon Z\to X$ and $g\colon
X\to Y$.

  1. (1)

If ${\mathscr{F}}^{\bullet}\in D^{b}(X;R)$ is ${\mathscr{S}}$-(weakly)
constructible, then its  Verdier dual

${\mathscr{D}}{\mathscr{F}}^{\bullet}:=R{\mathscr{H}}om^{\bullet}({\mathscr{F}}^{\bullet},{\mathbb{D}}_{X}^{\bullet})$

is also ${\mathscr{S}}$-(weakly) constructible, with

$f^{!}\left({\mathscr{D}}{\mathscr{F}}^{\bullet}\right)\simeq{\mathscr{D}}\left(f^{*}{\mathscr{F}}^{\bullet}\right)\quad\text{resp.,}\quad
Rg_{*}\left({\mathscr{D}}{\mathscr{F}}^{\bullet}\right)\simeq{\mathscr{D}}\left(Rg_{!}{\mathscr{F}}^{\bullet}\right)\>.$

  2. (2)

If ${\mathscr{F}}^{\bullet}\in D^{b}(X;R)$ is constructible, then  biduality
holds:

(2)
${\mathscr{F}}^{\bullet}\simeq{\mathscr{D}}{\mathscr{D}}{\mathscr{F}}^{\bullet}$

so that ${\mathscr{F}}^{\bullet}\in D^{b}(X;R)$ is
(${\mathscr{S}}$-)constructible iff its Verdier dual
${\mathscr{D}}{\mathscr{F}}^{\bullet}\in D^{b}(X;R)$ is
(${\mathscr{S}}$-)constructible. Moreover

$f^{*}\left({\mathscr{D}}{\mathscr{F}}^{\bullet}\right)\simeq{\mathscr{D}}\left(f^{!}{\mathscr{F}}^{\bullet}\right)\quad\text{resp.,}\quad
Rg_{!}\left({\mathscr{D}}{\mathscr{F}}^{\bullet}\right)\simeq{\mathscr{D}}\left(Rg_{*}{\mathscr{F}}^{\bullet}\right)\>.$

Here, for the last isomorphism one has to assume that
$Rg_{!}\left({\mathscr{D}}{\mathscr{F}}^{\bullet}\right)$ is constructible
(e.g., as in the algebraic context).

Note that already for a point space $X=\\{pt\\}$ the biduality result uses our
assumption that $R$ is a noetherian and commutative ring of finite global
dimension (see also [67, Exercise I.30]). Similarly, Verdier duality commutes
for constructible sheaf complexes with external tensor products (see, e.g.,
[113, Corollary 2.0.4] in a more general context).

######  Proposition 2.12.

Let $X_{i}$ be a complex algebraic (or analytic) variety with
${\mathscr{F}}^{\bullet}_{i}\in D^{b}_{c}(X;R)$ for $i=1,2$. Then

${\mathscr{D}}\left({\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}\right)\simeq{\mathscr{D}}\left({\mathscr{F}}_{1}^{\bullet}\right)\overset{L}{\boxtimes}{\mathscr{D}}\left({\mathscr{F}}_{2}^{\bullet}\right)\>.$

In particular, ${\mathbb{D}}_{X_{1}\times
X_{2}}^{\bullet}\simeq{\mathbb{D}}_{X_{1}}^{\bullet}\overset{L}{\boxtimes}{\mathbb{D}}_{X_{2}}^{\bullet}$,
so that one gets the classical Künneth formula for homology, e.g., for $R$ a
field:

$H_{*}(X_{1}\times X_{2};R)\simeq H_{*}(X_{1};R)\otimes H_{*}(X_{2};R)\>,$

and in the algebraic context also for Borel Moore homology, e.g., for $R$ a
field:

$H^{BM}_{*}(X_{1}\times X_{2};R)\simeq H^{BM}_{*}(X_{1};R)\otimes
H^{BM}_{*}(X_{2};R)\>.$

The general calculus of constructible sheaves also includes finiteness results
for the cohomology (with compact support) of constructible sheaf complexes.

######  Corollary 2.13.

Assume that ${\mathscr{F}}^{\bullet}\in D^{b}_{c}(X;R)$ and that either

  1. (a)

$X$ is a complex algebraic variety, or

  2. (b)

$X$ is an analytic space and ${\rm supp}({\mathscr{F}}^{\bullet})$ is compact.

Then the hypercohomology groups $H^{i}(X;{\mathscr{F}}^{\bullet})$ and
$H_{c}^{i}(X;{\mathscr{F}}^{\bullet})$ are finite type $R$-modules for every
$i\in\mathbb{Z}$ (which are zero for $|i|$ large enough).

Assume $X$ is a compact analytic space with a Whitney stratification
${\mathscr{S}}$ so that $j\colon U\hookrightarrow X$ is the inclusion of the
open complement of a closed union of strata. If ${\mathscr{G}}^{\bullet}\in
D^{b}_{{\mathscr{S}}|_{U}-c}(U;R)$, then also the hypercohomology groups
$H^{i}(U;{\mathscr{G}}^{\bullet})$ and $H_{c}^{i}(U;{\mathscr{G}}^{\bullet})$
are finite type $R$-modules for every $i\in\mathbb{Z}$ (which are zero for
$|i|$ large enough).

With ${\mathscr{F}}^{\bullet}\in D^{b}_{c}(X;R)$ as in the above corollary, we
make the following.

######  Definition 2.14.

Assume $R$ is a field. The (compactly supported) Euler characteristic of
${\mathscr{F}}^{\bullet}\in D^{b}_{c}(X;R)$ is defined as:

$\chi_{(c)}(X,{\mathscr{F}}^{\bullet}):=\chi(H^{*}_{(c)}(X;{\mathscr{F}}^{\bullet})):=\sum_{i\in\mathbb{Z}}(-1)^{i}\dim_{R}H_{(c)}^{i}(X;{\mathscr{F}}^{\bullet}).$

(Here, we use the notation $\chi_{(c)}$ and $H_{(c)}^{i}$ to indicate that the
definition applies to the compactly supported Euler characteristic $\chi_{c}$
by using $H_{c}^{i}$, as well as to the usual Euler characteristic $\chi$ by
using $H^{i}$.)

As it will be explained later on (see Example 4.37, and also [113, Section 2.3
and Section 6.0.6]), in this complex context we have the equality:

(3) $\chi(X,{\mathscr{F}}^{\bullet})=\chi_{c}(X,{\mathscr{F}}^{\bullet}).$

Moreover, this Euler characteristic depends only on the associated
constructible function

(4) $\chi_{stalk}\left({\mathscr{F}}^{\bullet}\right)\in CF(X)$

given by the stalkwise Euler characteristic
$\chi_{stalk}\left({\mathscr{F}}^{\bullet}\right)(x):=\chi\left({\mathscr{F}}^{\bullet}_{x}\right)$
for $x\in X$, with $CF(X)$ the corresponding abelian group of constructible
functions given by (locally) finite ${\mathbb{Z}}$-linear combinations of
indicator functions $1_{Z}$, for $Z\subset X$ a closed irreducible algebraic
(or analytic) subset of $X$. Similarly for the abelian group
$CF_{{\mathscr{S}}}(X)$ of ${\mathscr{S}}$-constructible functions given by
(locally) finite ${\mathbb{Z}}$-linear combinations of indicator functions
$1_{S}$ or $1_{\bar{S}}$ for $S\in{\mathscr{S}}$ a stratum, i.e.,
${\mathbb{Z}}$-valued functions which are constant on all strata
$S\in{\mathscr{S}}$. This induces a surjective group homomorphism

(5) $\chi_{stalk}\colon\>K_{0}\left(D^{b}_{({\mathscr{S}}-)c}(X;R)\right)\to
CF_{({\mathscr{S}})}(X)\>,$

with

(6)
$\chi_{c}(X,{\mathscr{F}}^{\bullet})=\sum_{S\in{\mathscr{S}}}\>\chi_{c}(S)\cdot\chi_{stalk}\left({\mathscr{F}}^{\bullet}\right)(S)$

for an ${\mathscr{S}}$-constructible complex ${\mathscr{F}}^{\bullet}$ on $X$
in the complex algebraic context (or analytic context with $X$ compact). Here
$\chi_{c}(S):=\chi_{c}(R_{S})=\chi(H_{c}^{*}(S;R))$ is the corresponding Euler
characteristic of a stratum $S\in{\mathscr{S}}$.

Let us also mention here the following (co)stalk calculation.

######  Proposition 2.15.

Let ${\mathscr{F}^{\bullet}}\in D_{c}^{b}(X;R)$, $x\in X$, and
$i_{x}\colon\\{x\\}\hookrightarrow X$ the inclusion. Then

(7)
$\mathscr{H}^{j}({\mathscr{F}^{\bullet}})_{x}\simeq{H}^{j}(i_{x}^{*}{\mathscr{F}^{\bullet}})\simeq
H^{j}(\mathring{B}_{\epsilon,x};\mathscr{F}^{\bullet}),$ (8)
${H}^{j}(i_{x}^{!}{\mathscr{F}^{\bullet}})\simeq{H}_{c}^{j}(\mathring{B}_{\epsilon,x};\mathscr{F}^{\bullet})\simeq{H}^{j}(\mathring{B}_{\epsilon,x},\mathring{B}_{\epsilon,x}\setminus{x};\mathscr{F}^{\bullet}),$

where $\mathring{B}_{\epsilon,x}$ is the intersection of $X$ with an open
small $\epsilon$-ball neighborhood of $x$ in some local embedding of $X$ in
$\mathbb{C}^{N}$. Here, $i_{x}^{*}{\mathscr{F}}^{\bullet}$ and
$i_{x}^{!}{\mathscr{F}}^{\bullet}$ are called the stalk and, respectively,
costalk of ${\mathscr{F}}^{\bullet}$ at $x$.

In fact, this will be an easy application of the Morse theoretical results
explained later on, since the proper real analytic function $r$ given by the
squared distance to $x$ (in these local coordinates) has no stratified
critical values in an interval $\,]0,\epsilon[\,$ for $\epsilon>0$ small
enough (see Lemma 3.3 and, e.g., [113, Lemma 5.1.1]).

###  2.2. Perverse sheaves

Perverse sheaves are an important class of constructible complexes, introduced
in [6] as a formalization of Kashiwara’s Riemann–Hilbert correspondence [65]
(see also [58]), which relates the topology of complex algebraic, resp.,
analytic varieties (intersection homology) and the algebraic, resp., analytic
theory of differential equations (holonomic $D$-modules). We recall their
definition below.

######  Definition 2.16.

(a) The perverse t-structure on $D_{c}^{b}(X;R)$ consists of the two strictly
full subcategories ${{}^{p}D}^{\leq 0}(X;R)$ and ${{}^{p}D}^{\geq 0}(X;R)$ of
$D_{c}^{b}(X;R)$ defined as:

${{}^{p}D}^{\leq 0}(X;R):=\\{{\mathscr{F}}^{\bullet}\in
D^{b}_{c}(X;R)\mid\dim{\rm supp}^{-j}({\mathscr{F}}^{\bullet})\leq j,\forall
j\in{\mathbb{Z}}\\},$ ${{}^{p}D}^{\geq 0}(X;R):=\\{{\mathscr{F}}^{\bullet}\in
D^{b}_{c}(X;R)\mid\dim{\rm cosupp}^{j}({\mathscr{F}}^{\bullet})\leq j,\forall
j\in{\mathbb{Z}}\\},$

where, for $i_{x}\colon\\{x\\}\hookrightarrow X$ denoting the point inclusion,
we define the $j$-th support and, respectively, the $j$-th cosupport of
${\mathscr{F}}^{\bullet}\in D^{b}_{c}(X;R)$ by:

${\rm supp}^{j}({\mathscr{F}}^{\bullet})=\overline{\\{x\in X\mid
H^{j}(i_{x}^{*}{\mathscr{F}}^{\bullet})\neq 0\\}},$ ${\rm
cosupp}^{j}({\mathscr{F}}^{\bullet})=\overline{\\{x\in X\mid
H^{j}(i_{x}^{!}{\mathscr{F}}^{\bullet})\neq 0\\}}.$

(For a constructible complex ${\mathscr{F}}^{\bullet}$, the sets
$\mathrm{supp}^{j}(\mathscr{F}^{\bullet})$ and
$\mathrm{cosupp}^{j}(\mathscr{F}^{\bullet})$ are closed algebraic (or
analytic) subvarieties of $X$, hence their dimensions are well defined.)
(b) For a given Whitney stratification ${\mathscr{S}}$ of $X$ this also
induces the perverse t-structure on $D_{{\mathscr{S}}-c}^{b}(X;R)$ with
${{}^{p}D}_{{\mathscr{S}}}^{\leq 0}(X;R)$, resp.,
${{}^{p}D}_{{\mathscr{S}}}^{\geq 0}(X;R)$ defined by the same (co)support
conditions.
(c) A (${\mathscr{S}}$-)constructible complex ${\mathscr{F}}^{\bullet}\in
D^{b}_{({\mathscr{S}}-)c}(X;R)$ is called a perverse sheaf on $X$ if

${\mathscr{F}}^{\bullet}\in
Perv_{({\mathscr{S}})}(X;R):={{}^{p}D}_{({\mathscr{S}})}^{\leq
0}(X;R)\cap{{}^{p}D}_{({\mathscr{S}})}^{\geq 0}(X;R).$

The category of (${\mathscr{S}}$-constructible) perverse sheaves is the heart
of the perverse t-structure, hence it is an abelian category, and it is stable
by extensions (see, e.g., [6, Theorem 1.3.6]).

######  Remark 2.17.

(a) The same definition also defines the perverse t-structure for
(${\mathscr{S}}$-)
weakly constructible complexes with heart the abelian category of
(${\mathscr{S}}$-)weakly constructible perverse sheaves.
(b) An algebraically (weakly) constructible complex is perverse if and only if
it is so when viewed as an analytically (weakly) constructible complex.
(c) If $R$ is a field, the Universal Coefficient Theorem can be used to show
that the Verdier duality functor ${\mathscr{D}}\colon D^{b}_{c}(X;R)\to
D^{b}_{c}(X;R)$ satisfies:

(9) ${\rm cosupp}^{j}({\mathscr{F}}^{\bullet})={\rm
supp}^{-j}({\mathscr{D}}{\mathscr{F}}^{\bullet}),$

In particular, ${\mathscr{D}}$ exchanges ${{}^{p}D}^{\leq 0}(X;R)$ and
${{}^{p}D}^{\geq 0}(X;R)$, so that it preserves (${\mathscr{S}}$-)
constructible perverse sheaves with field coefficients.

Recall here that two subcategories ${{}^{p}D}^{\leq 0}(X;R)$ and
${{}^{p}D}^{\geq 0}(X;R)$ of $D_{({\mathscr{S}}-w)c}^{b}(X;R)$ define a
t-structure just means (see, e.g., [6, Definition 1.3.1]):

  1. (1)

${\rm{Hom}}_{D^{b}(X;R)}({\mathscr{F}}^{\bullet},{\mathscr{G}}^{\bullet}[-1])=0$
for all ${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$ and
${\mathscr{G}}^{\bullet}\in{{}^{p}D}^{\geq 0}(X;R)$.

  2. (2)

${{}^{p}D}^{\leq 0}(X;R)$ is stable under $[1]$, and ${{}^{p}D}^{\geq 0}(X;R)$
is stable under $[-1]$.

  3. (3)

For any ${\mathscr{E}}^{\bullet}\in D_{({\mathscr{S}}-w)c}^{b}(X;R)$ there is
a distinguished triangle

${\mathscr{F}}^{\bullet}\longrightarrow{\mathscr{E}}^{\bullet}\longrightarrow{\mathscr{G}}^{\bullet}[-1]\overset{[1]}{\longrightarrow}$

for some ${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$ and
${\mathscr{G}}^{\bullet}\in{{}^{p}D}^{\geq 0}(X;R)$.

Then it is enough to check these properties for a fixed Whitney stratification
${\mathscr{S}}$, where they can be proved by induction on $\dim X$ via the
gluing of t-structures as in [6, Corollary 2.1.4, Proposition 2.1.14]. Here it
is important to note that the conditions
${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$ and, resp.,
${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\geq 0}(X;R)$ can also be described in
terms of a fixed Whitney stratification of $X$ for which
${\mathscr{F}}^{\bullet}$ is ${\mathscr{S}}$-(weakly) constructible. Indeed,
the perverse t-structure can be characterized as follows:

######  Theorem 2.18.

Assume ${\mathscr{F}}^{\bullet}\in D^{b}_{(w)c}(X;R)$ is (weakly)
constructible with respect to a Whitney stratification ${\mathscr{S}}$ of $X$.
Then:

  * (i)

stalk vanishing:

${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$ $\iff$ $\forall
S\in{\mathscr{S}}$, $\forall x\in S$:
$H^{j}(i_{x}^{*}{\mathscr{F}}^{\bullet})=0$ for all $j>-\dim S$.

  * (ii)

costalk vanishing:

${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\geq 0}(X;R)$ $\iff$ $\forall
S\in{\mathscr{S}}$, $\forall x\in S$:
$H^{j}(i_{x}^{!}{\mathscr{F}}^{\bullet})=0$ for all $j<\dim S$.

Of course the isomorphism class of the stalk
$i_{x}^{*}{\mathscr{F}}^{\bullet}$ for $x\in S$ as above does not depend on
the choice of the point $x\in S$. In fact, if $i_{S}\colon S\hookrightarrow X$
denotes the inclusion of the stratum $S$, the stalk vanishing condition is
equivalent to

  * (i’)

${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$ $\iff$ $\forall
S\in{\mathscr{S}}$: ${\mathscr{H}}^{j}(i_{S}^{*}{\mathscr{F}}^{\bullet})=0$
for all $j>-\dim S$.

Similarly for the costalk $i_{x}^{!}{\mathscr{F}}^{\bullet}$ for $x\in S$,
since this costalk is also isomorphic to
$k_{x}^{!}i_{S}^{!}{\mathscr{F}}^{\bullet}$ with
$k_{x}\colon\\{x\\}\hookrightarrow S$ the inclusion of the point $\\{x\\}$
into $S$. But $i_{S}^{!}{\mathscr{F}}^{\bullet}$ has locally constant
cohomology sheaves so that $k_{x}^{!}i_{S}^{!}{\mathscr{F}}^{\bullet}[2\dim
S]\simeq k_{x}^{*}i_{S}^{!}{\mathscr{F}}^{\bullet}$ for all $x\in S$. And then
the costalk condition is equivalent to

  * (ii’)

$\;{\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\geq 0}(X;R)$ $\iff$ $\forall
S\in{\mathscr{S}}$: ${\mathscr{H}}^{j}(i_{S}^{!}{\mathscr{F}}^{\bullet})=0$
for all $j<-\dim S$.

######  Example 2.19.

Assume $X$ is of pure complex dimension with $c\colon X\to pt$ the constant
map to a point space. As usual, for an $R$-module $M$ we denote by
$M_{X}=c^{*}M$ the constant $R$-sheaf on $X$ with stalk $M_{x}=M$ for all
$x\in X$. Then:

  * (a)

${M}_{X}[\dim X]\in{{}^{p}D}^{\leq 0}(X;R)$. More generally, if
${\mathscr{L}}$ is a local system on $X$, then ${\mathscr{L}}[\dim
X]\in{{}^{p}D}^{\leq 0}(X;R)$ (since this is a local condition).

  * (b)

If $X$ is smooth with the trivial stratification, and ${\mathscr{L}}$ is a
local system on $X$, then ${\mathscr{L}}[\dim X]$ is perverse on $X$ (since in
this case $i_{x}^{!}{\mathscr{L}}[2\dim X]\simeq i_{x}^{*}{\mathscr{L}}$ for
all $x\in X$).

  * (c)

The intersection ($IC$) complexes on $X$ of Goresky-MacPherson [46] (for the
middle perversity) are examples of perverse sheaves. For a given Whitney
stratification ${\mathscr{S}}$ of $X$ these are defined by further imposing,
for all strata $S\in{\mathscr{S}}$ with $\dim S<\dim X$, the stronger
stalk/costalk vanishing conditions in Theorem 2.18 obtained by replacing $>$
with $\geq$ and $<$ with $\leq$. Such an $IC$ complex on $X$ is
${\mathscr{S}}$-weakly constructible and determined by its restriction to the
top dimensional strata, which is (isomorphic to) a shifted local system
${\mathscr{L}}[\dim X]$, so we may denote it unambiguously by
$IC_{X}({\mathscr{L}})$. It is ${\mathscr{S}}$-constructible if
${\mathscr{L}}$ has finitely generated stalks. If ${\mathscr{L}}$ is the
constant sheaf $R$ on the top dimensional strata, we use the notation
$IC_{X}$. In the end, these perverse $IC$-complexes do not depend on the
chosen Whitney stratification ${\mathscr{S}}$, but only on the generically
defined local system ${\mathscr{L}}$ (on the complement of a closed algebraic
or analytic subset of $X$ of dimension smaller than $\dim X$).

  * (d)

If $X$ is a local complete intersection then ${R}_{X}[\dim X]$ is a perverse
sheaf on $X$ (see Example 4.29 and, e.g., [113, Example 6.0.11]). More
generally, if ${\mathscr{L}}$ is a local system on $X$, then
${\mathscr{L}}[\dim X]$ is perverse on $X$. By (a) one only has to show that
${\mathscr{L}}[\dim X]\in{{}^{p}D}^{\geq 0}(X;R)$. Since this is a local
condition in the classical topology, we can assume ${\mathscr{L}}=M_{X}$ is a
constant sheaf, with $i\colon X\hookrightarrow X^{\prime}$ the inclusion of a
closed analytic subset defined as the zero set of $k$ holomorphic functions on
a pure dimensional complex manifold $X^{\prime}$ with $\dim X^{\prime}-k=\dim
X$. Then $M_{X}=i^{*}M_{X^{\prime}}$ with $M_{X^{\prime}}[\dim
X^{\prime}]\in{{}^{p}D}^{\geq 0}(X^{\prime};R)$ by (b), so that the claim
follows from Proposition 4.28 below (see, e.g., [113, Proposition 6.0.2]).

To simplify the formulation of some results, let us recall the following.

######  Definition 2.20.

For $n\in{\mathbb{Z}}$, define the following two strictly full subcategories
${{}^{p}D}^{\leq n}_{({\mathscr{S}})}(X;R)$ and ${{}^{p}D}^{\geq
n}_{({\mathscr{S}})}(X;R)$ of $D_{({\mathscr{S}}-)c}^{b}(X;R)$, resp.,
$D_{({\mathscr{S}}-)wc}^{b}(X;R)$ by

${{}^{p}D}^{\leq n}_{({\mathscr{S}})}(X;R):={{}^{p}D}^{\leq
0}_{({\mathscr{S}})}(X;R)[-n]\quad\text{and}\quad{{}^{p}D}^{\geq
n}_{({\mathscr{S}})}(X;R):={{}^{p}D}^{\geq 0}_{({\mathscr{S}})}(X;R)[-n]$

so that ${{}^{p}D}^{\leq-1}_{({\mathscr{S}})}(X;R)\subset{{}^{p}D}^{\leq
0}_{({\mathscr{S}})}(X;R)$ and ${{}^{p}D}^{\geq
1}_{({\mathscr{S}})}(X;R)\subset{{}^{p}D}^{\geq 0}_{({\mathscr{S}})}(X;R)$.

######  Remark 2.21.

The two subcategories ${{}^{p}D}^{\leq n}_{({\mathscr{S}})}(X;R)$ and
${{}^{p}D}^{\geq n}_{({\mathscr{S}})}(X;R)$ define a shifted perverse
t-structure on $D_{({\mathscr{S}}-)c}^{b}(X;R)$ and
$D_{({\mathscr{S}}-)wc}^{b}(X;R)$. But for most of the results of this paper
(especially those proved by stratified Morse theory later on), we only need
the following obvious properties (as in [113, Chapter VI]):

  1. (1)

The zero object belongs to ${{}^{p}D}^{\leq n}_{({\mathscr{S}})}(X;R)$ and
${{}^{p}D}^{\geq n}_{({\mathscr{S}})}(X;R)$.

  2. (2)

${{}^{p}D}^{\leq n}_{({\mathscr{S}})}(X;R)$ and ${{}^{p}D}^{\geq
n}_{({\mathscr{S}})}(X;R)$ are stable by extensions.

  3. (3)

${{}^{p}D}^{\leq n}(X;R)$ is stable under $[1]$, and ${{}^{p}D}^{\geq n}(X;R)$
is stable under $[-1]$.

######  Example 2.22.

On a point space $X=\\{pt\\}$ one gets:

  1. (1)

${{}^{p}D}^{\leq n}(\\{pt\\};R)$ is given by the bounded complexes of
$R$-modules, whose cohomology is concentrated in degree $\leq n$. This
condition is stable under shifting a complex to the left (i.e., the shift
$[1]$).

  2. (2)

${{}^{p}D}^{\geq n}(\\{pt\\};R)$ is given by the bounded complexes of
$R$-modules, whose cohomology is concentrated in degree $\geq n$. This
condition is stable under shifting a complex to the right (i.e., the shift
$[-1]$).

######  Lemma 2.23.

Let $f\colon X\to Y$ be a morphism of complex algebraic (or analytic)
varieties, whose fiber dimension is bounded by $d\in{\mathbb{N}}_{0}$. Then:

  * (a)

$f^{!}$ maps ${{}^{p}D}^{\geq n}(Y;R)$ into ${{}^{p}D}^{\geq n-d}(X;R)$.

  * (b)

$f^{*}$ maps ${{}^{p}D}^{\leq n}(Y;R)$ into ${{}^{p}D}^{\leq n+d}(X;R)$.

This follows from the definitions using $i_{x}^{!}f^{!}\simeq i_{f(x)}^{!}$
and $i_{x}^{*}f^{*}\simeq i_{f(x)}^{*}$ for all $x\in X$. As an example, one
can take for $f$ a locally closed inclusion or an unramified covering map,
both of which have fiber dimension $d=0$.

######  Example 2.24.

Let $X$ be a complex algebraic (or analytic) variety of dimension $d=\dim X$,
with $c\colon X\to pt$ a constant map. Then

${\mathbb{D}}_{X}^{\bullet}=c^{!}R\in{{}^{p}D}^{\geq-d}(X;R)\quad\text{and}\quad
R_{X}=c^{*}R\in{{}^{p}D}^{\leq d}(X;R)\>.$

######  Corollary 2.25.

Let $f\colon X\to Y$ be a smooth morphism (i.e., a submersion) of complex
algebraic (or analytic) varieties, with constant relative (or fiber) dimension
$d\in{\mathbb{N}}_{0}$. Then $f^{!}\simeq f^{*}[2d]$ so that

  * (a)

$f^{*}$ maps ${{}^{p}D}^{\geq n}(Y;R)$ into ${{}^{p}D}^{\geq n+d}(X;R)$.

  * (b)

$f^{!}$ maps ${{}^{p}D}^{\leq n}(Y;R)$ into ${{}^{p}D}^{\leq n-d}(X;R)$.

In particular $f^{!}[-d]\simeq f^{*}[d]$ maps $Perv(Y;R)$ into $Perv(X;R)$.

######  Example 2.26.

Let $f\colon X\to Y$ be a smooth morphism (i.e., a submersion) of complex
algebraic (or analytic) varieties, with constant relative (or fiber) dimension
$d$. Assume $f$ is surjective and $Y$ (and then also $X$) is pure dimensional.
Then

$f^{*}IC_{Y}({\mathscr{L}})[d]\simeq IC_{X}(f^{*}{\mathscr{L}})$

for a generically defined local system ${\mathscr{L}}$ on $Y$, with
$f^{*}{\mathscr{L}}$ the corresponding generically defined local system on $X$
defined by pullback.

The following result will be very important for the applications of the
stratified Morse theory for constructible sheaves in the next sections.

######  Proposition 2.27.

Let $Y\hookrightarrow M$ be a closed complex algebraic (or analytic)
subvariety of an ambient complex algebraic (or analytic) manifold $M$. Assume
$N\hookrightarrow M$ is a closed complex algebraic (or analytic) submanifold
of constant codimension $d=\dim M-\dim N$, which is transversal to a Whitney
stratification ${\mathscr{S}}$ of $Y$ (i.e., $N$ is transversal to all strata
$S\in{\mathscr{S}}$). Then $X:=Y\cap N$ gets an induced Whitney stratification
${\mathscr{S}}^{\prime}$ with strata $S^{\prime}$ the connected components of
the intersections $S\cap N$ for $S\in{\mathscr{S}}$, with $\dim S\cap N=\dim
S-d$ for all $S\in{\mathscr{S}}$ (and $S\cap N\neq\emptyset$). Let $i\colon
X=Y\cap N\hookrightarrow Y$ be the (stratified) closed inclusion. Then

  * (a)

$i^{*}$ maps ${{}^{p}D}_{{\mathscr{S}}}^{\leq n}(Y;R)$ into ${{}^{p}D}^{\leq
n-d}_{{\mathscr{S}}^{\prime}}(X;R)$.

  * (b)

$i^{*}$ maps ${{}^{p}D}^{\geq n}_{{\mathscr{S}}}(Y;R)$ into ${{}^{p}D}^{\geq
n-d}_{{\mathscr{S}}^{\prime}}(X;R)$.

In particular $i^{*}[-d]$ maps $Perv_{{\mathscr{S}}}(Y;R)$ into
$Perv_{{\mathscr{S}}^{\prime}}(X;R)$.

###### Proof.

Consider the following cartesian diagram of closed inclusions for
$S\in{\mathscr{S}}$:

$\begin{CD}S^{\prime}:=S\cap N@>{i_{S^{\prime}}}>{}>X=Y\cap N\\\
@V{i^{\prime}}V{}V@V{}V{i}V\\\ <EMAIL_ADDRESS>
Then (a) follows from $i^{*}_{S^{\prime}}i^{*}\simeq i^{\prime*}i^{*}_{S}$ for
checking the stalk vanishing condition (i’). Similarly (b) follows from the
base change isomorphism

(10) $i^{!}_{S^{\prime}}i^{*}{\mathscr{F}}^{\bullet}\simeq
i^{\prime*}i^{!}_{S}{\mathscr{F}}^{\bullet}\quad\text{for
${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-wc}(X;R)$}$

for checking the costalk vanishing condition (ii’). ∎

######  Example 2.28.

Consider the context of Proposition 2.27, with $Y$ (and therefore also
$X=Y\cap N$) pure dimensional. Let ${\mathscr{L}}$ be a local system on the
open subset $U$ of $Y$ given by the top dimensional stratum of ${\mathscr{S}}$
(which is then dense in $Y$). Similarly $U\cap N$ is open and dense in $X$.
Then

$i^{*}IC_{Y}({\mathscr{L}})[-d]\simeq IC_{X}(i^{\prime*}{\mathscr{L}})\>,$

with $i^{\prime}\colon U\cap N\to U$ the induced inclusion.

In the base change isomorphism (10) used above, we can even assume that
$S\hookrightarrow Y$ is a closed stratum, by restriction to the open
complement of $\partial S$. Then it is a special case (with $Y^{\prime}=S$) of
the following more general result (see, e.g., [113, Proposition 4.3.1 and
Remark 4.3.6]).

######  Theorem 2.29 (Base change isomorphisms).

Let $Y\hookrightarrow M$ be a closed complex algebraic (or analytic)
subvariety of an ambient complex algebraic (or analytic) manifold $M$. Assume
$N\hookrightarrow M$ is a closed complex algebraic (or analytic) submanifold,
which is transversal to a Whitney stratification ${\mathscr{S}}$ of $Y$. Let
$i\colon Y^{\prime}\hookrightarrow Y$ be the inclusion of a closed union of
strata of ${\mathscr{S}}$, with $j\colon U:=Y\backslash
Y^{\prime}\hookrightarrow Y$ the inclusion of the open complement with its
induced stratification ${\mathscr{S}}|_{U}$. Consider the cartesian diagram

$\begin{CD}X^{\prime}:=Y^{\prime}\cap N@>{i^{\prime}}>{}>X:=Y\cap
N@<{j^{\prime}}<{}<U^{\prime}:=U\cap N\\\
@V{}V{k^{\prime}}V@V{}V{k}V@V{}V{k^{\prime\prime}}V\\\
Y^{\prime}<EMAIL_ADDRESS>
Then one has the following base change isomorphisms:

(11) $k^{*}Rj_{*}{\mathscr{F}}^{\bullet}\simeq
Rj^{\prime}_{*}k^{\prime\prime*}{\mathscr{F}}^{\bullet}\quad\text{for
${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}|_{U}-wc}(U;R)$}$

and

(12) $k^{\prime*}i^{!}{\mathscr{F}}^{\bullet}\simeq
i^{\prime!}k^{*}{\mathscr{F}}^{\bullet}\quad\text{for
${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-wc}(Y;R)$.}$

Next we study the relation between the perverse t-structure and external
tensor products.

######  Proposition 2.30.

Let $X_{i}$ be complex algebraic (or analytic) varieties ($i=1,2$). Then

  * (a)

The external tensor product $\overset{L}{\boxtimes}$ induces

$\overset{L}{\boxtimes}:\>{{}^{p}D}^{\leq n}(X_{1};R)\times{{}^{p}D}^{\leq
m}(X_{2};R)\to{{}^{p}D}^{\leq n+m}(X_{1}\times X_{2};R)\>.$

  * (b)

Assume $R$ is a field. Thent $\overset{L}{\boxtimes}$ induces

$\overset{L}{\boxtimes}:\>{{}^{p}D}^{\geq n}(X_{1};R)\times{{}^{p}D}^{\geq
m}(X_{2};R)\to{{}^{p}D}^{\geq n+m}(X_{1}\times X_{2};R)\>.$

In particular, if $R$ is a field, $\overset{L}{\boxtimes}$ induces

$\overset{L}{\boxtimes}:\>Perv(X_{1};R)\times Perv(X_{2};R)\to
Perv(X_{1}\times X_{2};R)\>.$

Property (a) follows from
$i_{(x_{1},x_{2})}^{*}(-\overset{L}{\boxtimes}-)\simeq
i_{x_{1}}^{*}(-)\overset{L}{\otimes}i_{x_{2}}^{*}(-)$ and the right exactness
of the tensor product $\otimes$. Property (b) is a consequence of the Künneth
isomorphism $i_{(x_{1},x_{2})}^{!}(-\overset{L}{\boxtimes}-)\simeq
i_{x_{1}}^{!}(-)\overset{L}{\otimes}i_{x_{2}}^{!}(-)$ and the exactness of the
tensor product $\otimes$ for $R$ a field.

######  Example 2.31.

Let $X_{i}$ be pure dimensional complex algebraic (or analytic) varieties
($i=1,2$), with $R$ a field. Then

$IC_{X_{1}}({\mathscr{L}}_{1})\overset{L}{\boxtimes}IC_{X_{2}}({\mathscr{L}}_{2})\simeq
IC_{X_{1}\times X_{2}}({\mathscr{L}}_{1}\boxtimes{\mathscr{L}}_{2})$

for a generically defined local system ${\mathscr{L}}_{i}$ on $X_{i}$
($i=1,2$).

######  Example 2.32.

Let $X$ be a complex algebraic (or analytic) variety, with $R$ a field. If
${\mathscr{F}}^{\bullet}\in Perv(X;R)$ and ${\mathscr{L}}$ is a locally
constant sheaf on $X$, then ${\mathscr{F}}^{\bullet}\otimes{\mathscr{L}}\in
Perv(X;R)$. In particular, if $X$ is also pure dimensional, one gets:

$IC_{X}({\mathscr{L}}_{1})\otimes{\mathscr{L}}_{2}\simeq
IC_{X}({\mathscr{L}}_{1}\otimes{\mathscr{L}}_{2}),$

with ${\mathscr{L}}_{1}$ a generically defined local system on $X$, and
${\mathscr{L}}_{2}$ a local system on all of $X$. As a special case, one also
gets:

$IC_{X}\otimes{\mathscr{L}}\simeq IC_{X}({\mathscr{L}})$

if $X$ is pure dimensional and ${\mathscr{L}}$ is a local system on $X$. These
assertions can be checked locally in the analytic topology (so it suffices to
assume that ${\mathscr{L}}_{2}$ is a constant sheaf), by using Proposition
2.30 and Example 2.31 in which we take $X_{2}$ to be a point space.

######  Corollary 2.33.

Let $X_{i}\hookrightarrow M$ be closed complex algebraic (or analytic)
subvarieties of the complex algebraic (or analytic) manifold $M$ of pure
dimension $d$ ($i=1,2$). Let ${\mathscr{S}}_{i}$ be Whitney stratifications of
$X_{i}$ ($i=1,2$) which are transversal in $M$, i.e., all strata
$S_{1}\in{\mathscr{S}}_{1}$ are transversal to all strata
$S_{2}\in{\mathscr{S}}_{2}$. This is equivalent to the diagonal embedding
$\Delta\colon M\to M\times M$ being transversal to the product stratification
${\mathscr{S}}_{1}\times{\mathscr{S}}_{2}$ of $X_{1}\times X_{2}$. In
particular, $X_{1}\cap X_{2}\simeq\left(X_{1}\times X_{2}\right)\cap\Delta(M)$
gets an induced Whitney stratification
${\mathscr{S}}_{1}\cap{\mathscr{S}}_{2}$ with strata the connected components
of the intersections $S_{1}\cap S_{2}$. Then one gets for the induced map
$\Delta\colon X_{1}\cap X_{2}\to X_{1}\times X_{2}$ the following:

  * (a)

The tensor product
$\overset{L}{-\otimes-}\simeq\Delta^{*}(-\overset{L}{\boxtimes-})$ induces

$\overset{L}{\otimes}:\>{{}^{p}D}^{\leq
n}_{{\mathscr{S}}_{1}}(X_{1};R)\times{{}^{p}D}^{\leq
m}_{{\mathscr{S}}_{2}}(X_{2};R)\to{{}^{p}D}^{\leq
n+m-d}_{{\mathscr{S}}_{1}\cap{\mathscr{S}}_{2}}(X_{1}\cap X_{2};R)\>.$

  * (b)

Assume $R$ is a field. Then the tensor product
$\overset{L}{-\otimes-}\simeq\Delta^{*}(-\overset{L}{\boxtimes-})$ induces

$\overset{L}{\otimes}:\>{{}^{p}D}^{\geq
n}_{{\mathscr{S}}_{1}}(X_{1};R)\times{{}^{p}D}^{\geq
m}_{{\mathscr{S}}_{2}}(X_{2};R)\to{{}^{p}D}^{\geq
n+m-d}_{{\mathscr{S}}_{1}\cap{\mathscr{S}}_{2}}(X_{1}\cap X_{2};R)\>.$

In particular, if $R$ is a field, $\left(-\overset{L}{\otimes}-\right)[-d]$
induces

$Perv_{{\mathscr{S}}_{1}}(X_{1};R)\times Perv_{{\mathscr{S}}_{2}}(X_{2};R)\to
Perv_{{\mathscr{S}}_{1}\cap{\mathscr{S}}_{2}}(X_{1}\cap X_{2};R)\>.$

######  Example 2.34.

In the context of Corollary 2.33, let $X_{i}$ be in addition pure dimensional
($i=1,2$), with $R$ a field. Then

$\left(IC_{X_{1}}({\mathscr{L}}_{1})\overset{L}{\otimes}IC_{X_{2}}({\mathscr{L}}_{2})\right)[-d]\simeq
IC_{X_{1}\cap X_{2}}({\mathscr{L}}_{1}\otimes{\mathscr{L}}_{2})$

for a local system ${\mathscr{L}}_{i}$ defined on the open dense subset
$U_{i}$ of $X_{i}$ given by the top dimensional stratum ($i=1,2$). Here
$X_{1}\cap X_{2}$ is also pure dimensional with
${\mathscr{L}}_{1}\otimes{\mathscr{L}}_{2}$ defined on the open dense subset
$U_{1}\cap U_{2}$ of $X_{1}\cap X_{2}$.

The existence of the perverse t-structure on $D^{b}_{(w)c}(X;R)$ implies the
existence of perverse truncation functors ${{}^{p}\tau}_{\leq
0},{{}^{p}\tau}_{\geq 0}$, which are adjoint to the inclusions

${{}^{p}D}^{\leq 0}(X;R)\hookrightarrow
D^{b}_{(w)c}(X;R)\hookleftarrow{{}^{p}D}^{\geq 0}(X;R)\>.$

In particular, for every $k\in{\mathbb{Z}}$, there are adjunction maps
$\mathscr{F}^{\bullet}\to{}^{p}\tau_{\geq k}\mathscr{F}^{\bullet}$ and
${}^{p}\tau_{\leq k}\mathscr{F}^{\bullet}\to\mathscr{F}^{\bullet}$ (see, e.g.,
[6, Proposition 1.3.3]). These perverse truncation functors can be used to
associate to any constructible complex ${\mathscr{F}}^{\bullet}\in
D^{b}_{(w)c}(X;R)$ its perverse cohomology sheaves defined as:

${{}^{p}{\mathscr{H}}}^{i}({\mathscr{F}}^{\bullet}):={{}^{p}\tau}_{\leq
0}{{}^{p}\tau}_{\geq 0}({\mathscr{F}}^{\bullet}[i])\in Perv(X;R).$

These are ${\mathscr{S}}$-(weakly) constructible for ${\mathscr{F}}^{\bullet}$
${\mathscr{S}}$-(weakly) constructible. It then follows that
${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$ if and only if
${{}^{p}{\mathscr{H}}}^{i}({\mathscr{F}}^{\bullet})=0$ for all $i>0$.
Similarly, ${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\geq 0}(X;R)$ if and only if
${{}^{p}{\mathscr{H}}}^{i}({\mathscr{F}}^{\bullet})=0$ for all $i<0$. In
particular, ${\mathscr{F}}^{\bullet}\in Perv(X;R)$ if and only if
${{}^{p}{\mathscr{H}}}^{i}({\mathscr{F}}^{\bullet})=0$ for all $i\neq 0$ and
${{}^{p}{\mathscr{H}}}^{0}({\mathscr{F}}^{\bullet})={\mathscr{F}}^{\bullet}$
(see, e.g., [6, Proposition 1.3.7]). Perverse cohomology sheaves can be used
to calculate the (hyper)cohomology groups of any $\mathscr{F}^{\bullet}\in
D_{(w)c}^{b}(X)$ via the perverse cohomology spectral sequence

(13)
$E_{2}^{i,j}=H^{i}(X;{{}^{p}{\mathscr{H}}^{j}}({\mathscr{F}}^{\bullet}))\Longrightarrow
H^{i+j}(X;{\mathscr{F}}^{\bullet}).$

######  Definition 2.35.

A functor $F\colon D_{1}\to D_{2}$ of triangulated categories with
t-structures is left t-exact if $F(D_{1}^{\geq 0})\subseteq D_{2}^{\geq 0}$,
right t-exact if $F(D_{1}^{\leq 0})\subseteq D_{2}^{\leq 0}$, and t-exact if
$F$ is both left and right t-exact.

######  Example 2.36.

The inclusion of full subcategories

$D^{b}_{{\mathscr{S}}-(w)c}(X;R)\subset D^{b}_{(w)c}(X;R)\quad\text{and}\quad
D^{b}_{({\mathscr{S}}-)wc}(X;R)\subset D^{b}_{({\mathscr{S}}-)c}(X;R)$

are t-exact with respect to the perverse t-structures.

######  Remark 2.37.

If $F$ is a t-exact functor, it restricts to a functor on the corresponding
hearts. More generally, if $F\colon D_{1}\to D_{2}$ is a functor of
triangulated categories with t-structures, and we let
$\mathcal{C}_{1},\mathcal{C}_{2}$ be the corresponding hearts with
$k_{i}\colon\mathcal{C}_{i}\hookrightarrow D_{i}$, then

${}^{p}F:=\ ^{t}H^{0}\circ F\circ
k_{1}\colon\mathcal{C}_{1}\rightarrow\mathcal{C}_{2}$

is called the perverse functor associated to $F$. (Here, if $\tau_{\leq 0}$
and $\tau_{\geq 0}$ are the truncation functors on a triangulated category $D$
with heart $\mathcal{C}$, we set ${}^{t}H^{0}:=\tau_{\geq 0}\tau_{\leq
0}=\tau_{\leq 0}\tau_{\geq 0}\colon D\to\mathcal{C}$.) In this paper we work
only with the perverse t-structure, so a t-exact functor preserves perverse
sheaves.

######  Example 2.38.

Let $X$ be a complex analytic (or algebraic) variety, and let $Z\subseteq X$
be a closed subset. Fix a Whitney stratification ${\mathscr{S}}$ of the pair
$(X,Z)$, i.e., $Z$ is a union of strata of ${\mathscr{S}}$. Then $Z$ and
$U:=X\setminus Z$ inherit Whitney stratifications as well, and if we denote by
$i\colon Z\hookrightarrow X$ and $j\colon U\hookrightarrow X$ the stratified
inclusion maps, then the functors $j^{*}=j^{!}$, $i^{!}$, $i^{*}$,
$i_{*}=i_{!}$, $j_{!}$ and $Rj_{*}$ preserve (weak) constructibility with
respect to the above fixed stratifications, with $j^{*}i_{!}=0$ and
$i^{!}Rj_{*}=0$. Moreover, the functors $j_{!}$, $i^{*}$ are right t-exact,
the functors $j^{!}=j^{*}$, $i_{*}=i_{!}$ are t-exact, and $Rj_{*}$, $i^{!}$
are left t-exact. Similarly for the functors $j^{*}=j^{!}$, $i^{!}$, $i^{*}$,
$i_{*}=i_{!}$, $j_{!}j^{*}$ and $Rj_{*}j^{*}$ (as well as $j_{!}$ and $Rj_{*}$
in the complex algebraic context), if we do not fix a Whitney stratification.

######  Example 2.39.

Let $f\colon X\to Y$ be a smooth morphism (i.e., a submersion) of complex
algebraic (or analytic) varieties, with constant relative (or fiber) dimension
$d$. Then $f^{!}[-d]\simeq f^{*}[d]$ is t-exact. Similarly, the external
tensor product $\overset{L}{\boxtimes}$ is t-exact in each variable in case
$R$ is a field.

######  Example 2.40.

Let $f\colon X\to Y$ be a finite map (i.e., proper, with finite fibers). Then
$Rf_{*}=f_{!}$ is t-exact (see Example 3.31). If, moreover, $X$ is pure
dimensional with $f$ surjective and generically bijective, then
$Rf_{*}IC_{X}\simeq f_{*}IC_{X}\simeq IC_{Y}$. The latter fact applies, in
particular, to the case when $X$ is the (algebraic) normalization of $Y$.

The perverse cohomology sheaf construction provides a way to get perverse
sheaves out of any (weakly) constructible complex. Another important method
for constructing perverse sheaves, the intermediate extension, will be
discussed below.

Let $j\colon U\hookrightarrow X$ be the inclusion of an open constructible
subset of the complex algebraic (or analytic) variety $X$, with $i\colon
Z=X\setminus U\hookrightarrow X$ the closed inclusion. We can also work with a
fixed Whitney stratification ${\mathscr{S}}$ of $X$ so that $U$ is an open
union of strata of ${\mathscr{S}}$. A complex ${\mathscr{F}}^{\bullet}\in
D^{b}_{(w)c}(X;R)$ is a (weakly) constructible extension of
${\mathscr{G}}^{\bullet}\in D^{b}_{(w)c}(U;R)$ if
$j^{*}{\mathscr{F}}^{\bullet}\simeq{\mathscr{G}}^{\bullet}$. In what follows,
we are interested to find perverse extensions of ${\mathscr{G}}^{\bullet}\in
Perv_{({\mathscr{S}}|_{U})}(U;R)$.

######  Definition 2.41.

The intermediate extension of the perverse sheaf ${\mathscr{G}}^{\bullet}\in
Perv_{({\mathscr{S}}|_{U})}(U;R)$ is the image $j_{!*}{\mathscr{G}}^{\bullet}$
in the abelian category $Perv_{({\mathscr{S}})}(X;R)$ of the morphism

${{}^{p}j}_{!}{\mathscr{G}}^{\bullet}:={{}^{p}{\mathscr{H}}}^{0}\left(j_{!}{\mathscr{G}}^{\bullet}\right)\to{{}^{p}{\mathscr{H}}}^{0}\left(Rj_{*}{\mathscr{G}}^{\bullet}\right)=:{{}^{p}j}_{*}{\mathscr{G}}^{\bullet}.$

Here,
${{}^{p}j}_{!}{\mathscr{G}}^{\bullet}\to{{}^{p}j}_{*}{\mathscr{G}}^{\bullet}$
is obtained by applying the functor ${{}^{p}{\mathscr{H}}}^{0}$ to the natural
morphism $j_{!}{\mathscr{G}}^{\bullet}\to Rj_{*}{\mathscr{G}}^{\bullet}$ in
$D^{b}_{(w)c}(X;R)$. In the analytic context we explicitly have to assume that
$j_{!}{\mathscr{G}}^{\bullet}$ (and then also
$Rj_{*}{\mathscr{G}}^{\bullet}\simeq Rj_{*}j^{*}j_{!}{\mathscr{G}}^{\bullet}$)
is (weakly) constructible, e.g., ${\mathscr{G}}^{\bullet}\in Perv(U;R)$ is
(weakly) constructible with respect to ${\mathscr{S}}|_{U}$.

######  Example 2.42.

If $X$ is of pure dimension $n$, and $j\colon U\hookrightarrow X$ is the
inclusion of a smooth open subset whose complement is an algebraic (or
analytic) subset of dimension $<\dim X$, then for a local system
${\mathscr{L}}$ on $U$ one has that ${\mathscr{L}}[n]\in Perv(U;R)$ and

$IC_{X}({\mathscr{L}})\simeq j_{!*}({\mathscr{L}}[n])\>.$

######  Example 2.43.

Let $X$ be smooth of pure dimension one and let $j\colon U\hookrightarrow X$
be the inclusion of a Zariski open and dense subset. If ${\mathscr{L}}$ is a
local system on $U$, then:

$IC_{X}({\mathscr{L}})\simeq
j_{!*}({\mathscr{L}}[1])\simeq(j_{*}{\mathscr{L}})[1]\>.$

The intermediate extension functor plays an important role in describing the
simple objects in the abelian category of (${\mathscr{S}}-)$constructible
perverse sheaves on $X$. Indeed, we have the following.

######  Proposition 2.44.

Consider the context of Definition 2.41 above.

  * (a)

The intermediate extension $j_{!*}{\mathscr{G}}^{\bullet}$ of
${\mathscr{G}}^{\bullet}\in Perv_{({\mathscr{S}}|_{U})}(U;R)$ has no non-
trivial sub-object and no non-trivial quotient object whose supports are
contained in $Z=X\setminus U$.

  * (b)

Moreover, if ${\mathscr{G}}^{\bullet}\in Perv_{({\mathscr{S}}|_{U})}(U;R)$ is
a simple constructible object then $j_{!*}{\mathscr{G}}^{\bullet}\in
Perv_{({\mathscr{S}})}(X;R)$ is also simple constructible.

Moreover, the following important result holds, see [6, Theorem 4.3.1] (where
the assumptions are needed for the use of biduality and the stability of
perverse sheaves under duality):

######  Theorem 2.45.

Let $X$ be a complex algebraic variety, assume that the coefficient ring $R$
is a field, and consider (constructible) perverse sheaves $Perv(X;R)\subset
D^{b}_{c}(X;R)$.

  1. (a)

The category of perverse sheaves $Perv(X;R)$ is artinian and noetherian, i.e.,
every perverse sheaf on $X$ admits an increasing finite filtration with
quotients simple perverse sheaves .

  2. (b)

The simple $R$-perverse sheaves on $X$ are the twisted intersection complexes
$IC_{\overline{V}}({\mathscr{L}})$ (regarded as complexes on $X$ via extension
by zero), where $V$ runs through the family of smooth connected constructible
subvarieties of $X$, ${\mathscr{L}}$ is a simple (i.e., irreducible) $R$-local
system of finite rank on $V$, and $\overline{V}$ is the closure of $V$ in $X$.

######  Remark 2.46.

A similar result as in Theorem 2.45 is also true for a compact analytic
variety, or if one works in the algebraic or analytic context with
${\mathscr{S}}$-constructible perverse sheaves
$Perv_{{\mathscr{S}}}(X;R)\subset D^{b}_{{\mathscr{S}}-c}(X;R)$ for a fixed
Whitney stratification ${\mathscr{S}}$ of $X$ with only finitely many strata.
In the later case one needs to use in (b) for $V$ only the strata
$S\in{\mathscr{S}}$.

######  Example 2.47.

Let $R$ be a field of coefficients, and let $X={\mathbb{C}}$,
$U={\mathbb{C}}^{*}$ with open inclusion $j\colon U\hookrightarrow X$, and
$Z=\\{0\\}$ with closed inclusion $i\colon Z\hookrightarrow X$. Let
${\mathscr{L}}$ be a local system of finite rank on $U$ with stalk $V$ and
monodromy automorphism $h\colon V\to V$. Then ${\mathscr{L}}[1]\in Perv(U;R)$
and $IC_{X}({\mathscr{L}})\simeq j_{*}{\mathscr{L}}[1]$. Moreover, it is an
instructive exercise to show that the following assertions hold:

  * (a)

$j_{!}{\mathscr{L}}[1]$ and $Rj_{*}{\mathscr{L}}[1]$ are perverse sheaves on
$X$.

  * (b)

There is a short exact sequence of perverse sheaves on $X$:

(14) $0\longrightarrow IC_{X}({\mathscr{L}})\longrightarrow
Rj_{*}{\mathscr{L}}[1]\longrightarrow i_{!}IC_{Z}(V_{h})\longrightarrow 0\ ,$

where $V_{h}=\mathrm{Coker}~{}(h-1)$. In particular, if the local system
${\mathscr{L}}$ is simple, then the perverse sheaf $Rj_{*}{\mathscr{L}}[1]$
admits the filtration

$IC_{X}({\mathscr{L}})\subset Rj_{*}{\mathscr{L}}[1]$

with simple quotients $IC_{Z}({V_{h}})$ and $IC_{X}({\mathscr{L}})$.

  * (c)

There is a short exact sequence of perverse sheaves on $X$:

(15) $0\longrightarrow i_{!}IC_{Z}(V^{h})\longrightarrow
j_{!}{\mathscr{L}}[1]\longrightarrow IC_{X}({\mathscr{L}})\longrightarrow 0\
,$

where $V^{h}=\mathrm{Ker}~{}(h-1)$. Hence, if the local system ${\mathscr{L}}$
is simple, then the perverse sheaf $j_{!}{\mathscr{L}}[1]$ admits the
filtration

$IC_{Z}(V^{h})\subset Rj_{*}{\mathscr{L}}[1]$

with simple quotients $IC_{X}({\mathscr{L}})$ and $IC_{Z}({V^{h}})$.

The next result describes the behavior of the intermediate extension with
respect to the dualizing functor (see, e.g., [92, Proposition 8.4.15]).

######  Proposition 2.48.

Consider the context of Definition 2.41 with coefficient ring $R$ a field, and
let ${\mathscr{G}}^{\bullet}\in Perv_{({\mathscr{S}}|_{U})}(U;R)$ be
(${\mathscr{S}}|_{U}$-)constructible. Then

(16) ${\mathscr{D}}(j_{!*}{\mathscr{G}}^{\bullet})\simeq
j_{!*}({\mathscr{D}}{\mathscr{G}}^{\bullet}).$

In particular, ${\mathscr{D}}IC_{X}\simeq IC_{X}$ for $X$ pure dimensional.

###  2.3. Strongly perverse sheaves. dual t-structure and rectified
homological depth

In this section we only consider (${\mathscr{S}}-$)constructible sheaves so
that biduality is available, with a dual perverse t-structure

${{}^{p^{+}}D}^{\leq 0}(X;R):=\mathcal{D}\left({{}^{p}D}^{\geq
0}(X;R)\right)\quad\text{and}\quad{{}^{p^{+}}D}^{\geq
0}(X;R):=\mathcal{D}\left({{}^{p}D}^{\leq 0}(X;R)\right)$

on $D^{b}_{({\mathscr{S}}-)c}(X;R)$. Here $\mathcal{D}$ denotes the Verdier
duality functor on $X$. But only for $R$ a Dedekind domain one can give a more
explicit description of this dual perverse t-structure. If $R$ s a field, the
Universal Coefficient Theorem yields that
${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\geq 0}(X;R)$ if and only if
$\mathcal{D}{\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$. More
generally, one has the following result (see also [6, Section 3.3]):

######  Proposition 2.49.

Assume that the ring $R$ is a Dedekind domain (e.g., a field or a PID). If
${\mathscr{F}}^{\bullet}\in D^{b}_{c}(X;R)$ is constructible with respect to a
Whitney stratification ${\mathscr{S}}$ of $X$, then
$\mathcal{D}{\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$, or
equivalently ${\mathscr{F}}^{\bullet}\in{{}^{p^{+}}D}^{\geq 0}(X;R)$, if and
only if the following two conditions are satisfied:

  * (i)

${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\geq 0}(X;R)$;

  * (ii)

for any stratum $S\in{\mathscr{S}}$ and any $x\in S$, the costalk cohomology
$H^{\dim S}(i_{x}^{!}{\mathscr{F}}^{\bullet})$ is torsion-free.

###### Proof.

Let $S\in{\mathscr{S}}$ and $x\in S$, with inclusion
$i_{x}\colon\\{x\\}\hookrightarrow X$. Properties of the dualizing functor and
the Universal Coefficient Theorem yield:

$H^{j}(i_{x}^{*}\mathcal{D}{\mathscr{F}}^{\bullet})\simeq
H^{j}(\mathcal{D}i_{x}^{!}{\mathscr{F}}^{\bullet})\simeq{\rm
Hom}(H^{-j}(i_{x}^{!}{\mathscr{F}}^{\bullet}),R)\oplus{\rm
Ext}(H^{-j+1}(i_{x}^{!}{\mathscr{F}}^{\bullet}),R).$

The desired equivalence can now be checked easily. In particular,

${{}^{p}D}^{\geq-1}(X;R)\subset{{}^{p^{+}}D}^{\geq
0}(X;R)\subset{{}^{p}D}^{\geq 0}(X;R).$

∎

######  Definition 2.50.

Assume that $R$ is a Dedekind domain. We say that ${\mathscr{F}}^{\bullet}\in
D^{b}_{c}(X;R)$ is strongly perverse if
${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$ and
$\mathcal{D}{\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)$. Equivalently,

${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 0}(X;R)\cap{{}^{p^{+}}D}^{\geq
0}(X;R).$

If $R$ is a field, the notions of perverse sheaf and strongly perverse sheaf
coincide. If $R$ is a Dedekind domain, then ${\mathscr{F}}^{\bullet}$ is
strongly perverse (with respect to ${\mathscr{S}}$) if and only if
${\mathscr{F}}^{\bullet}$ is perverse and property (ii) above holds (i.e.,
costalks of ${\mathscr{F}}^{\bullet}$ in the lowest possible degree are
torsion-free on each stratum).

######  Example 2.51.

Let $X$ be a pure dimensional complex algebraic or analytic variety, and let
$R$ be a Dedekind domain. Then $IC_{X}({\mathscr{L}})$ is strongly perverse if
the generically defined local system ${\mathscr{L}}$ has finitely generated
and torsion-free stalks (since then condition (ii) only needs to be checked
for the top dimensional strata $S$, with $H^{\dim
S}(i_{x}^{!}{\mathscr{L}}[\dim X])\simeq{\mathscr{L}}_{x}$ for $x$ in such a
stratum $S$).

Strongly perverse sheaves are related to the notion of rectified homological
depth, which we now define.

Let $X$ be a complex algebraic or analytic variety. Following [113, Definition
6.0.4], we make the following.

######  Definition 2.52.

The rectified homological depth ${\mathrm{rHd}}(X,R)$ of $X$ with respect to
the commutative base ring $R$ is $\geq d$ (for some $d\in{\mathbb{Z}}$) if

(17) $\mathcal{D}({R}_{X}[d])\in{{}^{p}D}^{\leq 0}(X;R)\>,$

or, equivalently,

(18) $R_{X}\in{{}^{p^{+}}D}^{\geq d}(X;R)\>.$

The rectified homological depth ${\mathrm{rHd}}(X,R)$ of $X\neq\emptyset$ with
respect to the commutative base ring $R$ is the maximum of such integers $d$.

As indicated in [113, p. 387], the above definition agrees with the notion of
rectified homological depth introduced by Hamm and Lê [56] (following an
earlier definition of Grothendieck, together with the corresponding
homotopical notion) in more geometric terms. In the following, let $\dim_{-}X$
be the minimum of the dimension of the irreducible components of $X$. So for
$X\neq\emptyset$ smooth, this is the minimum of the dimension of the connected
components of $X$. Similarly, $\dim_{-}X=\dim_{-}X_{reg}$, for $X_{reg}$ the
open dense regular part of $X$.

######  Example 2.53.

  * (a)

One always has ${\mathrm{rHd}}(X,R)\leq\dim_{-}X$, and
${\mathrm{rHd}}(X,R)=\dim_{-}X$ if $X$ is smooth and nonempty (since then
${\mathbb{D}}_{X}^{\bullet}\simeq R_{X}[2\dim X]$ with $\dim X$ viewed as a
locally constant function). Moreover, ${\mathrm{rHd}}(X,R)=\dim X$ implies
that $X$ is pure dimensional (by looking at the regular part).

  * (b)

If $X$ is a pure-dimensional local complete intersection, then
${\mathrm{rHd}}(X,R)=\dim X$ (see Example 4.29 and, e.g., [113, Example
6.0.11]).

In view of Example 2.19(a) and Definition 2.50, one has the following
equivalence (see also [113, (6.14)]):

######  Proposition 2.54.

Let $R$ be a Dedekind domain. For any nonempty complex algebraic or analytic
variety $X$ one has:

${\mathrm{rHd}}(X,R)=\dim X$ $\iff$ $X$ is pure-dimensional and $R_{X}[\dim
X]$ is strongly perverse.

As a consequence, Proposition 2.49 yields the following.

######  Corollary 2.55.

Let $X$ be a nonempty pure-dimensional complex algebraic or analytic variety
with a Whitney stratification ${\mathscr{S}}$.

  * (a)

If $R$ is a field, then:

${\mathrm{rHd}}(X,R)=\dim X$ $\iff$ $R_{X}[\dim X]$ is perverse.

  * (b)

If $R$ is a Dedekind domain (e.g., a PID), then ${\mathrm{rHd}}(X,R)=\dim X$
if and only if the following two conditions are satisfied:

    * (i)

$R_{X}[\dim X]$ is perverse.

    * (ii)

for any stratum $S\in{\mathscr{S}}$ and any $x\in S$ with
$i_{x}\colon\\{x\\}\hookrightarrow X$, the costalk cohomology $H^{\dim
S}(i_{x}^{!}R_{X}[\dim X])$ is torsion-free.

Let us finish this section with citing the following result from [6, Section
3.3].

######  Proposition 2.56.

Assume that the ring $R$ is a Dedekind domain (e.g., a field or a PID). If
${\mathscr{F}}^{\bullet}\in D^{b}_{c}(X;R)$ is constructible with respect to a
Whitney stratification ${\mathscr{S}}$ of $X$, then
$\mathcal{D}{\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\geq 0}(X;R)$, or
equivalently ${\mathscr{F}}^{\bullet}\in{{}^{p^{+}}D}^{\leq 0}(X;R)$, if and
only if the following two conditions are satisfied:

  * (i)

${\mathscr{F}}^{\bullet}\in{{}^{p}D}^{\leq 1}(X;R)$;

  * (ii)

for any stratum $S\in{\mathscr{S}}$ and any $x\in S$, the stalk cohomology
$H^{-\dim S+1}(i_{x}^{*}{\mathscr{F}}^{\bullet})$ is a torsion module.

In particular, ${{}^{p}D}^{\leq 0}(X;R)\subset{{}^{p^{+}}D}^{\leq
0}(X;R)\subset{{}^{p}D}^{\leq 1}(X;R)$.

Assume $R$ is a Dedekind domain. Let us denote by
$D^{b}_{({\mathscr{S}}-)tc}(X;R)$ the full triangulated subcategory of
$({\mathscr{S}}$-)constructible sheaf complexes ${\mathscr{F}}^{\bullet}\in
D^{b}_{({\mathscr{S}}-)c}(X;R)$, with all stalk cohomology modules
$H^{k}(i_{x}^{*}{\mathscr{F}}^{\bullet})$ a torsion module for all $x\in
X,k\in{\mathbb{Z}}$. These are also stable by the usual truncation functors.
Similarly, in the context of Example 2.38, all considered functors preserve
the subcategory $D^{b}_{{\mathscr{S}}-tc}(-;R)$ of these torsion
${\mathscr{S}}$-constructible sheaf complexes. For $i_{!}=i_{*}$ and
$j^{*}=j^{!}$ this is trivial, and for $i^{!}$ it follows from the
corresponding property for $Rj_{*}$ and the standard distinguished triangle

$i_{*}i^{!}\longrightarrow id\longrightarrow Rj_{*}j^{*}\longrightarrow\>.$

But the stalk cohomology modules of the open push-forward $Rj_{*}$ can be
expressed in terms of the compact link [113, Remark 4.4.2], and by the Künneth
formula they vanish after tensoring with the quotient field $Q(R)$ of $R$.
Then the usual perverse t-structure and its truncation functors restrict to
the perverse t-structure

(19) $\left({{}^{p}D}^{\leq 0}_{{\mathscr{S}}-t}(X;R),{{}^{p}D}^{\geq
0}_{{\mathscr{S}}-t}(X;R)\right)\quad\text{on
$D^{b}_{{\mathscr{S}}-tc}(X;R)$,}$

and the corresponding heart of ${\mathscr{S}}$-constructible torsion perverse
sheaves

$Perv_{{\mathscr{S}}-t}(X;R)\>.$

######  Example 2.57.

Let $X$ be a pure dimensional complex algebraic or analytic variety with a
Whitney stratification ${\mathscr{S}}$, and let $R$ be a Dedekind domain. Then
$IC_{X}({\mathscr{L}})$ is a ${\mathscr{S}}$-constructible torsion perverse
sheaf if the local system ${\mathscr{L}}$ defined on the open top dimensional
strata has only finitely generated torsion stalks.

######  Corollary 2.58.

Assume $R$ is a Dedekind domain. Then one gets for the shifted Verdier duality
functor $({\mathscr{D}}(-))[1]$:

  * (a)

$({\mathscr{D}}(-))[1]$ maps ${{}^{p}D}^{\leq 0}_{{\mathscr{S}}-t}(X;R)$ into
${{}^{p}D}^{\geq 0}_{{\mathscr{S}}-t}(X;R)$.

  * (b)

$({\mathscr{D}}(-))[1]$ maps ${{}^{p}D}^{\geq 0}_{{\mathscr{S}}-t}(X;R)$ into
${{}^{p}D}^{\leq 0}_{{\mathscr{S}}-t}(X;R)$.

In particular, the category $Perv_{{\mathscr{S}}-t}(X;R)$ of
${\mathscr{S}}$-constructible torsion perverse sheaves is stable under the
shifted Verdier duality functor $({\mathscr{D}}(-))[1]$.

######  Example 2.59.

Assume $R$ is a Dedekind domain. Then on a point space $X=\\{pt\\}$ the
shifted duality functor $({\mathscr{D}}(-))[1]$ is given by

$R{\mathscr{H}}om^{\bullet}(-,R)[1]\simeq{\mathscr{H}}om^{\bullet}(-,Q(R)/R):\>D^{b}_{tc}(\\{pt\\};R)\to
D^{b}_{tc}(\\{pt\\};R)\>,$

preserving finitely generated torsion modules (viewed as complexes
concentrated in degree zero). Here $Q(R)$ is the quotient field of $R$ so that
the short exact sequence

$0\longrightarrow R\longrightarrow Q(R)\longrightarrow Q(R)/R\longrightarrow
0$

gives the injective resolution $[Q(R)\longrightarrow Q(R)/R]$ of $R$ used for
calculating $R{\mathscr{H}}om^{\bullet}(-,R)$.

In fact this example is also used on a general $X$ for showing that the
Verdier duality functor ${\mathscr{D}}$ preserves
$D^{b}_{({\mathscr{S}}-)tc}(X;R)$, together with
$i_{x}^{*}{\mathscr{D}}\simeq{\mathscr{D}}i_{x}^{!}$ for all $x\in X$. For
applications and a discussion of related results see also [20, 48, 90, 91,
119].

##  3\. Stratified Morse theory for constructible sheaves

In this section, we explain the basic results from [113] about stratified
Morse theory in the framework of (weakly) constructible sheaves in the complex
context, continuing and extending the recent survey of Goresky [49] in this
handbook series (as well as the survey of Massey [88]). We follow the notions
of the geometric stratified Morse theory of Goresky-MacPherson so that one can
easily compare our results with those of [47]. Moreover, we also explain some
relations to the general micro-local sheaf theory of Kashiwara-Schapira [67].
But unlike most of these references, we do not need a global embedding into an
ambient complex manifold (except for the index theorems for characteristic
cycles of constructible sheaves and the comparison with the notion of micro-
support from [67]).

###  3.1. Morse functions, local and normal Morse data

We work with a fixed complex algebraic (or analytic) variety $X$ with a given
Whitney stratification ${\mathscr{S}}$.

######  Definition 3.1.

A function $f\colon X\to{\mathbb{R}}$ is a $C^{k}$-differentiable function
(with $2\leq k\leq\infty$) if, for any $x\in X$, there exists a local
embedding $(X,x)\hookrightarrow({\mathbb{C}}^{n},x)$ such that $f$ is induced
by restriction of a $C^{k}$-function germ
$\hat{f}\colon({\mathbb{C}}^{n},x)\to({\mathbb{R}},f(x))$.

Of course, a complex algebraic (or analytic) function or morphism $h\colon
X\to{\mathbb{C}}$ is by definition locally given by restriction of a complex
algebraic (or analytic) function germ
$\hat{h}\colon({\mathbb{C}}^{n},x)\to({\mathbb{C}},h(x))$.

######  Definition 3.2.

A point $x\in X$ is called a stratified critical point of $f\colon
X\to{\mathbb{R}}$, resp., $h\colon X\to{\mathbb{C}}$ (with respect to
${\mathscr{S}}$) if $f\colon S\to{\mathbb{R}}$, resp., $h\colon
S\to{\mathbb{C}}$ is not a submersion at the point $x$ belonging to the
stratum $S\in{\mathscr{S}}$. Let $\mathrm{Sing}(f|_{S})$, resp.,
$\mathrm{Sing}(h|_{S})$ be the set of critical points of $f$, resp., $h$ in
the stratum $S\in{\mathscr{S}}$, with

$\mathrm{Sing}_{{\mathscr{S}}}(f):=\bigcup_{S\in{\mathscr{S}}}\mathrm{Sing}(f|_{S})\quad\text{resp.,}\quad\mathrm{Sing}_{{\mathscr{S}}}(h):=\bigcup_{S\in{\mathscr{S}}}\mathrm{Sing}(h|_{S})\>.$

Then $\mathrm{Sing}_{{\mathscr{S}}}(f)$ (resp.,
$\mathrm{Sing}_{{\mathscr{S}}}(h)$) is a closed (complex algebraic or
analytic) subset of $X$ by the Whitney a-condition.

The following result shows that for ${\mathscr{S}}$-weakly constructible sheaf
complexes the local change of cohomology with respect to $f$ is located at the
critical points $x\in\mathrm{Sing}_{{\mathscr{S}}}(f)$ (see, e.g., [113,
Example 5.1.1]).

######  Lemma 3.3.

Let $f\colon X\to\,]c,d[\,\subset{\mathbb{R}}$ be a proper differentiable
function, and assume ${\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-wc}(X;R)$ is a ${\mathscr{S}}$-weakly constructible
complex.

  1. (1)

If $f$ is a stratified submersion at a point $x\in\\{f=e\\}$ (with
$e\in\,]c,d[\,$, and here $f$ does not need to be proper), then

$\bigl{(}R\Gamma_{\\{f\geq e\\}}({\mathscr{F}}^{\bullet})\bigr{)}_{x}\simeq
0\simeq\bigl{(}R\Gamma_{\\{f\leq
e\\}}({\mathscr{F}}^{\bullet})\bigr{)}_{x}\>.$

  2. (2)

If $f$ is a stratified submersion at all points of $\\{f=e\\}$ (for
$e\in\,]c,d[\,$), then

$R\Gamma(\\{f=e\\},R\Gamma_{\\{f\geq e\\}}({\mathscr{F}}^{\bullet}))\simeq
0\simeq R\Gamma(\\{f=e\\},R\Gamma_{\\{f\leq
e\\}}({\mathscr{F}}^{\bullet}))\>.$

  3. (3)

If $f$ is a stratified submersion at all points of $\\{a<f<b\\}$ (with
$]a,b[\,\subset\,]c,d[\,$), then all cohomology sheaves of
$(Rf_{*}{{\mathscr{F}}^{\bullet}})|_{\,]a,b[\,}$ are locally constant.

These results can be proved by induction on the dimension of $X$ without using
the “first isotopy Lemma of Thom”. Property 1. above just means that a Whitney
b-regular stratification satisfies the “local stratified acyclicity” property
form [113, Definition 4.0.3]. In the embedded context of a global closed
embedding $k\colon X\hookrightarrow M$ into a complex manifold it also implies
the following estimate [113, Corollary 4.0.3] of the micro-support of
$Rk_{*}{\mathscr{F}}^{\bullet}$ in the sense of Kashiwara-Schapira [67,
Definition 5.1.2] (with $Rk_{*}=k_{*}=k_{!}$ the extension by zero):

(20) $\mu
supp(Rk_{*}{\mathscr{F}}^{\bullet})\subset\bigcup_{S\in{\mathscr{S}}}\>T^{*}_{S}M\hookrightarrow
T^{*}M\quad\text{for any ${\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-wc}(X;R)$.}$

Here, $\mu supp({\mathscr{G}}^{\bullet})\subset T^{*}M$ is by definition the
complement of the largest open subset $U\subset T^{*}M$ such that
$R\Gamma_{\\{f\geq f(x)\\}}({\mathscr{G}}^{\bullet})_{x}\simeq 0$ for any
differentiable function germ $f\colon(M,x)\to({\mathbb{R}},f(x))$ with
$df_{x}\in U$ (see [67, Definition 5.1.2]). We also use the notation
$T^{*}_{S}M$ for the conormal bundle of the stratum $S$ in $M$, i.e., for the
kernel of the natural vector bundle epimorphism $T^{*}M|_{S}\to T^{*}S$ dual
to the inclusion $TS\hookrightarrow TM|_{S}$. Let us denote by
$T^{*}_{{\mathscr{S}}}M$ this union of conormal spaces of strata
$S\in{\mathscr{S}}$. Then the Whitney a-condition for ${\mathscr{S}}$ is
equivalent to the fact that $T^{*}_{{\mathscr{S}}}M\hookrightarrow T^{*}M$ is
closed in $T^{*}M$. Moreover it is conic in the sense that it is invariant
under the natural ${\mathbb{C}}^{*}$-action on the cotangent bundle. The
inclusion of the micro-support in (20) implies that $\mu
supp(Rk_{*}{\mathscr{F}}^{\bullet})\subset T^{*}M$ is a closed complex
algebraic (or analytic) conic Lagrangian subset of $T^{*}M$, and (20) is even
equivalent to ${\mathscr{S}}$-weak constructibility (see, e.g., [113, Lemma
4.1]).

######  Lemma 3.4.

Let ${\mathscr{G}}^{\bullet}\in D^{b}(M;R)$ be given with $\mu
supp({\mathscr{G}}^{\bullet})\subset T^{*}_{{\mathscr{S}}}M$. Then
$supp({\mathscr{G}}^{\bullet})\subset X$ and ${\mathscr{G}}^{\bullet}|_{X}$ is
${\mathscr{S}}$-weakly constructible.

######  Example 3.5.

Assume $X=M$ with ${\mathscr{G}}^{\bullet}\in D^{b}(M;R)$ be given. Then $\mu
supp({\mathscr{G}}^{\bullet})\subset T^{*}_{M}M$ the zero section of $T^{*}M$
if and only if all cohomology sheaves
${\mathscr{H}}^{i}({\mathscr{G}}^{\bullet})$ are locally constant on $M$.

This Example can be proved directly with the “non-characteristic deformation
Lemma” of Kashiwara (see, e.g., [113, Proposition 4.1.1]). Moreover, it can be
used as a substitute for the “first isotopy Lemma of Thom”.

If one does not want to fix a complex Whitney stratification ${\mathscr{S}}$,
then one gets the following micro-local characterization of weak
constructibility (see, e.g., [113, Theorem 4.0.1] and [67, Theorem 8.5.5]).

######  Theorem 3.6.

Let $M$ be a complex algebraic (or analytic) manifold. Then
${\mathscr{G}}^{\bullet}\in D^{b}(M;R)$ is a weakly constructible complex
${\mathscr{G}}^{\bullet}\in D^{b}_{wc}(M;R)$ in the complex algebraic (or
analytic) sense if and only if $\mu supp({\mathscr{G}}^{\bullet})\subset
T^{*}M$ is a closed complex algebraic (or analytic) conic Lagrangian subset of
$T^{*}M$.

Let us now come back to the general non-embedded context with ${\mathscr{S}}$
a fixed Whitney stratification of $X$. Then Lemma 3.3 implies (see, e.g.,
[113, Corollary 5.1.1]).

######  Corollary 3.7.

Let $f\colon X\to\,]c,d[\,\subset{\mathbb{R}}$ be a proper differentiable
function, and assume ${\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-wc}(X;R)$ is a ${\mathscr{S}}$-weakly constructible
complex. If $f$ is a stratified submersion at all points of $\\{a\leq f\leq
b\\}\cap\\{f\neq e\\}$ (with $[a,b]\subset\,]c,d[\,$ and $e\in\,]a,b[\,$),
then one has distinguished triangles

(21) $R\Gamma(\\{f=e\\},R\Gamma_{\\{f\geq e\\}}({\mathscr{F}}^{\bullet}))\to
R\Gamma(\\{f\leq b\\},{\mathscr{F}}^{\bullet})\to R\Gamma(\\{f\leq
a\\},{\mathscr{F}}^{\bullet})\to\>,$

and

(22) $R\Gamma_{c}(\\{f<a\\},{\mathscr{F}}^{\bullet})\to
R\Gamma_{c}(\\{f<b\\},{\mathscr{F}}^{\bullet})\to
R\Gamma(\\{f=e\\},R\Gamma_{\\{f\leq e\\}}({\mathscr{F}}^{\bullet}))\to\>.$

Here, $R\Gamma(\\{f=e\\},R\Gamma_{\\{f\geq e\\}}({\mathscr{F}}^{\bullet}))$ is
the cohomological counterpart of the “coarse Morse datum” as defined in [47,
p.62, Definition 3.4]. In particular,

$R\Gamma(\\{f=e\\},R\Gamma_{\\{f\leq e\\}}({\mathscr{F}}^{\bullet})\simeq
R\Gamma(\\{-f=-e\\},R\Gamma_{\\{-f\geq-e\\}}({\mathscr{F}}^{\bullet}))$

is the “coarse Morse datum” of $-f$. So this “coarse Morse datum” of $-f$
corresponds to the “relative cohomology with compact support” in the triangle
(22). But the cohomology with compact support is in some sense “dual” to the
cohomology with closed support. So this reflects the “duality” observed in
[47, p.27, 2.7]. In the context of stratified Morse theory for constructible
sheaves, this is a special form of Poincaré-Verdier duality (see Equation
(25)).

The next step is to consider a differentiable function $f$ with isolated
stratified critical points in $\\{f=e\\}$, and to localize the “coarse Morse
datum” at these critical points. In terms of sheaf theory, this is quite easy
(since $R\Gamma_{\\{f\geq e\\}}({\mathscr{F}}^{\bullet})|_{\\{f=e\\}}$ is
supported on the stratified critical points).

######  Lemma 3.8.

Let $f\colon X\to\,]c,d[\,\subset{\mathbb{R}}$ be a proper differentiable
function, and assume ${\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-wc}(X;R)$ is a ${\mathscr{S}}$-weakly constructible
complex. Assume $f$ is a stratified submersion at all points of $\\{a\leq
f\leq b\\}$ except for finitely many $x_{i}\in\\{f=e\\}$ (with
$[a,b]\subset\,]c,d[\,$ and $e\in\,]a,b[\,$). Then

(23) $R\Gamma(\\{f=e\\},R\Gamma_{\\{f\geq
e\\}}{\mathscr{F}}^{\bullet}))\simeq\bigoplus_{i}\>\left(R\Gamma_{\\{f\geq
e\\}}({\mathscr{F}}^{\bullet})\right)_{x_{i}}\>.$

The stalk complex

(24) $LMD({\mathscr{F}}^{\bullet},f,x):=\bigl{(}R\Gamma_{\\{f\geq
f(x)\\}}({\mathscr{F}}^{\bullet})\bigr{)}_{x}$

is the sheaf theoretic counterpart of the “local Morse datum” of [47, p.63,
Definition 3.5.2]. The “duality” between $f$ and $-f$ discussed above is
closely related to Verdier duality for the local Morse data in the following
form of [113, Equation (5.54) on p.314]:

(25) ${\mathscr{D}}(LMD({\mathscr{F}}^{\bullet},-f,x))\simeq
LMD({\mathscr{D}}({\mathscr{F}}^{\bullet}),f,x)\quad\text{for
${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-wc}(X;R)$.}$

Let us now recall two notions from [47].

######  Definition 3.9.

Fix a point $x$ in the stratum $S\in{\mathscr{S}}$.

  1. (1)

$x$ is called a stratified Morse critical point of the differentiable function
germ $f\colon(X,x)\to{\mathbb{R}}$ (with respect to ${\mathscr{S}}$), if
$f|_{S}$ has at $x$ a Morse critical point in the classical sense (i.e., its
Hessian at $x$ is non-degenerate), and $f=\hat{f}|_{X}$ is induced in some
local embedding $(X,x)\hookrightarrow({\mathbb{C}}^{n},x)$ by a differentiable
function germ $\hat{f}\colon({\mathbb{C}}^{n},x)\to({\mathbb{R}},f(x))$ such
that the covector $d\hat{f}_{x}$ is non-degenerate in the sense of [47, p.44,
Definition 1.8], i.e., it does not vanish on any generalized tangent space

$\tau:=\lim_{x_{n}\to x}\;T_{x_{n}}S^{\prime}\quad\text{at $x$,}$

with $x_{n}$ a sequence in a stratum $S^{\prime}\in{\mathscr{S}}$ with
$S\subset\partial S^{\prime}$.

  2. (2)

A normal slice $N$ at $x\in S\hookrightarrow X\hookrightarrow{\mathbb{C}}^{n}$
is a closed complex submanifold germ $N$ of $({\mathbb{C}}^{n},x)$, with
$N\cap S=\\{x\\}$ such that $N$ intersects $S$ transversally at $x$ (so that
$\hbox{\rm codim}\,N=\dim S$).

Note that a normal slice $N$ intersects all strata transversally near $x$ by
Whitney a-regularity. Therefore $N\cap X$ gets an induced Whitney
stratification ${\mathscr{S}}|_{N}$ near $x$ with strata the connected
components of intersections $S^{\prime}\cap N\neq\emptyset$
($S^{\prime}\in{\mathscr{S}}$) and $\\{x\\}=S\cap N$ now a point stratum.

Similarly, a stratified Morse critical point of $f$ is an isolated stratified
critical point of $f$. Finally, if $f=\hat{f}|_{X}$ is induced in some local
embedding $(X,x)\hookrightarrow({\mathbb{C}}^{n},x)$ by a differentiable
function germ $\hat{f}\colon({\mathbb{C}}^{n},x)\to({\mathbb{R}},f(x))$ such
that the covector $d\hat{f}_{x}$ is non-degenerate, then one can find such an
$\hat{f}$ in any local embedding $(X,x)\hookrightarrow({\mathbb{C}}^{m},x)$.

######  Example 3.10.

Let $S$ be an open stratum of $X$. Then $x\in S$ is a stratified Morse
critical point of $f$ if and only if $f|_{S}$ has at $x$ a classical Morse
critical point. If $S=\\{x\\}$ is a point stratum, then $x$ is by definition a
stratified critical point for any differentiable function $f\colon
X\to{\mathbb{R}}$. It is stratified Morse critical point if and only if
$f=\hat{f}|_{X}$ is induced in some local embedding
$(X,x)\hookrightarrow({\mathbb{C}}^{n},x)$ by a differentiable function germ
$\hat{f}\colon({\mathbb{C}}^{n},x)\to({\mathbb{R}},f(x))$ such that the
covector $d\hat{f}_{x}$ is non-degenerate.

######  Example 3.11.

Let $X=M$ be a complex manifold, with $M^{\prime}\hookrightarrow M$ a closed
complex submanifold of positive codimension and the Whitney stratification
${\mathscr{S}}$ given by the connected components of $M^{\prime}$ and
$M\backslash M^{\prime}$. Then a differentiable function $f\colon
M\to{\mathbb{R}}$ has a stratified Morse critical point at $x\in M^{\prime}$
if and only if $x$ is not a critical point of $f\colon M\to{\mathbb{R}}$ (this
is equivalent to $df_{x}$ is nondegenrate) and $x\in M^{\prime}$ is a
classical Morse critical point for $f\colon M^{\prime}\to{\mathbb{R}}$.

Let us now state the first main result from stratified Morse theory for
constructible sheaves (see, e.g., [113, Theorem 5.0.1]).

######  Theorem 3.12.

Let $X$ be a complex algebraic (or analytic) variety with a Whitney
stratification ${\mathscr{S}}$, and let ${\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-wc}(X;R)$ be a given ${\mathscr{S}}$-weakly constructible
complex. Let $f\colon(X,x)\to{\mathbb{R}}$ be a differentiable function germ
with a stratified Morse critical point at $x\in S$ for some stratum
$S\in{\mathscr{S}}$. Take a normal slice $N\subset{\mathbb{C}}^{n}$ at $x$ in
some local embedding $(X,x)\hookrightarrow({\mathbb{C}}^{n},x)$.

  1. (1)

Then one has an isomorphism

$\bigl{(}R\Gamma_{\\{f\geq
f(x)\\}}({\mathscr{F}}^{\bullet})\bigr{)}_{x}\simeq\bigl{(}R\Gamma_{\\{f|_{N\cap
X}\geq f(x)\\}}({\mathscr{F}}^{\bullet}|_{N\cap X})\bigr{)}_{x}[-\tau]\>,$

with $\tau$ the  Morse index of $f|_{S}$ (i.e., its Hessian at $x$ has exactly
$\tau$ negative eigenvalues).

  2. (2)

The isomorphism class of

$\bigl{(}R\Gamma_{\\{f|_{N\cap X}\geq f(x)\\}}({\mathscr{F}}^{\bullet}|_{N\cap
X})\bigr{)}_{x}$

is independent of the choice of such a local embedding and normal slice $N$.
Moreover, this isomorphism class only depends on the stratum
$S\in{\mathscr{S}}$, but not on the point $x\in S$ or the function germ
$f=\hat{f}|_{X}$ with $d\hat{f}_{x}$ non-degenerate in such a local embedding.

The isomorphism class of

(26) $NMD({\mathscr{F}}^{\bullet},S):=\bigl{(}R\Gamma_{\\{f|_{N\cap X}\geq
f(x)\\}}({\mathscr{F}}^{\bullet}|_{N\cap X})\bigr{)}_{x}$

is the sheaf theoretic counterpart of the “normal Morse data” of [47, p.65,
Definition 3.6.1]. In the (locally) embedded context $k\colon X\hookrightarrow
M$, $NMD({\mathscr{F}}^{\bullet},S)$ is also isomorphic to the “micro-local
type of $Rk_{*}{\mathscr{F}}^{\bullet}$ at the non-degenerate covector
$\omega=d\hat{f}_{x}$” in the sense of Kashiwara-Schapira [67, Proposition
6.6.1(ii)] (as shown in [113, Equation (5.52) on p.311]). Part 1. of Theorem
3.12 corresponds to [47, p.8, Theorem SMT Part B; p.65, Main Theorem 3.7], and
part 2. to [47, p.93, Theorem 7.5.1] (compare also with [47, p.223,
Proposition 6.A.1]).

Let us consider again the embedded context with $k\colon X\hookrightarrow M$ a
closed embedding into a complex algebraic (or analytic) manifold, and
${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-wc}(X;R)$. By using the
estimate (20) and the involutivity of $\mu
supp(Rk_{*}{\mathscr{F}}^{\bullet})$ [67, Theorem 6.5.4] one can show that
$\mu supp(Rk_{*}{\mathscr{F}}^{\bullet})$ is a union of closures of conormal
bundles $T^{*}_{S}M$ of some strata $S\in{\mathscr{S}}$. Then the explicit
characterization of the micro-suport $\mu supp(Rk_{*}{\mathscr{F}}^{\bullet})$
is given as follows (see, e.g., [113, Proposition 5.0.1]).

######  Proposition 3.13.

Let ${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-wc}(X;R)$ be given. Then

$\overline{T^{*}_{S}M}\subset\mu supp(Rk_{*}{\mathscr{F}}^{\bullet})\iff
NMD({\mathscr{F}}^{\bullet},S)\not\simeq 0\>.$

For the applications of stratified Morse theory for constructible sheaves, it
is important to get information about this normal Morse datum
$NMD({\mathscr{F}}^{\bullet},S)$ for a stratum $S\in{\mathscr{S}}$. In our
complex context this can be obtained in the following way. Since it is a local
study, we can assume that for a given point $x\in S$ we have (locally) a
closed embedding $(X,x)\hookrightarrow({\mathbb{C}}^{n},x=0)$.

We already know that the isomorphism class of the “normal Morse datum”
$NMD({\mathscr{F}}^{\bullet},S)$ only depends on the non-degenerate covector
$\omega=d\hat{f}_{x}\in T^{*}_{S}{\mathbb{C}}^{n}|_{x}$. So one can use a
holomorphic function germ $g\colon({\mathbb{C}}^{n},x)\to({\mathbb{C}},0)$
with $d\hat{f}_{x}=d(Re(g))_{x}$. Also choose a local normal slice $N$ at
$x\in S\hookrightarrow X\hookrightarrow{\mathbb{C}}^{n}$, with $\\{x\\}=S\cap
N$ a point stratum of the induced Whitney stratification ${\mathscr{S}}|_{N}$
of $X\cap N$. Then one gets (see Remark 4.18 and also, e.g., [113, Proposition
5.0.3]).

######  Proposition 3.14.

Let $k_{x}\colon\\{x\\}\to N\cap X$ be the inclusion of this point stratum,
and ${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-wc}(X;R)$ be given. Then
there exists two distinguished triangles

(27)
$\begin{CD}R\Gamma(l_{X},{\mathscr{F}}^{\bullet})[-1]@>{}>{}>NMD({\mathscr{F}}^{\bullet},S)@>{}>{}>k_{x}^{*}({\mathscr{F}}^{\bullet}|_{N\cap
X})@>{}>{}>\>,\end{CD}$

and

(28) $\begin{CD}k_{x}^{!}({\mathscr{F}}^{\bullet}|_{N\cap
X})@>{}>{}>NMD({\mathscr{F}}^{\bullet},S)@>{}>{}>R\Gamma(l_{X},\partial
<EMAIL_ADDRESS>
Moreover, the isomorphism classes of $R\Gamma(l_{X},{\mathscr{F}}^{\bullet})$
and $R\Gamma(l_{X},\partial l_{X},{\mathscr{F}}^{\bullet})$ only depend on the
stratum $S$ of $X$ (but on no other choice like the local embedding
$(X,x)\hookrightarrow({\mathbb{C}}^{n},x=0)$, or the choice of $N$ or $g$ as
above).

Here we use the following notation (with $0<|w|\ll\delta\ll 1$ and
$r(z):=\sum_{1\leq i\leq n}z_{i}\bar{z}_{i}$ the square of the local distance
to $x=0\in{\mathbb{C}}^{n}$)

(29) $(l_{X},\partial l_{X}):=(X\cap N\cap\\{r\leq\delta,g=w\\},X\cap
N\cap\\{r=\delta,g=w\\})$

for the complex link of $X$ with respect to $S\in{\mathscr{S}}$ (see [47,
p.161, Definition.2.2]). In other words, this is the local Milnor fiber (with
its boundary) (see, e.g., [74] or [113, Example 1.1.2]) of the holomorphic
function germ $g\colon(X\cap N,x)\to({\mathbb{C}},0)$, which has an isolated
stratified critical point in $x$ with respect to ${\mathscr{S}}|_{N}$ (since
$dg_{x}$ is a non-degenerate covector).

######  Example 3.15.

Assume $S\in{\mathscr{S}}$ is an open stratum in $X$. Then for $x\in S$ and a
normal slice $N$ at $x$ one gets $X\cap N=\\{x\\}$, so that the corresponding
complex link is empty, $l_{X}=\emptyset$, with

$NMD({\mathscr{F}}^{\bullet},S)\simeq
k_{x}^{*}({\mathscr{F}}^{\bullet}|_{N\cap X})\simeq
i_{x}^{*}{\mathscr{F}}^{\bullet}$

just the stalk of ${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-wc}(X;R)$
at $x$.

######  Example 3.16.

We state here cohomological counterparts of some results of [47].

  1. (1)

If we take ${\mathscr{F}}^{\bullet}={\mathbb{Z}}_{X}$, then we get by the
first distinguished triangle (27) for the cohomology of the “normal Morse
datum” $NMD({\mathscr{F}}^{\bullet},S)$ for $x\in S$:

$\bigl{(}R^{k}\Gamma_{\\{Re(g)|_{N\cap X}\geq
0\\}}({\mathscr{F}}^{\bullet}|_{N\cap
X})\bigr{)}_{x}\simeq\begin{cases}{\mathbb{Z}}&\text{for $k=0$ and
$l_{X}=\emptyset$,}\\\ 0&\text{for $k\neq 0$ and $l_{X}=\emptyset$,}\\\
\widetilde{H}^{k-1}(l_{X};{\mathbb{Z}})&\text{for
$l_{X}\neq\emptyset$.}\end{cases}$

This corresponds to the “first part of the fundamental theorem of complex
stratified Morse theory” [47, p.16, Theorem CSMT Part A, p.166 Corollary
2.4.1].

  2. (2)

Consider now the case ${\mathscr{F}}^{\bullet}=Rj_{*}{\mathbb{Z}}_{U}$ for
$j\colon U\hookrightarrow X$ the inclusion of an open subspace, which is a
union of strata. Assume $x\not\in U$. Then one gets by the second
distinguished triangle (28) for the cohomology of the “normal Morse datum”
$NMD({\mathscr{F}}^{\bullet},S)$ for $x\in S$:

$\bigl{(}R^{k}\Gamma_{\\{Re(g)|_{N\cap X}\geq
0\\}}({\mathscr{F}}^{\bullet}|_{N\cap X})\bigr{)}_{x}\simeq
H^{k-1}(l_{X},\partial l_{X};Rj_{*}{\mathbb{Z}}_{U})\>.$

Moreover, by the base change isomorphism (11) one gets

$H^{k-1}(l_{X},\partial l_{X};Rj_{*}{\mathbb{Z}}_{U})\simeq
H^{k-1}(l_{U},\partial l_{U};{\mathbb{Z}})\>,$

with $(l_{U},\partial l_{U}):=(l_{X},\partial l_{X})\cap U$ the complex link
of $U$. This corresponds to the “second part of the fundamental theorem of
complex stratified Morse theory” [47, p.18, Theorem CSMT Part B, p.169,
Corollary 2.6.1].

Proposition 3.14 and the corresponding distinguished triangles for $dg_{x}\in
T^{*}_{S}{\mathbb{C}}^{n}$ a non-degenerate covector are related to the theory
of the nearby and vanishing cycle functors of Deligne, as will be explained
and discussed in later sections (see Remark 4.18):

(30) $NMD({\mathscr{F}}^{\bullet},S)\simeq\left(\,\varphi_{g|_{N\cap
X}}[-1]({\mathscr{F}}^{\bullet}|_{N\cap X})\,\right)_{x}$

and

(31) $R\Gamma(l_{X},{\mathscr{F}}^{\bullet})\simeq\left(\,\psi_{g|_{N\cap
X}}({\mathscr{F}}^{\bullet}|_{N\cap X})\,\right)_{x}\>,$

resp.,

(32) $R\Gamma(l_{X},\partial l_{X},{\mathscr{F}}^{\bullet})\simeq
k_{x}^{!}\left(\,\psi_{g|_{N\cap X}}({\mathscr{F}}^{\bullet}|_{N\cap
X})\,\right)\>.$

One also has for a holomorphic function germ $g\colon(X,x)\to({\mathbb{C}},0)$
and ${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-wc}(X;R)$ an isomorphism
(see Corollary 4.14 or, e.g., [113, Corollary 1.1.1])

(33) $LMD({\mathscr{F}}^{\bullet},Re(g),x)=\bigl{(}\,R\Gamma_{\\{Re(g)\geq
0\\}}({\mathscr{F}}^{\bullet})\,\bigr{)}_{x}\simeq\left(\,\varphi_{g}[-1]({\mathscr{F}}^{\bullet})\,\right)_{x}\>.$

######  Example 3.17 (Normal Morse data for products).

Let $X_{i}$ be a complex algebraic (or analytic) variety, with
${\mathscr{F}}_{i}^{\bullet}$ weakly constructible with respect to a Whitney
stratification ${\mathscr{S}}_{i}$ of $X_{i}$, for $i=1,2$. Then
${\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet}$
is weakly constructible with respect to the product Whitney stratification
${\mathscr{S}}_{1}\times{\mathscr{S}}_{2}$ of $X_{1}\times X_{2}$ (with strata
$S_{1}\times S_{2}$ for $S_{1}\in{\mathscr{S}}_{1}$ and
$S_{2}\in{\mathscr{S}}_{2}$). Moreover

$NMD\left({\mathscr{F}}_{1}^{\bullet}\overset{L}{\boxtimes}{\mathscr{F}}_{2}^{\bullet},S_{1}\times
S_{2}\right)\simeq
NMD({\mathscr{F}}^{\bullet}_{1},S_{1})\overset{L}{\otimes}NMD({\mathscr{F}}^{\bullet}_{2},S_{2})\>.$

Let $N_{i}\subset{\mathbb{C}}^{n_{i}}$ be normal slices at $x_{i}\in S_{i}$,
with a corresponding non-degenerate covector $dg_{i,x_{i}}\in
T^{*}_{S_{i}}{\mathbb{C}}^{n_{i}}$ for a holomorphic function germ
$g_{i}\colon(\mathbb{C}^{n_{i}},x_{i})\to({\mathbb{C}},0)$ as above ($i=1,2$).
Then $N=N_{1}\times
N_{2}\subset{\mathbb{C}}^{n_{1}}\times{\mathbb{C}}^{n_{2}}$ is a normal slice
at $(x_{1},x_{2})\in S_{1}\times S_{2}$, with
$dg_{(x_{1},x_{2})}=\left(dg_{1,x_{1}},dg_{2,x_{2}}\right)$ a corresponding
non-degenerate covector defined by
$g(z_{1},z_{2}):=g_{1}(z_{1})+g_{2}(z_{2})$. Example 3.17 is then by (30) a
very special case of the Thom-Sebastiani Theorem for vanishing cycles stated
in Theorem 4.30.

###  3.2. Perverse sheaf description via normal Morse data

The distinguished triangles of Proposition 3.14 can be used to get (inductive)
information about the “normal Morse data”. The advantage in the complex
context comes from the fact that $X\cap N\cap\\{g=w\\}$ is a complex analytic
space (of lower dimension) with an induced Whitney stratification. Moreover,
the distance function $r$ used for the stratified Morse theory on $X\cap
N\cap\\{g=w\\}$ (in the induction step) is now strongly plurisubharmonic so
that one has nice estimates for the Morse index (of an approximation by a
Morse function) of $r$ and $-r$.

######  Definition 3.18.

A $C^{k}$-function $f\colon X\to{\mathbb{R}}$ ($k\geq 2$) is called q-convex
in $x\in X$, if one can choose
$\hat{f}\colon({\mathbb{C}}^{n},x)\to({\mathbb{R}},f(x))$ in a local embedding
$(X,x)\hookrightarrow({\mathbb{C}}^{n},x)$ with $\hat{f}|_{X}=f$ so that its
Levi form $L_{\hat{f}}(x)$ at the point $x$ has at most $q$ non-positive
eigenvalues. Here the Levi form of $\hat{f}$ at the point $x$ is the Hermitian
form given by the matrix of partial derivatives:

$L_{\hat{f}}(x):=\left(\;\frac{\partial^{2}\hat{f}}{\partial
z_{j}\,\partial\bar{z}_{k}}(x)\;\right)_{1\leq j,k\leq n}\>.$

$f$ is called q-convex (in a subset $Z\subset X$) if it is q-convex in all
points (of $Z$).

For example, a strongly plurisubharmonic $C^{k}$-function on an open subset
$U$ of ${\mathbb{C}}^{n}$ (with $k\geq 2$) is $0$-convex in all of $U$. In
particular, the distance function

$r\colon{\mathbb{C}}^{n}\to{\mathbb{R}}^{\geq
0}\,;\>r(z):=\sum_{i=1}^{n}\>z_{i}\bar{z}_{i}\quad\text{is $0$-convex in
${\mathbb{C}}^{n}$.}$

We next state the second main result for stratified Morse theory of
constructible sheaves in the complex context (see [113, Theorem 6.0.1] for a
more general formulation). We include here a complete proof based on Theorem
3.12, for explaining the use of stratified Morse theory and because similar
ideas can also be used later on for other important results.

######  Theorem 3.19.

Let ${\mathscr{S}}$ be a Whitney stratification of the complex algebraic (or
analytic) variety $X$, and consider a $C^{\infty}$-function $f\colon
X\to{\mathbb{R}}$. Assume $a<b$ are stratified regular values of $f$ with
respect to ${\mathscr{S}}$ such that $\\{a\leq f\leq b\\}$ is compact. Let
${\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-(w)c}(X;R)$ be given.

  1. (1)

Suppose $f$ is q-convex in $\\{a\leq f\leq b\\}$, and let
$q^{\prime}:=\min\\{q,\dim X\\}$.

${\mathscr{F}}^{\bullet}\in\;^{p}D^{\leq
n}_{{\mathscr{S}}}(X;R)\>\Rightarrow\>R\Gamma(\\{a\leq f\leq
b\\},\\{f=a\\},{\mathscr{F}}^{\bullet})\in\;^{p}D^{\leq
n+q^{\prime}}(\\{pt\\};R)\>.$

If $R$ is a Dedekind domain and ${\mathscr{F}}^{\bullet}$ is
${\mathscr{S}}$-constructible, then also

${\mathscr{F}}^{\bullet}\in\;^{p^{+}}D^{\leq
n}_{{\mathscr{S}}}(X;R)\>\Rightarrow\>R\Gamma(\\{a\leq f\leq
b\\},\\{f=a\\},{\mathscr{F}}^{\bullet})\in\;^{p^{+}}D^{\leq
n+q^{\prime}}(\\{pt\\};R)\>.$

  2. (2)

Suppose $-f$ is q-convex in $\\{a\leq f\leq b\\}$, and let
$q^{\prime}:=\min\\{q,\dim X\\}$.

${\mathscr{F}}^{\bullet}\in\;^{p}D^{\geq
n}_{{\mathscr{S}}}(X;R)\>\Rightarrow\>R\Gamma(\\{a\leq f\leq
b\\},\\{f=a\\},{\mathscr{F}}^{\bullet})\in\;^{p}D^{\geq
n-q^{\prime}}(\\{pt\\};R)\>.$

If $R$ is a Dedekind domain and ${\mathscr{F}}^{\bullet}$ is
${\mathscr{S}}$-constructible, then also

${\mathscr{F}}^{\bullet}\in\;^{p^{+}}D^{\geq
n}_{{\mathscr{S}}}(X;R)\>\Rightarrow\>R\Gamma(\\{a\leq f\leq
b\\},\\{f=a\\},{\mathscr{F}}^{\bullet})\in\;^{p^{+}}D^{\geq
n-q^{\prime}}(\\{pt\\};R)\>.$

Here $R\Gamma(\\{a\leq f\leq b\\},\\{f=a\\},{\mathscr{F}}^{\bullet})\in
D^{b}_{c}(\\{pt\\};R)$ in case ${\mathscr{F}}^{\bullet}$ is
${\mathscr{S}}$-constructible.

###### Proof.

We prove the claim by induction over $\dim X$, where the case $\dim X=0$ is
trivial (since the conditions to be checked contain the zero object and are
stable by finite direct sums).

After approximating $f$ by a stratified Morse function (see [8]), we can
assume that $f$ has only finitely many stratified Morse critical points in
$\\{a\leq f\leq b\\}$. If the approximation is close enough, then also this
new $\pm f$ is q-convex. This implies the following important estimate for the
Morse index $\tau(f|_{S})$ of $f|_{S}$ for a stratum $S\in{\mathscr{S}}$ in
terms of $q^{\prime}:=\min\\{q,\dim(X)\\}$ (compare with [47, p.191, Lemma
4.A.2]):

(34) $\begin{cases}\tau(f|_{S})\leq\dim S+q^{\prime},&\quad\text{if $f$ is
q-convex.}\\\ \tau(f|_{S})=2\dim S-\tau(-f|_{S})\geq\dim
S-q^{\prime},&\quad\text{if $-f$ is q-convex.}\end{cases}$

Then we apply Lemma 3.8. Since the conditions to be checked contain the zero
object and are stable by extensions, we only have to show that the “local
Morse datum”

$LMD({\mathscr{F}}^{\bullet},f,x)=(R\Gamma_{\\{f\geq
f(x)\\}}({\mathscr{F}}^{\bullet}))_{x}$

of $f$ at a stratified Morse critical point $x$ in a stratum
$S\in{\mathscr{S}}$ belongs to $\;{}^{p^{(+)}}D^{\leq
n+q^{\prime}}(\\{pt\\};R)$ for statement 1. (resp., belongs to
$\;{}^{p^{(+)}}D^{\geq n+q^{\prime}}(\\{pt\\};R)$ for statement 2.).

By Theorem 3.12 we get

$LMD({\mathscr{F}}^{\bullet},f,x)\simeq
NMD({\mathscr{F}}^{\bullet},S)[-\tau],$

with $NMD({\mathscr{F}}^{\bullet},S)$ the “normal Morse datum” of
${\mathscr{F}}^{\bullet}$ in the stratum $S$ and $\tau$ the Morse index of
$f|_{S}$. Here we can use the estimate (34) for this Morse index. Since the
conditions to be checked contain the zero object and are stable by extensions
as well as shifting by $[1]$ (resp., $[-1]$) for the first (resp., second)
result, we only have to show that (with $s:=\dim S$):

(35) $NMD({\mathscr{F}}^{\bullet},S)\in\begin{cases}\;{}^{p^{(+)}}D^{\leq
n-s}(\\{pt\\};R)\quad&\text{in part 1. of the theorem.}\\\
\;{}^{p^{(+)}}D^{\geq n-s}(\\{pt\\};R)\quad&\text{in part 2. of the
theorem.}\end{cases}$

By Proposition 3.14 we have two distinguished triangles:

$\begin{CD}R\Gamma(l_{X},{\mathscr{F}}^{\bullet})[-1]@>{}>{}>NMD({\mathscr{F}}^{\bullet},S)@>{}>{}>k_{x}^{*}({\mathscr{F}}^{\bullet}|_{N\cap
X})@>{}>{}>\>,\end{CD}$

and

$\begin{CD}k_{x}^{!}({\mathscr{F}}^{\bullet}|_{N\cap
X})@>{}>{}>NMD({\mathscr{F}}^{\bullet},S)@>{}>{}>R\Gamma(l_{X},\partial
l_{X},{\mathscr{F}}^{\bullet})[-1]@>{}>{}>\>,\end{CD}$

where we use, as before, the notation (with $0<|w|\ll\delta\ll 1$)

$(l_{X},\partial l_{X}):=(X\cap N\cap\\{r\leq\delta,g=w\\},X\cap
N\cap\\{r=\delta,g=w\\})$

for the complex link of $X$ with respect to $S\in{\mathscr{S}}$. Also,
$k_{x}\colon\\{x\\}\hookrightarrow N\cap X$ is the inclusion, and $r$ is the
0-convex distance function $r(z):=\sum_{i=1}^{n}\>z_{i}\bar{z}_{i}$ in our
fixed local embedding $(X,x)\hookrightarrow({\mathbb{C}}^{n},x=0)$.

But $N\cap\\{g=w\\}$ is transversal to ${\mathscr{S}}$ near $x$ (for $0<|w|\ll
1$). So

$L:=X\cap N\cap\\{g=w\\}$

gets an induced Whitney stratification ${\mathscr{S}}|_{L}$. Moreover, for
$0<|w|\ll\delta\ll 1$, $\delta$ is a regular value of $r$ with respect to
${\mathscr{S}}|_{L}$, since $dg_{x}$ is non-degenerate. Then we can use the
induction hypothesis with $q=0$ for

$\begin{cases}{\mathscr{F}}^{\bullet}|_{L},\>r\quad&\text{and
$[a,b]:=[-1,\delta]$ in part 1. of the theorem.}\\\
{\mathscr{F}}^{\bullet}|_{L},\>-r\quad&\text{and $[a,b]:=[-\delta,1]$ in part
2. of the theorem.}\end{cases}$

Note that

${\mathscr{F}}^{\bullet}|_{L}[-1]\in\;^{p^{(+)}}D^{\leq
n-s}_{{\mathscr{S}}|_{L}}(L;R)\quad\text{or}\quad\;^{p^{(+)}}D^{\geq
n-s}_{{\mathscr{S}}|_{L}}(L;R)$

by transversality and Proposition 2.27 (or its proof), with $\hbox{\rm
codim}\,L=s+1$.

By the induction hypothesis we get

$R\Gamma(l_{X},{\mathscr{F}}^{\bullet})[-1]\in\;^{p^{(+)}}D^{\leq
n-s}(\\{pt\\};R)$

or

$R\Gamma(l_{X},\partial
l_{X},{\mathscr{F}}^{\bullet})[-1]\in\;^{p^{(+)}}D^{\geq n-s}(\\{pt\\};R)\>.$

But, by definition, $k_{x}^{*}({\mathscr{F}}^{\bullet}|_{N\cap
X})\in\;^{p^{(+)}}D^{\leq n-s}(\\{pt\\};R)$, which implies part 1. of the
theorem by the first distinguished triangle above. Similarly

${\mathscr{F}}^{\bullet}|_{X\cap N}\in\;^{p^{(+)}}D^{\geq
n-s}_{{\mathscr{S}}|_{N}}(X\cap N;R)$

by Proposition 2.27 (or its proof), so that
$k_{x}^{!}({\mathscr{F}}^{\bullet}|_{N\cap X})\in\;^{p^{(+)}}D^{\geq
n-s}(\\{pt\\};R)$. This implies part 2. of the theorem by the second
distinguished triangle above. The last claim of Theorem 3.19 follows by
inspection of the proof. ∎

######  Corollary 3.20.

Let $\bar{X}$ be a compact complex algebraic (or analytic) variety of
dimension $d$, with ${\mathscr{F}}^{\bullet}\in D^{b}_{wc}(\bar{X};R)$ (resp.,
${\mathscr{F}}^{\bullet}\in D^{b}_{c}(\bar{X};R)$ and $R$ is a Dedekind domain
in case we consider the dual perverse t-structure). Let $j\colon
X\hookrightarrow\bar{X}$ be the inclusion of the open dense complement of a
closed algebraic (or analytic) subset $Z\subset\bar{X}$ (so that $d=\dim
X=\dim\bar{X}$). Then:

  1. (1)

$j^{*}{\mathscr{F}}^{\bullet}\in\;^{p^{(+)}}D^{\leq
n}(X;R)\>\Rightarrow\>Rj_{!}j^{*}{\mathscr{F}}^{\bullet}\in\;^{p^{(+)}}D^{\leq
n}(\bar{X};R)\>$

$\Rightarrow\>R\Gamma_{c}(X,j^{*}{\mathscr{F}}^{\bullet})\simeq
R\Gamma(\bar{X},Rj_{!}j^{*}{\mathscr{F}}^{\bullet})\in\;^{p^{(+)}}D^{\leq
n+d}(\\{pt\\};R)\>.$

  2. (2)

$j^{*}{\mathscr{F}}^{\bullet}\in\;^{p^{(+)}}D^{\geq
n}(X;R)\>\Rightarrow\>Rj_{*}j^{*}{\mathscr{F}}^{\bullet}\in\;^{p^{(+)}}D^{\geq
n}(\bar{X};R)\>$

$\Rightarrow\>R\Gamma(X,j^{*}{\mathscr{F}}^{\bullet})\simeq
R\Gamma(\bar{X},Rj_{*}j^{*}{\mathscr{F}}^{\bullet})\in\;^{p^{(+)}}D^{\geq
n-d}(\\{pt\\};R)\>.$

Note that by a theorem of Nagata (see, e.g., [82, 23]) such a compactification
is always available in the complex algebraic context.

The (method of) proof of Theorem 3.19 also implies the following.

######  Proposition 3.21.

Let ${\mathscr{S}}$ be a Whitney stratification of the complex algebraic (or
analytic) variety $X$, and consider a differentiable function $f\colon
X\to{\mathbb{R}}$. Assume $a<b$ are stratified regular values of $f$ with
respect to ${\mathscr{S}}$ such that $\\{a\leq f\leq b\\}$ is compact. Let
$T\subset D^{b}(\\{pt\\};R)$ be a fixed “null system”, i.e., a full
triangulated subcategory stable by isomorphisms.

  1. (1)

Denote by $D^{b}_{({\mathscr{S}}-)T-stalk}(X;R)$ the induced “null system”
given by all complexes ${\mathscr{F}}^{\bullet}\in
D^{b}_{({\mathscr{S}}-)wc}(X;R)$ with stalks
$i_{x}^{*}{\mathscr{F}}^{\bullet}\in T$ for all $x\in X$. Then

${\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-T-stalk}(X;R)\>\Rightarrow\>R\Gamma(\\{a\leq f\leq
b\\},\\{f=a\\},{\mathscr{F}}^{\bullet})\in T\>.$

  2. (2)

Denote by $D^{b}_{({\mathscr{S}}-)T-costalk}(X;R)$ the induced “null system”
given by all ${\mathscr{F}}^{\bullet}\in D^{b}_{({\mathscr{S}}-)wc}(X;R)$ with
costalks $i_{x}^{!}{\mathscr{F}}^{\bullet}\in T$ for all $x\in X$. Then

${\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-T-costalk}(X;R)\>\Rightarrow\>R\Gamma(\\{a\leq f\leq
b\\},\\{f=a\\},{\mathscr{F}}^{\bullet})\in T\>.$

######  Corollary 3.22.

Let $\bar{X}$ be a compact complex algebraic (or analytic) variety, with
${\mathscr{F}}^{\bullet}\in D^{b}_{wc}(\bar{X};R)$. Let $j\colon
X\hookrightarrow\bar{X}$ be the inclusion of the open complement of a closed
algebraic (or analytic) subset $Z\subset\bar{X}$, and fix a given “null
system” $T\subset D^{b}(\\{pt\\};R)$.

  1. (1)

$j^{*}{\mathscr{F}}^{\bullet}\in
D^{b}_{T-stalk}(X;R)\>\Rightarrow\>Rj_{!}j^{*}{\mathscr{F}}^{\bullet}\in
D^{b}_{T-stalk}(\bar{X};R)\>$

$\Rightarrow\>R\Gamma_{c}(X,j^{*}{\mathscr{F}}^{\bullet})\simeq
R\Gamma(\bar{X},Rj_{!}j^{*}{\mathscr{F}}^{\bullet})\in T\>.$

  2. (2)

$j^{*}{\mathscr{F}}^{\bullet}\in
D^{b}_{T-costalk}(X;R)\>\Rightarrow\>Rj_{*}j^{*}{\mathscr{F}}^{\bullet}\in
D^{b}_{T-costalk}(\bar{X};R)\>$

$\Rightarrow\>R\Gamma(X,j^{*}{\mathscr{F}}^{\bullet})\simeq
R\Gamma(\bar{X},Rj_{*}j^{*}{\mathscr{F}}^{\bullet})\in T\>.$

######  Example 3.23.

Let us mention here some important examples of such a “null system” $T\subset
D^{b}(\\{pt\\};R)$.

  1. (1)

$T=D^{b}_{c}(\\{pt\\};R)$ are the complexes with finitely generated
cohomology. Then $D^{b}_{({\mathscr{S}}-)T-stalk}(X;R)$ is the category of
(${\mathscr{S}}$-)constructible sheaf complexes.

  2. (2)

Assume $R$ is a Dedekind domain and $T=D^{b}_{tc}(\\{pt\\};R)$ are the
complexes with finitely generated torsion cohomology. Then
$D^{b}_{({\mathscr{S}}-)T-stalk}(X;R)$ is the category of
(${\mathscr{S}}$-)constructible torsion sheaf complexes.

  3. (3)

Assume $R$ is a field and $T=D^{b}_{c,\chi=0}(\\{pt\\};R)$ are the complexes
with finitely generated cohomology whose Euler characteristic is zero. Then
$D^{b}_{({\mathscr{S}}-)T-stalk}(X;R)$ is the category of
(${\mathscr{S}}$-)constructible sheaf complexes ${\mathscr{F}}^{\bullet}\in
D^{b}_{({\mathscr{S}}-)c}(X;R)$ with associated
(${\mathscr{S}}$-)constructible function

$\chi_{stalk}\left({\mathscr{F}}^{\bullet}\right)=0\in
CF_{({\mathscr{S}})}(X)\>,$

i.e., whose stalkwise Euler characteristic
$\chi_{stalk}\left({\mathscr{F}}^{\bullet}\right)(x):=\chi\left({\mathscr{F}}^{\bullet}_{x}\right)$
vanishes for all $x\in X$. Then $[{\mathscr{F}}^{\bullet}]\in
K_{0}\left(D^{b}_{({\mathscr{S}}-)c}(X;R)\right)$ is in the kernel of

$\chi_{stalk}\colon\>K_{0}\left(D^{b}_{({\mathscr{S}})-c}(X;R)\right)\to
CF_{({\mathscr{S}})}(X)\>\iff{\mathscr{F}}^{\bullet}\in
D^{b}_{({\mathscr{S}}-)T-stalk}(X;R)\>.$

######  Example 3.24.

Let $\bar{X}$ be a compact complex algebraic (or analytic) variety, with a
given Whitney stratification ${\mathscr{S}}$. Let
${\mathscr{F}}^{\bullet},{\mathscr{G}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-c}(\bar{X};R)$ and assume $R$ a field. Let $j\colon
X\hookrightarrow\bar{X}$ be the inclusion of the open complement of a closed
algebraic (or analytic) subset $Z\subset\bar{X}$, which is a union of strata
$S\in{\mathscr{S}}$, with ${\mathscr{S}}|_{X}$ the induced Whitney
stratification of $X$. Then
$\chi_{stalk}(j^{*}{\mathscr{F}}^{\bullet})=\chi_{stalk}(j^{*}{\mathscr{G}}^{\bullet})$
as a constructible function implies

$\chi_{c}(X,j^{*}{\mathscr{F}}^{\bullet})=\chi_{c}(X,j^{*}{\mathscr{G}}^{\bullet})\>.$

So the global Euler characteristic with compact support
$\chi_{c}(X,j^{*}{\mathscr{F}}^{\bullet})$ only depends on the underlying
constructible function $\chi_{stalk}(j^{*}{\mathscr{F}}^{\bullet})=\alpha\in
CF_{{\mathscr{S}}}(X)$, with

(36) $\chi_{c}(X,{\mathscr{F}}^{\bullet})=\int_{X}\alpha
d\chi:=\sum_{S\in{\mathscr{S}}|_{X}}\>\chi_{c}(S)\cdot\alpha(S)\>.$

Here $\chi_{c}(S):=\chi_{c}(S,R_{S})=\chi(H_{c}^{*}(S;R))$ is the
corresponding Euler characteristic of a stratum $S\in{\mathscr{S}}|_{X}$.

Similar considerations apply if $X$ is given as a relatively compact open
subset $X=\\{f<b\\}$ (resp., a compact subset $X=\\{f\leq b\\}$) for a proper
differentiable function $f\colon\bar{X}\to[a,d[\;\subset{\mathbb{R}}$, with
$b<d$ a stratified regular value of $f$ with respect to ${\mathscr{S}}$. Here
$\bar{X}$ does not need to be compact, with strata $S$ of
${\mathscr{S}}|_{\\{f<b\\}}$ given by the connected components of
$S^{\prime}\cap\\{f<b\\}$ for $S^{\prime}\in{\mathscr{S}}$. Similarly for the
decomposition ${\mathscr{S}}|_{\\{f\leq b\\}}$ of $\\{f\leq b\\}$, but here
the parts $S^{\prime}\cap\\{f\leq b\\}$ for $S^{\prime}\in{\mathscr{S}}$ are
complex manifolds with possible non-empty boundary $S^{\prime}\cap\\{f=b\\}$.

The proof of Theorem 3.19 implies directly the following characterizations
(see, e.g., [113, Corollary 6.0.2]).

######  Corollary 3.25.

Let ${\mathscr{S}}$ be a Whitney stratification of the complex algebraic (or
analytic) variety $X$, and consider ${\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-wc}(X;R)$ (resp., ${\mathscr{F}}^{\bullet}\in
D^{b}_{c}(X;R)$ and $R$ is a Dedekind domain in case we consider the dual
perverse t-structure). Then we have:

  1. (1)

${\mathscr{F}}^{\bullet}\in\,^{p^{(+)}}D^{\leq 0}_{{\mathscr{S}}}(X;R)$ $\iff$
$NMD({\mathscr{F}}^{\bullet},S)[-\dim(S)]\in\,^{p^{(+)}}D^{\leq
0}(\\{pt\\};R)$

for all strata $S\in{\mathscr{S}}$.

  2. (2)

${\mathscr{F}}^{\bullet}\in\,^{p^{(+)}}D^{\geq 0}_{{\mathscr{S}}}(X;R)$ $\iff$
$NMD({\mathscr{F}}^{\bullet},S)[-\dim(S)]\in\,^{p^{(+)}}D^{\geq
0}(\\{pt\\};R)$

for all strata $S\in{\mathscr{S}}$.

  3. (3)

Let $T\subset D^{b}(\\{pt\\};R)$ be a fixed “null system”. Then

$\displaystyle{\mathscr{F}}^{\bullet}\in D^{b}_{{\mathscr{S}}-T-stalk}(X;R)$
$\displaystyle\iff$ $\displaystyle NMD({\mathscr{F}}^{\bullet},S)\in
T\quad\text{for all strata $S\in{\mathscr{S}}$}$ $\displaystyle\iff$
$\displaystyle{\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-T-costalk}(X;R)\>.$

###### Proof.

The implications $\Rightarrow$ are already contained in (35) from the proof of
Theorem 3.19. For the other implications $\Leftarrow$ in case of a point
stratum $S=\\{x\\}$ one applies the proof of Theorem 3.19 for the (co)stalk
description

$i_{x}^{*}{\mathscr{F}}^{\bullet}\simeq
R\Gamma(\\{r\leq\delta\\},{\mathscr{F}}^{\bullet})\simeq R\Gamma(\\{-1\leq
r\leq\delta\\},\\{r=-1\\},{\mathscr{F}}^{\bullet})$

and

$i_{x}^{!}{\mathscr{F}}^{\bullet}\simeq
R\Gamma_{c}(\\{r<\delta\\},{\mathscr{F}}^{\bullet})\simeq
R\Gamma(\\{-\delta\leq-r\leq
1\\},\\{-r=-\delta\\},{\mathscr{F}}^{\bullet})\>,$

with $0<\delta$ small and $r$ the 0-convex distance function
$r(z):=\sum_{i=1}^{n}\>z_{i}\bar{z}_{i}$ in some fixed local embedding
$(X,x)\hookrightarrow({\mathbb{C}}^{n},x=0)$. The general case is reduced to a
point stratum via intersecting with a normal slice $N$ to $S\in{\mathscr{S}}$
at a given point $x\in S$. Here, the normal Morse data
$NMD({\mathscr{F}}^{\bullet},S^{\prime})=NMD({\mathscr{F}}^{\bullet}|_{N},S^{\prime}\cap
N)$ for $S^{\prime}\in{\mathscr{S}}$ close to $x$ does not change. ∎

######  Example 3.26.

Let ${\mathscr{S}}$ be a Whitney stratification of the complex algebraic (or
analytic) variety $X$, with ${\mathscr{F}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-wc}(X;R)$. Then ${\mathscr{F}}^{\bullet}$ is a perverse
sheaf if and only if

$H^{i}\left(NMD({\mathscr{F}}^{\bullet},S)[-\dim(S)]\right)=0\quad\text{for
$i\neq 0$ and all strata $S\in{\mathscr{S}}$.}$

If, moreover, ${\mathscr{F}}^{\bullet}$ is constructible and $R$ a Dedekind
domain, then ${\mathscr{F}}^{\bullet}$ is a strongly perverse sheaf if and
only if, in addition,

$H^{0}\left(NMD({\mathscr{F}}^{\bullet},S)[-\dim(S)]\right)\quad\text{ is
finitely generated and torsion-free }$

for all strata $S\in{\mathscr{S}}$.

######  Example 3.27.

Let ${\mathscr{S}}$ be a Whitney stratification of the complex algebraic (or
analytic) variety $X$, with
${\mathscr{F}}^{\bullet},{\mathscr{G}}^{\bullet}\in
D^{b}_{{\mathscr{S}}-c}(X;R)$ and $R$ a field. Then
$\chi_{stalk}({\mathscr{F}}^{\bullet})=\chi_{stalk}({\mathscr{G}}^{\bullet})$
as an ${\mathscr{S}}$-constructible function if and only if

$\chi\left(NMD({\mathscr{F}}^{\bullet},S)\right)=\chi\left(NMD({\mathscr{G}}^{\bullet},S)\right)\quad\text{for
all strata $S\in{\mathscr{S}}$.}$

So the Euler characteristic
$\chi\left(NMD({\mathscr{F}}^{\bullet},S)\right)=:\chi\left(NMD(\alpha,S)\right)$
for $S\in{\mathscr{S}}$ depends only on the underlying
${\mathscr{S}}$-constructible function
$\alpha=\chi_{stalk}({\mathscr{F}}^{\bullet})$.

######  Remark 3.28.

Corollary 3.25 also shows that the corresponding properties of the normal
Morse data do not depend on the choice of the Whitney stratification
${\mathscr{S}}$ of $X$ with ${\mathscr{F}}^{\bullet}$ ${\mathscr{S}}$-(weakly)
constructible. The given characterization of the perverse t-structure in terms
of normal Morse data is equivalent (in an embedded context) to [67, Theorem
10.3.12], if one identifies the normal Morse data with the corresponding
“micro-local type” in the sense of Kashiwara-Schapira [67, Proposition