# Hyperbolic localization of the Donaldson-Thomas sheaf

Pierre Descombes

# Hyperbolic localization of the Donaldson-Thomas sheaf

Pierre Descombes

###### Abstract.

In this paper we prove a toric localization formula in cohomological Donaldson
Thomas theory. Consider a $-1$-shifted symplectic algebraic space with a
$\mathbb{C}^{\ast}$ action leaving the $-1$-shifted symplectic form invariant.
This includes the moduli space of stable sheaves or complexes of sheaves on a
Calabi-Yau threefold with a $\mathbb{C}^{\ast}$-invariant Calabi-Yau form, or
the intersection of two $\mathbb{C}^{\ast}$-invariant Lagrangians in a
symplectic space with a $\mathbb{C}^{\ast}$-invariant symplectic form. In this
case we express the restriction of the Donaldson-Thomas perverse sheaf (or
monodromic mixed Hodge module) defined by Joyce et al. to the attracting
variety as a sum of cohomological shifts of the DT perverse sheaves on the
$\mathbb{C}^{\ast}$ fixed components. This result can be seen as a
$-1$-shifted version of the Białynicki-Birula decomposition for smooth
schemes.

###### Contents

  1. 1 Introduction
  2. 2 Classical Białynicki-Birula decomposition
    1. 2.1 Hyperbolic localization
    2. 2.2 Compatibility with smooth morphisms
  3. 3 Critical Białynicki-Birula decomposition
    1. 3.1 The functor of vanishing cycles
    2. 3.2 Functoriality of the vanishing cycles
    3. 3.3 Hyperbolic localization of the perverse sheaf of vanishing cycles
    4. 3.4 Compatibility with smooth maps
    5. 3.5 Compatibility with Thom-Sebastiani isomorphism
  4. 4 Białynicki-Birula decomposition on a d-critical algebraic space
    1. 4.1 D-critical algebraic space
    2. 4.2 Gluing the Donaldson-Thomas sheaves
    3. 4.3 Białynicki-Birula decomposition
    4. 4.4 Compatibility with smooth pullbacks
  5. 5 Technical lemmas
    1. 5.1 Trivialization of torus-equivariant embeddings
    2. 5.2 Smooth torus-equivariant morphism of d-critical scheme

##  1\. Introduction

### Overview

A $-1$ shifted symplectic space $X$, such as the moduli space of stable
sheaves or complexes of sheaves on a Calabi-Yau threefold or an intersection
of two Lagrangians, is informally an algebraic space where the obstruction
space is dual to the tangent space. Building on ideas of Kontsevich-Soibelman,
Joyce and collaborators have proved that such a space admits a canonical
d-critical structure $s$, providing étale charts of $X$ as the critical locus
of a function in a smooth space, and defined a perverse sheaf $P_{X,s}$ (and
furthermore a monodromic mixed Hodge module on this perverse sheaf) on such
$X$ once one has fixed an orientation data. The Euler number of the cohomology
of this so-called Donaldson-Thomas sheaf gives the numerical Donaldson-Thomas
invariants defined by Thomas. The numerical DT invariant localizes under a
torus action, leaving the shifted symplectic structure invariant as the Euler
number of a smooth space, i.e. the numerical DT invariant of a space $X$ with
a $\mathbb{C}^{\ast}$ action is the signed sum of the invariants of the
$\mathbb{C}^{\ast}$ fixed components. On a smooth space $U$, the Białynicki-
Birula decomposition gives a refinement of the localization formula of the
Euler number, which can be expressed using the hyperbolic localization functor
as

(1.1) $\displaystyle
p_{!}\eta^{\ast}\mathbb{Q}_{U}\simeq\bigoplus_{\pi\in\Pi}\mathbb{L}^{d^{+}_{\pi}}\mathbb{Q}_{U^{0}_{\pi}}$

where $\mathbb{Q}_{Y}$ is the constant sheaf on $Y$ (or the constant mixed
Hodge module), $\mathbb{L}^{1/2}$ is the shift $[-1]$ of a complex (or the
underlying monodromic mixed Hodge structure), $\eta$ is the inclusion of the
attracting variety $U^{+}$ of $U$, $p$ is the projection of the attracting
variety on the $\mathbb{C}^{\ast}$ fixed locus with connected component
decomposition $U^{0}:=\bigsqcup_{\pi\in\Pi}U^{0}_{\pi}$ and $d^{+}_{\pi}$ is
the number of contracting weights in the action of $\mathbb{C}^{\ast}$ on the
tangent space of $X$ in $X^{0}_{\pi}$. For an algebraic space $X$ with
$\mathbb{C}^{\star}$-invariant d-critical structure $s$ and orientation, the
fixed components $X^{0}_{\pi}$ have natural d-critical structures
$s^{0}_{\pi}$ and orientation. In this paper we prove a similar hyperbolic
localization formula for the Donaldson-Thomas perverse sheaf (and the
monodromic mixed Hodge module) defined using these oriented d-critical
structures:

(1.2) $\displaystyle
p_{!}\eta^{\ast}P_{X,s}\simeq\bigoplus_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}P_{X^{0}_{\pi},s^{0}_{\pi}}$

where ${\rm Ind}_{\pi}$ is the signed number of contracting weights in the
tangent obstruction complex of $X$ on the component $X^{0}_{\pi}$. We define
the cohomological DT invariant $[X]^{vir}:=[H_{c}(X,P_{X,s})]$ as the class of
hypercohomology with compact support of $P_{X,s}$ in the Grothendieck group of
monodromic mixed Hodge structures (in particular, one can extract a virtual
Hodge polynomial from it). To compute the cohomological DT invariants of $X$
one is reduced to computing the cohomological DT invariants of the fixed
components (which in nice cases are isolated fixed points), and the
cohomological DT invariants of the open complement $X-\eta(X^{+})$ of the
attracting locus (which in nice cases is simpler than the whole moduli space):

(1.3) $\displaystyle[X]^{vir}=\sum_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}[X^{0}_{\pi}]^{vir}+[X-\eta(X^{+})]^{vir}$

An example where this procedure can be carried out completely is given by
framed invariants of toric quivers, as studied in [15]. If $X$ is projective,
or more generally if each point of $X$ is in the attracting locus of a fixed
component (one says that the $\mathbb{C}^{\ast}$ action is circle-compact), we
obtain a formula similar to the localization formula for K-theoretic
invariants:

(1.4) $\displaystyle[X]^{vir}=\sum_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}[X^{0}_{\pi}]^{vir}$

### The Donaldson-Thomas perverse sheaf

Donaldson-Thomas theory was first developed to count sheaves on Calabi-Yau
threefolds. In [35], Thomas defined the numerical Donaldson-Thomas invariants,
giving the virtual Euler number of the moduli space of stable coherent sheaves
on a Calabi-Yau threefold, using the perfect obstruction theory given by the
Serre duality on the Ext spaces of the sheaves. In [6], Behrend gave a new
interpretation of these invariants: expressing the moduli space locally as the
critical locus of a potential, the numerical Donaldson-Thomas invariant is
given by an Euler characteristic weighted at each point by the Milnor number.
In [23] and [22], Kontsevich and Soibelman sketched the definition of a
cohomological refinement of this counting, with value in the abelian category
of monodromic mixed Hodge modules (MMHM), using the functor of vanishing
cycles of this potential, in a partially conjectural framework.

In papers [21], [11],[10] and [7], Joyce and collaborators have developed a
rigorous cohomological Donaldson-Thomas theory, using the language of
$-1$-shifted symplectic structures in derived geometry introduced in [29].
Informally, derived geometry replaces the notion of tangent space $T_{X}$ by a
tangent complex $\mathbb{T}_{X}$, where $(\mathbb{T}_{X})_{0}$ gives the
tangent directions, $(\mathbb{T}_{X})_{1}$ gives the obstructions,
$(\mathbb{T}_{X})_{2}$, $(\mathbb{T}_{X})_{3}$, . … give the higher
obstructions, and $(\mathbb{T}_{X})_{-1}$ gives the infinitesimal
automorphisms. A $-1$-shifted symplectic structure gives a pairing between
$\mathbb{T}_{X}$ and $\mathbb{T}_{X}[-1]$: in particular, it pairs tangent
directions with obstructions. In [29] it was shown that moduli stacks of
complexes on a Calabi-Yau 3-fold, and intersections of two Lagrangians in a
symplectic space, naturally have a $-1$-shifted symplectic structure.

In [11] it was shown that $-1$-shifted derived algebraic space can be written
étale locally as the critical locus of a functional on a smooth scheme, i.e.
can be covered by critical charts of the form $(R,U,f,i)$, where $R\subset X$
is open, $U$ is a smooth scheme, $f:U\to\mathbb{C}$ is a regular map, and
$i:R\to U$ is a closed embedding such that $i(R)={\rm Crit}(f)$, and there is
a way to compare intersecting critical charts. More formally, such an
algebraic space has a d-critical structure $s$, a notion we will introduce
below. In [10], Joyce and collaborators constructed the Donaldson-Thomas
perverse sheaf $P_{X,s}$ carrying a monodromic mixed Hodge module for a
d-critical algebraic space $(X,s)$, with additional data called the
orientation $K_{X}^{1/2}$. Its restriction to a critical $(R,U,f,i)$ is given
by $\mathcal{PV}_{U,f}$, the perverse sheaf (or monodromic mixed Hodge module)
of vanishing cycles defined by applying the vanishing cycles functor
$\phi_{f}$ of $f$ to $\mathcal{IC}_{U}$, the intersection cohomology complex
of $U$ with its natural mixed Hodge module structure, and then restricting to
$R$. The orientation is used to glue the perverse sheaves on intersections of
critical charts. The numerical Donaldson-Thomas invariant defined in [35] is
then, as expected, the Euler number of the cohomology of this perverse sheaf.
We will use the formalism of d-critical structures in this article, but not
the formalism of derived geometry and shifted symplectic structures.

### Hyperbolic localization

The aim of this paper is to provide a way to compute the cohomological
Donaldson-Thomas invariants by localization, namely, given a d-critical
algebraic space $X$ with a $\mathbb{C}^{\ast}$ action and a
$\mathbb{C}^{\ast}$ invariant d-critical structure, express the invariants of
$X$ in terms of the invariants of the $\mathbb{C}^{\ast}$ fixed algebraic
space $X^{0}$. Graber and Pandharipande proved a torus localization formula
for numerical Donaldson-Thomas invariants in [17]. An analogue formula was
derived in [5] using the alternative definition with weighted Euler
characteristic: the numerical Donaldson-Thomas invariant of $X$ is the sum of
those of each component of $X^{0}$ weighted by a sign given by the parity of
the dimension of the normal space to the component. Thus, numerical Donaldson-
Thomas invariants localize under torus action exactly like the Euler numbers
of smooth spaces.

In [8], Białynicki-Birula proved that, on a smooth scheme $X$ with
$\mathbb{C}^{\ast}$ action, the attracting subset of $X$ (ı.e. the subset of
points $x\in X$ such that $\lim_{t\to 0}t.x$ exists) admits a kind of cellular
decomposition. Each cell of this so-called Białynicki-Birula decomposition is
an affine fiber bundle on a component of the fixed variety $X^{0}$, whose
dimension is given by the number of contracting $\mathbb{C}^{\ast}$-weights in
the tangent space of $X^{0}$. Thus one can compute the cohomology of the
attracting subset of $X$ in terms of the cohomology of $X^{0}$. In [9] and
[16], Braden and Drinfeld gave a functorial reformulation of the Białynicki-
Birula decomposition that can be extended to singular algebraic spaces. For
any algebraic space $X$ with $\mathbb{C}^{\ast}$ action one can define the
$\mathbb{C}^{\ast}$ fixed space $X^{0}$ and also the contracting (resp.
repelling) space $X^{\pm}$ in an obvious way using the functor of points. Then
there is the so-called hyperbolic localization diagram:

${X}$${X^{\pm}}$${X^{0}}$$\scriptstyle{\eta^{\pm}_{X}}$$\scriptstyle{p^{\pm}_{X}}$

where $\eta^{\pm}_{X}$ is the inclusion of the attracting (resp. repelling)
subset, and $p^{+}_{X}$ (or $p^{-}_{X}$) sends $x$ to $\lim_{t\to 0}x$ (or
$\lim_{t\to\infty}x$). This diagram is easy to describe if $X$ is an affine
scheme: it suffices to find a $\mathbb{C}^{\ast}$-equivariant covering of $X$
by affine spaces. The existence of such a covering is not at all obvious, and
is far too restrictive in Zariski topology. Fortunately, a deep result of [2]
and [1] shows that quasi-separated algebraic spaces of finite type locally
have a $\mathbb{C}^{\ast}$-equivariant covering by affine spaces. Under these
assumptions $X^{0}$ and $X^{\pm}$ are then algebraic spaces. The functors
$(p^{\pm}_{X})_{!}(\eta^{\pm}_{X})^{\ast}$ are called hyperbolic localization
functors. From [16] there is a natural morphism built from morphisms of the
six-functor formalism in the derived category of constructible sheaves, or the
derived category of monodromic mixed Hodge modules:

(1.5) $\displaystyle
S_{X}:\mathbb{D}(p^{-}_{X})_{!}(\eta^{-}_{X})^{\ast}\mathbb{D}\to(p^{+}_{X})_{!}(\eta^{+}_{X})^{\ast}$

This is by [16] and [31] an isomorphism when these functors are applied to
$\mathbb{C}^{\ast}$-equivariant complexes, i.e. complexes equivariant under
the action of $\mathbb{C}^{\ast}$, in a sense we will explain below.

Assume now that $X$ is smooth, and denote by $\Pi$ the cocharacters of
$\mathbb{C}^{\ast}$, and for $\pi\in\Pi$ $X^{0}_{\pi}$ the union of the
connected components of $X^{0}$, where the $\mathbb{C}^{\ast}$ action on
$T_{X^{0}}$ has the cocharacter $\pi$. For $\pi\in\Pi$, denote by
$d^{0}_{\pi},d^{+}_{\pi},d^{-}_{\pi}$ the number of invariants, resp.
contracting, resp. repelling weights in $\pi$, and define ${\rm
Ind}_{\pi}:=p^{+}_{\pi}-p^{-}_{\pi}$. The Białynicki-Birula decomposition for
smooth schemes says that $X^{\pm}$ is a disjoint union of affine fiber bundles
$X^{\pm}_{\pi}$ of dimension $d^{\pm}_{\pi}$ over $X^{0}_{\pi}$. Using the
hyperbolic localization functors, this can be reformulated as

$\displaystyle(p^{\pm}_{X})_{!}(\eta^{\pm}_{X})^{\ast}\mathbb{Q}_{U}$
$\displaystyle=\bigoplus_{\pi\in\Pi}\mathbb{L}^{d^{\pm}_{\pi}}\mathbb{Q}_{U^{0}_{\pi}}$
(1.6)
$\displaystyle\Rightarrow(p^{\pm}_{X})_{!}(\eta^{\pm}_{X})^{\ast}\mathcal{IC}_{U}$
$\displaystyle=\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\mathcal{IC}_{U^{0}_{\pi}}$

where $\mathbb{Q}_{Y}$ and $\mathcal{IC}_{Y}$ are respectively the constant
sheaf and the intersection complex of $Y$, or their natural mixed Hodge
modules. This isomorphism sends $S_{X}$ to the self-duality isomorphism of the
intersection complex.

### The main result

For a $\mathbb{C}^{\ast}$-equivariant oriented d-critical algebraic space $X$
satisfying the assumptions under which the hyperbolic localization functors
were defined, we will prove that $X^{0}$ has a natural d-critical structure
$s^{0}$ and a natural orientation $K_{X^{0}}^{1/2}$ on $(X^{0},s^{0})$ induced
by $K_{X}^{1/2}$. The aim of this article is to prove that an analogue of
(1.6) holds for the Donaldson-Thomas perverse sheaf (res. monodromic mixed
Hodge module) on a d-critical oriented algebraic space :

######  Theorem 1.1.

(Theorem 4.2) For $X$ a quasi-separated locally of finite type algebraic space
with an action of a one-dimensional torus $\mathbb{C}^{\ast}$ and
$\mathbb{C}^{\ast}$-equivariant d-critical structure $s$ and orientation,
there are natural isomorphisms of perverse sheaves and monodromic mixed Hodge
modules:

(1.7)
$\displaystyle\beta^{\pm}_{X,s}:(p_{X}^{\pm})_{!}(\eta_{X}^{\pm})^{\ast}P_{X,s}\overset{\simeq}{\to}\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}P_{X^{0}_{\pi},s^{0}_{\pi}}$

where ${\rm Ind}_{\pi}$ is the signed number of contracting weights in the
$\mathbb{C}^{\ast}$ action on the tangent obstruction complex of $X$ at
$X^{0}_{\pi}$.

### Sketch of the proof

The main ingredient of the proof of this theorem is the commutation of the
hyperbolic localization and vanishing cycles functor proved in [31]. On a
critical chart $(R,U,f,i)$, denoting by $(R^{0},U^{0},f^{0},i^{0})$ the fixed
critical chart, combining with the classical Białynicki-Birula decomposition
of $U$ (1.6), one obtains isomorphisms in the derived category of
constructible sheaves with monodromy, or of monodromic mixed Hodge modules:

(1.8)
$\displaystyle\beta^{\pm}_{U,f}:(p_{R}^{\pm})_{!}(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f}\to\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}}$

This is the content of Proposition 3.3. We need to show that these
isomorphisms behave well with respect to pullback by étale maps of the
critical locus: this is the content of Proposition 3.4. We also need to show
that they behave well with respect to the Thom-Sebastiani isomorphism of [24]:

(1.9) $\displaystyle\mathcal{TS}_{U,f,V,g}:\mathcal{PV}_{U\times V,f\boxplus
g}\simeq\mathcal{PV}_{U,f}\boxtimes\mathcal{PV}_{S,g}$

This is the content of Proposition 3.5.

Now consider a d-critical oriented algebraic space $(X,s)$. The isomorphism
$\beta^{\pm}_{X,s}$ is an isomorphism in the abelian category
$\bigoplus_{\pi}{\rm Perv}(X^{0}_{\pi})[-{\rm Ind}_{\pi}]$ of shifted perverse
sheaves, or the abelian category $\bigoplus_{\pi}{\rm MMHM}(X^{0}_{\pi})[-{\rm
Ind}_{\pi}]$ of monodromic mixed Hodge modules, which form stacks for the
étale topology on $X$, so it suffices to define them as $\beta^{\pm}_{U,f}$ on
any critical chart $(R,U,f,i)$, and to show that they agree on intersections
of critical charts, compatibility with monodromy and self-duality can be
checked on critical charts. Note that the tangent obstruction complex of $X$
on $R$ is quasi-isomorphic with $0\to T_{U}\to T^{\ast}_{U}\to 0$, so ${\rm
Ind}_{\pi}$ is the signed number of contracting weights in the tangent-
obstruction complex of $X$ at $X^{0}_{\pi}$.

For d-critical algebraic spaces, Joyce proved in [21] that two critical charts
intersecting at a point can be related in an étale neighbourhood of that point
by stabilization by quadratic forms, that is, by embeddings
$(U,f)\hookrightarrow(U\times E,f\boxplus q)$, where $q$ is a non-degenerate
$\mathbb{C}^{\ast}$-invariant quadratic form on a
$\mathbb{C}^{\ast}$-equivariant vector space $E$. In [10], Joyce et al. glue
the perverse sheaves of vanishing cycles using the isomorphism:

${\mathcal{PV}_{U\times E,f\boxplus
q}}$${\mathcal{PV}_{U,f}\boxtimes\mathcal{PV}_{E,q}}$${\mathcal{PV}_{U,f}}$$\scriptstyle{\mathcal{TS}_{U,f,E,q}}$$\scriptstyle{\simeq}$

where they use the isomorphism $\mathcal{PV}_{E,q}\simeq\mathbb{Q}$ with a
choice of orientation on $E$ (this choice is at the root of orientation
problems in cohomological DT theory). To prove that the isomorphisms
$\beta^{\pm}$ agree on the intersection of critical charts, we prove that they
are compatible with stabilization, which is done using compatibility with the
Thom-Sebastiani isomorphism and a careful comparison of orientations.

In this paper we use the functorial properties of the six-functor formalism in
the derived categories $D^{b}_{c}(Y)$ of bounded constructible complexes over
various algebraic spaces $Y$, or, more formally, a $(\infty,1)$-categorical
enhancement of them. When we say that a diagram commutes from the naturality
of the six-functor formalism, we imply that it commutes up to a natural
2-isomorphism. Since the final diagrams lie in the abelian heart
$\bigoplus_{\pi}{\rm Perv}(\mathcal{X}^{0}_{\pi})[-{\rm Ind}_{\pi}]$ of
$D^{b}_{c}(X^{0})$, which is a classical category, they really commute.
Moreover, it is assumed without specification that all algebraic spaces are
quasi-separated and locally of finite type.

For $X$ an algebraic space, the category ${\rm MHM}(X)$ of mixed Hodge modules
introduced in [32] is an abelian category with a faithful and exact functor
$rat:{\rm MHM}(X)\to{\rm Perv}(X)$. In particular, because $rat$ is faithful,
the commutativity of a diagram can be checked at the level of perverse
sheaves. Moreover, since a mixed Hodge module $M$ is the data of filtrations
on the $\mathcal{D}$ module associated with the complexification of the
perverse sheaf $rat(M)$ by the Riemann-Hilbert correspondence, a morphism of
mixed Hodge modules is an isomorphism if the underlying morphism of perverse
sheaves is an isomorphism. There is a six-functor formalism on the derived
category of mixed Hodge modules which, under $rat$, projects to the six-
functor formalism on $D^{b}_{c}(X)$, the derived category of constructible
complexes. Furthermore, since $rat$ is exact, a morphism between complexes of
mixed Hodge modules is an isomorphism in the derived category if the
underlying morphism of constructible complexes is an isomorphism in the
derived category. Thus, the extension of our results to monodromic mixed Hodge
modules is straightforward: our morphisms are easily defined using the six-
functor formalism, so we used the same definition to extend them to morphisms
in the derived category of monodromic mixed Hodge modules. The hard work is to
check that these morphisms are isomorphisms or that they give commutative
diagrams, but this can be checked at the level of perverse sheaves as
explained above.

### Analytic version

In [10] there is also an analytic version of cohomological Donaldson-Thomas
theory, for analytic oriented d-critical analytic spaces which are locally the
critical locus of a holomorphic function. Unfortunately, we lack crucial
results on $\mathbb{C}^{\ast}$ actions on analytic spaces. If an analytic
space with $\mathbb{C}^{\ast}$ action has a $\mathbb{C}^{\ast}$ equivariant
analytic (resp. smooth) cover by an affine analytic space, we show that the
fixed and attracting/repelling varieties are analytic spaces, hence the
hyperbolic localization functors and the morphism $S_{X}$ are defined.
However, we do not know under which generality such a
$\mathbb{C}^{\ast}$-equivariant cover exists. Moreover, we need an analytic
version of [31, Theo B] which says that $S_{X}$ is an isomorphism when applied
to $\mathbb{C}^{\ast}$-equivariant complexes. Furthermore, according to [10,
Remark 2.20], the six-functor formalism is not fully defined for mixed Hodge
modules in the analytic case, so it might be difficult to lift a localization
result from the level of perverse sheaves to the level of mixed Hodge modules.
Once such results are available, all the results of this article should extend
directly to the analytic case.

### Case with semistable objects

Here we consider that we have $-1$-shifted symplectic spaces, which is the
case for the moduli space of stable objects in CY3 categories as proved in
[12]. If one has semistable objects that are not stable, the stack
$\mathfrak{X}$ of semistable objects is $-1$-shifted symplectic from [12].
Joyce and collaborators then built a perverse sheaf $P_{\mathfrak{X}}$ on the
$-1$-shifted symplectic stack with orientation data in [7]. If one had a nice
six-functor formalism for mixed Hodge modules on stack, one could similarly
define a DT mixed Hodge module on any $-1$-shifted symplectic stack, but such
a theory is still lacking.

Under very general assumptions, in this setting there is a good moduli space
$JH:\mathfrak{X}\to X$ from [3] which is not $-1$-shifted symplectic if there
are semistable objects which are not stable. As suggested by Toda in [36], one
can mimic the definition of [14] to define a BPS perverse sheaf:

(1.10)
$\displaystyle\mathcal{BPS}:=^{p}\mathcal{H}^{1}(JH_{!}P_{\mathfrak{X}})$

where ${}^{p}\mathcal{H}^{1}$ denotes the first perverse cohomology. However,
a nice integrality formalism generalising the results of [14] is still
missing.

It seems plausible that, once such a formalism is established, a similar
localization formula holds for the BPS perverse sheaf of mixed Hodge modules,
extending localization techniques to the case with semistable objects, and
this is the subject of future work. A localization result for the perverse
sheaf $P_{\mathfrak{X}}$ on the stack was proved in the first version of this
paper, but we have removed it for the clarity of the present paper. We have
only kept the subsection 4.4 proving the compatibility of the localization
isomorphism with smooth maps of d-critical loci, which was instrumental in
proving the localization formula for stacks, because it may be useful in the
future, and do not use any stack formalism.

### Relations to other works

The idea of using hyperbolic localization to obtain a localization formula in
cohomological DT theory was first formulated by Balazs Szendroi in [34,
Section 8.4]. It was applied in [27, Section 6] and in [30, Section 8.3],
where it was used in a specific example of framed representations of quivers
with potential. In [31] Timo Richarz proved the commutativity of hyperbolic
localization with the vanishing cycles functor in a more general way. We used
this result to establish the formula (1.3) for any critical locus of a
potential in [15], and the extension of this result to $-1$-shifted symplectic
spaces was suggested to us by Richard Thomas.

In [13] the authors defined a Donaldson-Thomas motive which glues the motive
of vanishing cycles defined by arc spaces. Motives glue in Zariski topology
and not in étale topology, so to prove a localization formula one must a
priori restrict to d-critical schemes admitting Zariski
$\mathbb{C}^{\ast}$-equivariant critical charts. According to [21, Prop 2.
43], the existence of such charts is equivalent to the existence of a
$\mathbb{C}^{\ast}$-equivariant Zariski cover (such a $\mathbb{C}^{\ast}$
action is then called ’good’), which holds for normal spaces from Sumihiro’s
theorem [33], but not in greater generality (see [21, Example 2. 46]). Davesh
Maulik has proved a formula similar to (1.4) for motivic DT invariants on
d-critical schemes with a good circle-compact $\mathbb{C}^{\ast}$ action, as
explained in [7, section 5.3], in an unpublished preprint (private
communication). A generalisation of this result for non-Archimedean geometry
was subsequently proved in [20, Theo 7.17]. For non-circle compact actions one
can compute the DT motive of the attracting variety, which is by definition
circle compact, in this way.

A toric localization similar to (1.4) also exists for K-theoretic DT
invariants as defined in [28]. The K-theoretic DT invariants are a refinement
of the numerical DT invariants defined for projective moduli spaces with
symmetric obstruction theories, developed in parallel to the motivic and
cohomological refinement of Kontsevich-Soibelman and Joyce and collaborators.
In general, they are expected to correspond to the $\chi_{y}$ genus of the
Hodge polynomial of cohomological DT theory, so in particular one replaces
$\mathbb{L}^{1/2}$ by $-y$ in K-theoretic formulae. If the moduli space $X$ is
not projective, but has a $\mathbb{C}^{\ast}$ action with projective fixed
components, it was suggested in [28] to use the equation (1.4) to define the
$K$-theoretic invariants of $X$. However, this definition depends on the
choice of the $\mathbb{C}^{\ast}$ action (this choice is called the choice of
the slope). The equation (1.3) in the non-projective case explains the origin
of this ambiguity: one computes by toric localization only the virtual
cohomology of the attracting variety, which is not the whole moduli space and
depends on the chosen $\mathbb{C}^{\ast}$-action.

This dependence on the slope was studied explicitly in [4] for the moduli
space of framed representations of a toric quiver, and this was related to the
ambiguity in the refined topological vertex of [18]. In this case there is a
two-dimensional torus invariant acting on the moduli space of framed
representations, scaling the arrows of the quiver by leaving the potential
invariant, so the space of slopes is $\mathbb{P}^{1}_{\mathbb{R}}$. The fixed
points can be described as molten crystals from [26]. In [4, Prop 3.3] it was
established that there is a wall and chamber structure on the space of slopes,
with the generating functions of framed invariants being constant in a chamber
and jumping at a wall, the wall corresponding to slopes where the weight of an
elementary cycle of the quiver becomes attracting or repelling. This is rather
strange at first sight, because inside a given chamber the cohomological
weight of a given molten crystal changes at many walls, but the final result
does not change: these walls are ’invisible’. In [15] it was established that
the attracting variety is the subspace of representations where the cycles
with repelling weights are nilpotent, so the attracting variety changes
exactly on the walls defined in [4], i.e. (1.3) give an explanation of this
wall and chamber structure. Moreover, using a nilpotent/invertible
decomposition for the unframed representation and a wall-crossing relation
between framed and unframed invariants, [15] obtained the full framed
generating series by multiplying the one obtained by localization by a
generating series of framed invariants where some cycles are imposed to be
invertible. The latter is easy to compute and has a universal closed formula
for all toric quivers. Note that in this case the moduli space is the critical
locus of the potential of the quiver, so one does not need all the subtleties
of gluing, i.e. one only needs the proposition 3.3.

### Acknowledgements

I thank Boris Pioline and Olivier Schiffmann, my PhD supervisors, and David
Rydh, Benjamin Hennion, Davesh Maulik, Timo Richarz, Matthieu Romagny, Balazs
Szendroi and Richard Thomas for interesting discussions and comments on a
preliminary version of the paper.

##  2\. Classical Białynicki-Birula decomposition

###  2.1. Hyperbolic localization

Consider an algebraic space $S$, and $X$ be an algebraic space over $S$ with a
relative $\mathbb{C}^{\ast}$ over $S$.

######  Definition 2.1.

An algebraic space $X$ over an algebraic space $S$ with a relative
$\mathbb{C}^{\ast}$ over $S$ is said to be étale locally (resp. analytical
locally) linearizable if there is a $\mathbb{C}^{\ast}$-equivariant étale
(resp. analytical) covering family $\\{U_{i}\to X\\}_{i}$ where the $U_{i}$
are affine $S$-algebraic spaces with $\mathbb{C}^{\ast}$-action.

According to [1, Cor 20.2], an algebraic space $X$ is then étale locally
linearizable over $S$ when $X$ and $S$ are quasi-separated, and $X$ is locally
of finite type over $S$, and we expect similarly that the fact of being
analytic locally linearizable is not too restrictive. We will here adapt the
setting of [31] to the complex analytic case. Consider the following functors
on the category of $S$-algebraic spaces:

$\displaystyle X^{0}:$ $\displaystyle Y\mapsto{\rm Hom}_{S}^{T}(Y,X)$
$\displaystyle X^{+}:$ $\displaystyle Y\mapsto{\rm
Hom}_{S}^{T}((\mathcal{A}^{1}_{Y})^{+},X)$ (2.1) $\displaystyle X^{-}:$
$\displaystyle Y\mapsto{\rm Hom}_{S}^{T}((\mathcal{A}^{1}_{Y})^{-},X)$

where the superscript $\mathbb{C}^{\ast}$ denotes the
$\mathbb{C}^{\ast}$-equivariant morphism, and
$T,(\mathcal{A}^{1}_{Y})^{+},(\mathcal{A}^{1}_{Y})^{-}$ has the trivial, resp.
usual, resp. opposite $\mathbb{C}^{\ast}$-action.

The structure morphism $(\mathcal{A}^{1}_{S})^{\pm}\to S$ is
$\mathbb{C}^{\ast}$-equivariant and defines then a morphism:

(2.2) $\displaystyle\zeta^{\pm}:X^{0}\to X^{\pm}$

The zero section of $(\mathcal{A}^{1}_{S})^{\pm}\to S$ defines a morphism:

(2.3) $\displaystyle p^{\pm}:X^{\pm}\to X^{0}$

such that $p^{\pm}\circ\zeta^{\pm}={\rm Id}_{X^{0}}$. Finally, the unit
section of $(\mathcal{A}^{1}_{S})^{\pm}\to S$ defines a morphism:

(2.4) $\displaystyle\eta^{\pm}:X^{\pm}\to X$

such that $\xi:=\eta^{+}\circ\zeta^{+}=\eta^{-}\circ\zeta^{-}$ is the
inclusion of the subfunctor $X^{0}\subset X$. Hence it defines the hyperbolic
localization diagram as in [31]:

${X^{0}}$${X^{+}}$${X}$${X^{+}\times_{X_{0}}X^{-}}$${X^{0}}$${X^{-}}$${X^{0}}$$\scriptstyle{\eta^{+}}$$\scriptstyle{p^{+}}$$\scriptstyle{{}^{\prime}\eta^{-}}$$\scriptstyle{{}^{\prime}\eta^{+}}$$\scriptstyle{\zeta^{+}}$$\scriptstyle{\zeta^{-}}$$\scriptstyle{j}$$\scriptstyle{\eta^{-}}$$\scriptstyle{p^{-}}$

######  Proposition 2.2.

For $X$ an $S$ algebraic space with an étale (resp. analytical)-locally
linearizable $\mathbb{C}^{\ast}$-action, $X^{0}$ is representable by a closed
algebraic (resp. analytic) subspace of $X$, and $X^{\pm}$ are representable as
$X^{0}$-affine algebraic (resp. algebraic) spaces, and $j$ is open and closed.

Proof: The version for algebraic spaces is [31, Prop 1.17]. We give a similar
proof in the complex analytic space. The proof of the proposition when $X$ is
affine goes exactly the same as in [31, Lem 1.9], namely $X^{0},X^{+},X^{-}$
are represented by the closed subscheme defined by the ideal of homogeneous
elements of degree zero (resp. strictly negative, resp. strictly positive
positive degree), and $X^{0}=X^{+}\times_{X_{0}}X^{-}$.

Suppose now the $U\to X$ is the embedding of a $\mathbb{C}^{\ast}$-invariant
open subset of $X$ in the analytic topology. One can adapt [31, Lem 1.10]: for
$Y/S$ a complex analytic space, an element $\phi\in(U\times_{X}X_{0})(Y)$
corresponds to a commutative diagram:

${U}$${Y}$${X}$$\scriptstyle{\tilde{f}}$$\scriptstyle{f}$

and, because $U\to X$ is an open immersion, $\tilde{f}$ is
$\mathbb{C}^{\ast}$-equivariant, hence $\phi\in U^{0}(T)$. Then
$U^{0}=U\times_{X}X^{0}$ a functors, and then, taking a
$\mathbb{C}^{\ast}$-equivariant analytic covering family $\\{U_{i}\to
X\\}_{i}$ where the $U_{i}$ are $S$-affine, one obtains a Cartesian
commutative diagram:

${\bigsqcup_{i}U^{0}_{i}}$${X^{0}}$${\bigsqcup_{i}U_{i}}$${X}$

Because the top left element is representable, and the vertical left arrow is
a closed immersion, one obtains that $X^{0}$ is representable as a close
analytic subspace of $X$.

For $U\to X$ the embedding of a $\mathbb{C}^{\ast}$-invariant open subset of
$X$ in the analytic topology, one can adapt [31, Lem 1.11]: for $Y/S$ a
complex analytic space, an element $\phi\in(U^{0}\times_{X_{0}}X^{\pm})(Y)$
corresponds to a commutative diagram of $\mathbb{C}^{\ast}$-equivariant
morphisms:

${Y}$${U}$${(\mathbb{A}^{1}_{Y})^{\pm}}$${X}$$\scriptstyle{f|_{Y}}$$\scriptstyle{f}$

where $Y\to(\mathbb{A}^{1}_{Y})^{\pm}$ is the zero section. The open subset
$f^{-1}(U)$ contains $Y$ and is $\mathbb{C}^{\ast}$ invariant in
$\mathbb{A}^{1}_{Y}$, hence is the whole $\mathbb{A}^{1}_{Y}$, i.e. $\phi\in
U^{\pm}(Y)$. Hence $U^{\pm}=U^{0}\times_{X_{0}}X^{\pm}$ as functors, and
taking a $\mathbb{C}^{\ast}$-equivariant analytic covering family $\\{U_{i}\to
X\\}_{i}$ where the $U_{i}$ are $S$-affine, one obtains a Cartesian
commutative diagram:

${\bigsqcup_{i}U^{\pm}_{i}}$${X^{\pm}}$${\bigsqcup_{i}U_{i}^{0}}$${X^{0}}$

Because the top left element is representable, and the vertical left arrow is
affine, one obtains that $X^{0}$ is representable as an affine analytic space
on $X^{0}$. Moreover, $X^{0}=X^{+}\times_{X_{0}}X^{-}$ is an open and closed
immersion because it is the case analytical locally.$\Box$

There is a natural transformation
$\mathbb{D}(\eta^{-})_{!}(p^{-})^{\ast}\mathbb{D}\simeq(\eta^{-})_{\ast}(p^{-})^{!}\to(\eta^{+})_{!}(p^{+})^{\ast}$
built using the six-functor formalism between derived functors
$D^{b}_{c}(X)\to D^{b}_{c}(X^{0})$, as explained in [31, Section 2.2], and we
can define the same similar transformation at the level of mixed Hodge
modules. We change slightly the terminology of [31], replacing the term
’monodromic’ by ’equivariant’ to avoid future confusions with monodromic mixed
Hodge modules. Let $a,p:T\otimes_{S}X\to X$ denote the action (resp.
projection). We say that a constructible complex $A\in D^{b}_{c}(X)$ is
naively $\mathbb{C}^{\ast}$-equivariant if there exists an isomorphism
$a^{\ast}A\simeq p^{\ast}A$ in $D^{b}_{c}(T\otimes_{S})$. Let us define
$D^{b}_{c}(X)^{\mathbb{C}^{\ast}-eq}$ to be the full subcategory strongly
generated by naively $\mathbb{C}^{\ast}$-equivariant complexes, i.e. generated
by a finite iteration of taking the cone of a morphism in $D^{b}_{c}(X)$. The
objects in $D^{b}_{c}(X)^{\mathbb{C}^{\ast}-eq}$ are called
$\mathbb{C}^{\ast}$-equivariant. The result [31, Theo B] can then be rephrased
as:

######  Proposition 2.3.

For $X$ an algebraic space with $\mathbb{C}^{\ast}$ action, the natural
morphism of constructible complexes (or complexes of mixed Hodge modules)
$S_{X}(A):\mathbb{D}(\eta^{-}_{X})_{!}(p^{-}_{X})^{\ast}\mathbb{D}(A)\to(\eta^{+}_{X})_{!}(p^{+}_{X})^{\ast}(A)$
is an isomorphism when $A$ is a $\mathbb{C}^{\ast}$-equivariant constructible
complex.

where we have used the fact that a morphism of complexes of mixed Hodge
modules is an isomorphism if the underlying morphism of constructible is an
isomorphism.

###  2.2. Compatibility with smooth morphisms

The hyperbolic localization diagram is functorial, namely for $\phi:X\to Y$
there are $\mathbb{C}^{\ast}$-equivariant morphisms $\phi^{\pm}:X^{\pm}\to
Y^{\pm}$ and morphisms $\phi^{0}:X^{0}\to Y^{0}$, obtained by composition with
$\phi$ using the functor description, such that all the squares commutes in
the following diagram:

${X}$${X^{\pm}}$${X^{0}}$${Y}$${Y^{\pm}}$${Y^{0}}$$\scriptstyle{\phi}$$\scriptstyle{\eta_{X}^{\pm}}$$\scriptstyle{p_{X}^{\pm}}$$\scriptstyle{\phi^{\pm}}$$\scriptstyle{\zeta_{X}^{\pm}}$$\scriptstyle{\phi^{0}}$$\scriptstyle{\eta_{Y}^{\pm}}$$\scriptstyle{p_{Y}^{\pm}}$$\scriptstyle{\zeta_{Y}^{\pm}}$

######  Proposition 2.4.

For a smooth $\mathbb{C}^{\ast}$-equivariant morphism $\phi:X\to Y$,
$\phi^{\pm}$ and $\phi^{0}$ are also smooth, and the natural map $X^{\pm}\to
X^{0}\times_{Y^{0}}Y^{\pm}$ is an affine fibre bundle.

Proof: Consider first that $\phi$ is étale (resp. the embedding of an open
subset in the analytic topology): then [31, Lem 1.10, 1.11] (resp. our proof
of the representability of $X^{0},X^{\pm}$) shows respectively that
$X^{0}=X\times_{Y}Y^{0}$ and that $X^{\pm}=X^{0}\times_{Y^{0}}Y^{\pm}$: in
particular, $\phi^{0}$, and therefore $\phi^{\pm}$, are étale (resp. the
embedding of an open subset in the analytic topology) by base change.

Consider now the case of an affine $\mathbb{C}^{\ast}$-equivariant fibration
$\phi:Y\times_{S}\mathbb{A}^{d}_{S}\to Y$ where the fiber has a linear
$\mathbb{C}^{\ast}$ action. Then
$\phi^{0}:Y^{0}\times_{S}(\mathbb{A}^{d}_{S})^{0}\to Y^{0}$ and
$\phi^{\pm}:Y^{\pm}\times_{S}(\mathbb{A}^{d}_{S})^{\pm}\to Y^{\pm}$ are
smooth, and:

$\displaystyle(Y\times_{S}\mathbb{A}^{d}_{S})^{\pm}=Y^{\pm}\times_{S}(\mathbb{A}^{d}_{S})^{\pm}$
(2.5) $\displaystyle\to$
$\displaystyle(Y\times_{S}\mathbb{A}^{d}_{S})^{0}\times_{Y^{0}}Y^{\pm}=Y^{\pm}\times_{S}(\mathbb{A}^{d}_{S})^{0}$

is affine.

Suppose now that $\phi$ is smooth. For $x\in X^{0}$, we can restricts to a
$\mathbb{C}^{\ast}$-invariant étale (resp. open in the analytic topology)
neighborhood of $\phi(x)$ in which $Y$ is affine, and further restricts to a
$\mathbb{C}^{\ast}$-invariant étale (resp. open in the analytic topology)
neighborhood of $x$ in $X$ which is affine. As argued in the proof of
proposition 2.43 of [21, Prop 2.43], because $\mathbb{C}^{\ast}$ is a torus,
an affine $\mathbb{C}^{\ast}$-equivariant space can be written as
$Spec(R[x_{1},...,x_{n}]/(h_{1},...,h_{r}))$ where the $x_{i}$ are
$\mathbb{C}^{\ast}$-equivariant coordinates, and the $h_{i}$ are
$\mathbb{C}^{\ast}$-equivariant polynomial. Then the
$\mathbb{C}^{\ast}$-equivariant smooth map $\phi:X\to Y$ is étale locally
given by the dual of the map of rings:

(2.6)
$\displaystyle\tilde{R}[x_{1},...,x_{n}]/(h_{1},...,h_{r})\to\tilde{R}[y_{1},...,y_{m}]/(k_{1},...,k_{s})$

where we replace the ring of polynomial in $n$ variables by the ring of
holomorphic functions in $n$ variables i the analytic setting, which can be
rewritten:

(2.7) $\displaystyle R\to R[y_{1},...,y_{m}]/(l_{1},...,l_{r+s})$

where $R:=\tilde{R}[x_{1},...,x_{n}]/(h_{1},...,h_{r})$ and the $l_{i}$ are
the $\mathbb{C}^{\ast}$-equivariant polynomials (resp. holomorphic functions)
$k_{1},...,k_{r},x_{1}-\phi^{\prime}(x_{1}),...,x_{n}-\phi^{\prime}(x_{n})$.
Denote by $c$ the rank of the matrix $(\frac{\partial l_{i}}{\partial
y_{j}}|_{x})_{ij}$ in the neighborhood of $x$, and up to reordering suppose
that $(\frac{\partial l_{i}}{\partial y_{j}}|_{x})_{0\leq i,j\leq c}$ is
invertible at $x$, hence in the neighborhood of $x$ it is still invertible,
i.e. $R[y_{c+1},...,y_{m}]\to R[y_{1},...,y_{m}]/(l_{1},...,l_{r+s})$ is étale
(resp. a local homeomorphism) and $R[y_{1},...,y_{m}]/(l_{1},...,l_{c})\to
R[y_{1},...,y_{m}]/(l_{1},...,l_{r+s})$ is a local homeomorphism, hence $\phi$
can be étale locally written as a composition of
$\mathbb{C}^{\ast}$-equivariant morphisms:

(2.8) $\displaystyle R\to R[y_{c+1},...,y_{m}]\to
S[y_{1},...,y_{m}]/(l_{1},...,l_{r+s})$

where the second map is étale (resp. a local homeomorphism). Because $X^{0}\to
Y^{0}$, $X^{\pm}\to Y^{\pm}$ and $X^{\pm}\to X^{0}\times_{Y^{0}}Y^{\pm}$ are
smooth for $X\to Y$ $\mathbb{C}^{\ast}$-equivariant and étale (resp. open
embeddings in the analytic topology) and for $X\to Y$ affine fibrations with
$\mathbb{C}^{\ast}$-linear fibers, then it is true for smooth maps because the
property of being smooth can be checked étale locally.

In order to show that the smooth morphism $q:X^{\pm}\to
X^{0}\times_{Y^{0}}Y^{\pm}$ is an affine fibre bundle, one can take
$X=X^{\pm}$, $Y=Y^{\pm}$. We provide here a relative version of the proof of
[19, Lem 7.2]. Consider the sheaf of ring $\mathcal{B}_{0}$ of $Y^{0}$ (resp.
$\mathcal{C}_{0}$ of $X^{0}$). Because $Y^{\pm}\to Y$ (resp. $X^{\pm}\to X$)
is affine, it corresponds to a sheaf of $\mathbb{Z}$-graded
$\mathcal{B}_{0}$\- (resp. $\mathcal{C}_{0})$-) algebra
$\mathcal{B}=\bigoplus_{i\geq 0}\mathcal{B}_{i}$ (resp.
$\mathcal{C}=\bigoplus_{i\geq 0}\mathcal{C}_{i}$). The morphism $q$ is dual to
$\mathcal{B}\otimes_{\mathcal{B}_{0}}\mathcal{C}_{0}\to\mathcal{C}$. Consider
a point $z\in\mathcal{B}\otimes_{\mathcal{B}_{0}}\mathcal{C}_{0}$ with residue
field $\kappa(x)$, and consider a minimal system $f_{1},...,f_{r}$ of
homogeneous generators of the ideal
$\mathcal{C}_{>0}/(\mathcal{B}_{>0})\otimes\kappa(z)$ of
$\mathcal{C}/(\mathcal{B}_{>0})\otimes\kappa(z)$. Shrink
$X^{0}\times_{Y^{0}}Y^{\pm}$ to an affine neighborhood $z$ so that those
generators lift to generators $F_{1},...,F_{r}$ of
$\mathcal{C}_{>0}/(\mathcal{B}_{>0})$. On the one hand, we have a natural
morphism
$p:\mathcal{B}\otimes_{\mathcal{B}_{0}}\mathcal{C}_{0}[F_{1},...,F-r]\to\mathcal{C}$.
Since $F_{1},...,F_{r}$ are homogeneous and generate the ideal
$\mathcal{C}/(\mathcal{B}_{>0})$, the morphism $p$ is surjective by induction
on the degrees.

On the other hand, note that $f_{1},...,f_{r}$ are linearly independent in the
relative cotangent space of $q$. As the fibers are smooth, we have
$r\leq\dim(q^{-1}(z))=\dim(Spec(\mathcal{C}\otimes\kappa(z)))$. Since $q$ is
flat, we have:

(2.9)
$\displaystyle\mathcal{C}\otimes\kappa(z)=(\mathcal{B}\otimes_{\mathcal{B}_{0}}\mathcal{C}_{0}[F_{1},...,F-r]/\ker(p))\otimes\kappa(z)=\frac{\kappa(z)[F_{1},...,F_{r}]}{\ker(p)\otimes\kappa(z)}$

The right hand side has dimension $r$ only if $\ker(p)\otimes\kappa(z)=0$.
Thus (after possibly shrinking $X^{0}\times_{Y^{0}}Y^{\pm}$ again), we have
$\ker(p)=0$ and
$\mathcal{C}\simeq\mathcal{B}\otimes_{\mathcal{B}_{0}}\mathcal{C}_{0}[F_{1},...,F-r]$,
so $X^{\pm}=\mathbb{A}^{r}\times X^{0}\times_{Y^{0}}Y^{\pm}$ locally on
$X^{0}\times_{Y^{0}}Y^{\pm}$, hence $X^{\pm}\to X^{0}\times_{Y^{0}}Y^{\pm}$ is
an affine fibre bundle.$\Box$

Denote now by $\Pi$ the set of cocharacters of $\mathbb{C}^{\ast}$, and for a
given cocharacter $\pi\in\Pi$, by $d^{+}_{\pi},d^{0}_{\pi},d^{-}_{\pi}$
respectively the number of $\mathbb{C}^{\ast}$-contracting, resp.
$\mathbb{C}^{\ast}$-invariants, resp. $\mathbb{C}^{\ast}$-repelling weights,
and by ${\rm Ind}_{\pi}=d^{+}_{\pi}-d^{-}_{\pi}$. For a smooth
$\mathbb{C}^{\ast}$-equivariant morphism $\phi:X\to Y$, denote by
$\phi^{0}_{\pi}$ the morphism $\phi^{0}$ restricted to the locus of $X^{0}$
where the $\mathbb{C}^{\ast}$-action on $X$ induces an action of cocharacter
$\pi$ on $T_{X/Y}$ (in particular it is smooth of relative dimension
$d^{0}_{\pi}$). It induces a decomposition into connected components, hence
$\phi^{0}=\bigsqcup_{\pi\in\Pi}\phi^{0}_{\pi}$, and similarly
$\phi^{+}=\bigsqcup_{\pi\in\Pi}\phi^{+}_{\pi}$.

######  Proposition 2.5.

For a smooth $\mathbb{C}^{\ast}$-equivariant morphism $\phi:X\to Y$ of
relative dimension $d$ there are natural isomorphisms in the six-functor
formalism for constructible complexes:

(2.10)
$\displaystyle(p_{X}^{\pm})_{!}(\eta_{X}^{\pm})^{\ast}\phi^{\ast}[d]\simeq\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}(\phi^{0}_{\pi})^{\ast}[d^{0}_{\pi}](p_{Y}^{\pm})_{!}(\eta_{Y}^{\pm})^{\ast}$

moreover, they commute with the duality $S_{X}$ and $S_{Y}$ in an obvious way.

In particular if $\phi$ is étale (resp. a local biholomorphism), $T_{X/Y}$ is
trivial, hence one has a natural isomorphism:

(2.11)
$\displaystyle(p_{X}^{\pm})_{!}(\eta_{X}^{\pm})^{\ast}\phi^{\ast}[d]\simeq(\phi^{0})^{\ast}[d](p_{Y}^{\pm})_{!}(\eta_{Y}^{\pm})^{\ast}$

Proof: Consider the following commutative diagram:

${X}$${X^{\pm}}$${X^{0}}$${Y^{\pm}\times_{Y^{0}}X^{0}}$${Y}$${Y^{\pm}}$${Y^{0}}$$\scriptstyle{\phi}$$\scriptstyle{\eta_{X}^{\pm}}$$\scriptstyle{p_{X}^{\pm}}$$\scriptstyle{\phi^{\pm}}$$\scriptstyle{f}$$\scriptstyle{\phi^{0}}$$\scriptstyle{\tilde{\phi}^{\pm}}$$\scriptstyle{\tilde{p}_{X}^{\pm}}$$\scriptstyle{\eta_{Y}^{\pm}}$$\scriptstyle{p_{Y}^{\pm}}$

One has the following sequence of isomorphisms:

$\displaystyle(p_{X}^{\pm})_{!}(\eta_{X}^{\pm})^{\ast}\phi^{\ast}[d]\simeq$
$\displaystyle(p_{X}^{\pm})_{!}(\phi^{\pm})^{\ast}[d](\eta_{Y}^{\pm})^{\ast}$
(2.12) $\displaystyle\simeq$
$\displaystyle(\tilde{p}_{X}^{\pm})_{!}f_{!}f^{\ast}(\tilde{\phi}^{\pm})^{\ast}[d](\eta_{Y}^{\pm})^{\ast}$

where we have used the commutativity of the left square in the first line, and
the commutativity of the two small triangles in the second line. The right
square of the above diagram admits an decomposition as a disjoint union of
those diagrams:

${X^{\pm}_{\pi}}$${X^{0}_{\pi}}$${Y^{\pm}\times_{Y^{0}}X^{0}_{\pi}}$${Y^{\pm}}$${Y^{0}}$$\scriptstyle{p_{X,\pi}^{\pm}}$$\scriptstyle{\phi^{\pm}_{\pi}}$$\scriptstyle{f_{\pi}}$$\scriptstyle{\phi^{0}_{\pi}}$$\scriptstyle{\tilde{\phi}^{\pm}_{\pi}}$$\scriptstyle{\tilde{p}_{X,\pi}^{\pm}}$$\scriptstyle{p_{Y}^{\pm}}$

hence one obtains:

$\displaystyle(\tilde{p}_{X,\pi}^{\pm})_{!}(f_{\pi})_{!}(f_{\pi})^{\ast}(\tilde{\phi}^{\pm}_{\pi})^{\ast}[d](\eta_{Y}^{\pm})^{\ast}\simeq$
$\displaystyle(\tilde{p}_{X,\pi}^{\pm})_{!}[-2d^{\pm}_{\pi}](\tilde{\phi}^{\pm}_{\pi})^{\ast}[d^{+}_{\pi}+d^{0}_{\pi}+d^{-}_{\pi}](\eta_{Y}^{\pm})^{\ast}$
(2.13) $\displaystyle\simeq$ $\displaystyle\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\phi^{0}_{\pi}[d^{0}_{\pi}](p^{\pm}_{Y})_{!}(\eta_{Y}^{\pm})^{\ast}$

where we have used the fact that $f_{\pi}$ is an affine fibration with
$d^{\pm}_{\pi}$-dimensional fiber in the first line and the base change in the
down right Cartesian triangle in the second line. Summing on the connected
components decomposition, one obtains a natural isomorphism in the six-functor
formalism:

(2.14)
$\displaystyle(p_{X}^{\pm})_{!}(\eta_{X}^{\pm})^{\ast}\phi^{\ast}[d]\simeq\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}(\phi^{0})^{\ast}[d^{0}_{\pi}](p_{Y}^{\pm})_{!}(\eta_{Y}^{\pm})^{\ast}$

$\Box$

Now considering $U$ a smooth $S$-algebraic space with
$\mathbb{C}^{\ast}$-action, one can consider the smooth
$\mathbb{C}^{\ast}$-equivariant morphism $\phi:U\to S$, where $S$ has the
trivial $\mathbb{C}^{\ast}$-action. The hyperbolic localization functor is
then trivial on $S$, and:

(2.15) $\displaystyle\mathcal{IC}_{U}$
$\displaystyle=\phi^{\ast}[d]\mathcal{IC}_{S}=\phi^{!}[-d]\mathcal{IC}_{S}$

in particular $\mathcal{IC}_{S}$ is $\mathbb{C}^{\ast}$-equivariant, hence
$\mathcal{IC}_{U}$ is also $\mathbb{C}^{\ast}$-equivariant, and self-Verdier
dual, hence the natural arrow
$\mathbb{D}(p_{U}^{-})_{!}(\eta_{U}^{-})^{\ast}\mathcal{IC}_{U}\to(p_{X}^{+})_{!}(\eta_{X}^{+})^{\ast}\mathcal{IC}_{U}$
is an isomorphism. Moreover, there is then an open and closed decomposition
$U^{0}=\bigsqcup_{\pi\in\Pi}U^{0}_{\pi}$, and:

(2.16) $\displaystyle\mathcal{IC}_{U^{0}_{\pi}}$
$\displaystyle=(\phi^{0}_{\pi})^{\ast}[d^{0}_{\pi}]\mathcal{IC}_{S}=(\phi^{0}_{\pi})^{!}[-d^{0}_{\pi}]\mathcal{IC}_{S}$

hence an absolute version of the last proposition gives:

######  Corollary 2.6.

For a smooth $\mathbb{C}^{\ast}$-equivariant $S$-algebraic (resp. analytic)
space $U$ there are natural isomorphisms in the six-functor formalism such
that the following diagram commutes:

${\mathbb{D}(p_{U}^{-})_{!}(\eta_{U}^{-})^{\ast}\mathcal{IC}_{U}}$${\bigoplus_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}\mathcal{IC}_{U^{0}_{\pi}}}$${(p_{U}^{+})_{!}(\eta_{U}^{+})^{\ast}\mathcal{IC}_{U}}$$\scriptstyle{S_{U}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

##  3\. Critical Białynicki-Birula decomposition

In this section and in the rest of the papers, all the algebraic spaces will
be assumed to be quasi-separated and locally of finite type over $\mathbb{C}$:
in particular, all the $\mathbb{C}^{\ast}$-actions will then be étale locally
linearizable, the proper pullback and proper pushout of any functor, and the
hyperbolic localization diagram, will be defined.

###  3.1. The functor of vanishing cycles

Consider an algebraic space $U$ with a regular function $f:U\to\mathbb{C}$. We
give a definition of the functor of vanishing cycles $\phi_{f}$, which is
equivalent to those exposed in definition 2.10 of [10]. Consider the following
commutative diagram of complex analytic spaces, where $U^{an}$ is the
analytification of $U$, and the square are Cartesian:

${U^{an}_{0}}$${U^{an}}$${\tilde{U}^{an}_{\theta}}$${\\{0\\}}$${\mathbb{C}}$${e^{i\theta}\\{v\in\mathbb{C}|\Re(v)\leq
0\\}}$$\scriptstyle{\iota}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{j_{\theta}}$$\scriptstyle{f}$

We define the functors:

(3.1)
$\displaystyle\phi_{f}^{\theta}:=\iota^{\ast}(j_{\theta})_{!}(j_{\theta})^{!}:D(X)\to
D(X_{0})$

There are natural isomorphisms
$\tilde{T}^{\theta}_{f}:\phi^{0}_{f}\to\phi^{\theta}_{f}$ induced by rotating
$\mathbb{C}$ counterclockwise by an angle $\theta$ around the origin. One
defines the vanishing cycles functor $\phi_{f}:=\phi^{0}_{f}$, and the natural
isomorphism $\tilde{T}^{2\pi}_{f}:\phi_{f}\to\phi_{f}$ is the monodromy of
vanishing cycle. Because $\phi_{f}$ is constructed using maps of analytic
spaces and not maps of $\mathbb{C}$-schemes, it is not direct that they maps
constructible complex of $U$ to constructible complex of $U_{0}$, but it is
the case, as explained in definition 2.10 of [10], i.e. it defines a functor
$\phi_{f}:D^{b}_{c}(U)\to D^{b}_{c}(U_{0})$, and it even defines functors
$\phi_{f}:{\rm Perv}(U)\to{\rm Perv}(U_{0})$.

Massey defines in [25] a natural isomorphism giving the commutation of the
vanishing cycles functor (considered as a functor $\phi_{f}:D^{b}_{c}(U)\to
D^{b}_{c}(U_{0})$) with the Verdier duality
$\mathbb{D}\phi_{f}\simeq\phi_{f}\mathbb{D}$. Consider the Cartesian diagram
of closed subspaces:

${U^{an}_{0}}$${\tilde{U}^{an}_{0}\cap\tilde{U}^{an}_{\pi}}$${\tilde{U}^{an}_{0}}$${\tilde{U}^{an}_{\pi}}$${U}$$\scriptstyle{q}$$\scriptstyle{\hat{j}_{\pi}}$$\scriptstyle{\hat{j}_{0}}$$\scriptstyle{m}$$\scriptstyle{j_{0}}$$\scriptstyle{j_{\pi}}$

Then Massey defines the following sequence of isomorphisms:

$\displaystyle\mathbb{D}\phi_{f}\simeq$
$\displaystyle\mathbb{D}q^{\ast}m^{\ast}(j_{0})_{!}(j_{0})^{!}$
$\displaystyle\simeq$
$\displaystyle\mathbb{D}q^{!}m^{\ast}(j_{0})_{!}(j_{0})^{!}$
$\displaystyle\simeq$
$\displaystyle\mathbb{D}q^{!}(\hat{j}_{0})^{!}(j_{\pi})^{\ast}$
$\displaystyle\simeq$ $\displaystyle
q^{\ast}(\hat{j}_{0})^{\ast}(j_{\pi})^{!}\mathbb{D}$ $\displaystyle\simeq$
$\displaystyle q^{\ast}m^{\ast}(j_{\pi})_{!}(j_{\pi})^{!}\mathbb{D}$ (3.2)
$\displaystyle\simeq$ $\displaystyle\phi^{\pi}_{f}\mathbb{D}$

where in the first and last line we have used the fact that $\iota=m\circ q$,
in the second line [25, Lem 2.1] which proves that the natural morphism
$q^{!}m^{\ast}(j_{0})_{!}(j_{0})^{!}\to
q^{\ast}m^{\ast}(j_{0})_{!}(j_{0})^{!}$ is an isomorphism and in the third and
fifth line the isomorphisms built in [25, Lem 2.2] using the six-functor
formalism in the Cartesian square of closed subspace of $X$, Using the fact
that $U^{an}=\tilde{U}^{an}_{0}\cup\tilde{U}^{an}_{\pi}$. The Massey
isomorphism is then defined by the composition:

${\mathbb{D}\phi_{f}}$${\phi^{\pi}_{f}\mathbb{D}}$${\phi_{f}\mathbb{D}}$$\scriptstyle{\simeq}$$\scriptstyle{(\tilde{T}_{f}^{\pi})^{-1}}$

and it commutes with the monodromy operator $\tilde{T}^{2\pi}_{f}$.

For two algebraic spaces $X,Y$ with projections $\pi_{1}:X\times Y\to
X,\pi_{2}:X\times Y\to Y$, for $F\in D^{b}_{c}(X),G\in D^{b}_{c}(Y)$, one
denotes:

(3.3) $\displaystyle F\boxtimes G:=\pi_{1}^{\ast}(F)\otimes\pi_{2}^{\ast}G$

For $f;X\to\mathbb{C},g:Y\to\mathbb{C}$, denote by $f\boxplus
g:=f\circ\pi_{1}+g\circ\pi_{2}:X\times Y\to\mathbb{C}$. For $j:Y\to X$ a
closed embedding, denote $R\Gamma_{Y}:=j_{!}j^{!}:D^{b}_{c}(X)\to
D^{b}_{c}(X)$.

Consider two algebraic spaces $U,V$ with a regular function
$f:U\to\mathbb{C},g:V\to\mathbb{C}$, denote by $k:U_{0}\times V_{0}\to(U\times
V)_{0}$ the embedding into the zero locus of $f\boxplus g$. Massey defines in
[24] a natural Thom-Sebastiani isomorphism:

(3.4) $\displaystyle\phi_{f}\boxtimes\phi_{g}\simeq k^{\ast}\phi_{f\boxplus
g}$

denote by $\iota:U_{0}\to U,\jmath:V_{0}\to V,q:(U\times V)_{0}\to U\times V$
the closed embeddings of the zero set of $f,g,f\boxplus g$. One has the closed
embedding $\tilde{U}_{0}\times\tilde{V}_{0}\to(\tilde{U\times V})_{0}$, and
[24, lem 1.2] shows that the natural arrow of the six-functor formalism
$R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}\to R\Gamma_{(\tilde{U\times
V})_{0}}$ is an isomorphism. The Thom-Sebastiani isomorphism is then defined
by the composition of natural isomorphisms of the six-functor formalism:

$\displaystyle\phi_{f}\boxtimes\phi_{g}=$
$\displaystyle\iota^{\ast}R\Gamma_{\tilde{U}_{0}}\boxtimes\jmath^{\ast}R\Gamma_{\tilde{V}_{0}}$
$\displaystyle\simeq$
$\displaystyle(\iota\times\jmath)^{\ast}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}$
$\displaystyle\overset{\simeq}{\to}$ $\displaystyle
k^{\ast}q^{\ast}R\Gamma_{\tilde{U\times V}_{0}}$ (3.5) $\displaystyle=$
$\displaystyle k^{\ast}\phi_{f\boxplus g}$

it is shown in [24] that this isomorphism commutes with monodromy and it can
be shown that it commutes with duality.

There is a theory of vanishing cycles functor for mixed Hodge modules, which
projects under $rat$ to the above theory for perverse sheaves, as exposed in
[10, sec 2.10]. There is a vanishing cycle functor:

(3.6) $\displaystyle\phi^{H}_{f}:{\rm MHM}(U)\to{\rm MMHM}(U_{0})$

It sends a mixed Hodge module on $U$ to a monodromic mixed Hodge module on
$U_{0}$, i.e. a mixed Hodge module with commuting actions of a unipotent
operator $T_{s}$ and a nilpotent operator $N$, giving respectively the
semisimple part and the logarithm of the unipotent part of the monodromy
operator. There is also a self duality isomorphism and a Thom-Sebastiani
isomorphism, and it is checked in [10, appendix A] that $rat$ sends them to
the corresponding isomorphisms of perverse sheaves. When a mixed Hodge module
$M\in{\rm MHM}(U)$ is polarized, i.e. is provided with an isomorphism
$\sigma:\mathbb{D}M\simeq M$, the self-duality isomorphism of $\phi^{H}_{f}$
provides them $\phi^{H}_{f}M$ with a strong polarization, i.e. an isomorphism
with its dual commuting with monodromy.

###  3.2. Functoriality of the vanishing cycles

The vanishing cycles functor has some functoriality properties. Consider a map
of algebraic spaces $\Phi:U\to V$, and a regular functions $f:V\to\mathbb{C}$,
$f=g\circ\Phi:U\to\mathbb{C}$, and denote by $\Phi_{0}:U_{0}\to V_{0}$ the
induced map on the zero locus.

######  Proposition 3.1.

There are natural morphisms built using the six-functor formalism for
constructible complexes or complexes of mixed Hodge modules:

$\displaystyle(\Phi_{0})^{\ast}\phi_{g}^{\theta}$
$\displaystyle\to\phi_{f}^{\theta}\Phi^{\ast}$ (3.7)
$\displaystyle(\Phi_{0})_{!}\phi_{f}^{\theta}$
$\displaystyle\to\phi_{g}^{\theta}\Phi_{!}$

They are compatible with monodromy and Thom-Sebastiani isomorphism, and are
compatible with composition and base change of $\Phi$.

Proof: We consider then the commutative diagram, where all the squares are
Cartesian:

${U^{an}_{0}}$${U^{an}}$${\tilde{U}^{an}_{\theta}}$${V^{an}_{0}}$${V^{an}}$${\tilde{V}^{an}_{\theta}}$${\\{0\\}}$${\mathbb{C}}$${e^{i\theta}\\{v\in\mathbb{C}|\Re(v)\leq
0\\}}$$\scriptstyle{\iota_{U}}$$\scriptstyle{\Phi_{0}}$$\scriptstyle{\Phi}$$\scriptstyle{j_{\theta,U}}$$\scriptstyle{\tilde{\Phi}_{\theta}}$$\scriptstyle{\iota_{V}}$$\scriptstyle{g}$$\scriptstyle{g}$$\scriptstyle{j_{\theta,V}}$$\scriptstyle{g}$

In the following of the proof, the superscript $"an"$ will be implicit for
readability. Then one has the following sequence of natural morphisms of the
six-functor formalism:

$\displaystyle(\Phi_{0})^{\ast}\phi_{g}^{\theta}=$
$\displaystyle\Phi_{0}^{\ast}(\iota_{V})^{\ast}(j_{\theta,V})_{!}(j_{\theta,V})^{!}$
$\displaystyle\simeq$
$\displaystyle(\iota_{U})^{\ast}\Phi^{\ast}(j_{\theta,V})_{!}(j_{\theta,V})^{!}$
$\displaystyle\simeq$
$\displaystyle(\iota_{U})^{\ast}(j_{\theta,U})_{!}\tilde{\Phi}^{\ast}(j_{\theta,V})^{!}$
$\displaystyle\to$
$\displaystyle(\iota_{U})^{\ast}(j_{\theta,U})_{!}(j_{\theta,U})^{!}\Phi^{\ast}$
(3.8) $\displaystyle=$ $\displaystyle\phi_{f}^{\theta}\Phi^{\ast}$

where the first and last lines are the definitions, the second line follows
from the commutativity of the left square, the third from the Cartesianity of
the right square, and then fourth from the commutativity of the right square.

$\displaystyle(\Phi_{0})_{!}\phi_{f}^{\theta}=$
$\displaystyle(\Phi_{0})_{!}(\iota_{U})^{\ast}(j_{\theta,U})_{!}(j_{\theta,U})^{!}$
$\displaystyle\simeq$
$\displaystyle(\iota_{V})^{\ast}\Phi_{!}(j_{\theta,U})_{!}(j_{\theta,U})^{!}$
$\displaystyle\simeq$ $\displaystyle(\iota_{V})^{\ast}(j_{\theta,VU})^{!}$
$\displaystyle\to$
$\displaystyle(\iota_{V})^{\ast}(j_{\theta,V})_{!}(j_{\theta,V})^{!}\Phi_{!}$
(3.9) $\displaystyle=$ $\displaystyle\phi_{g}^{\theta}\Phi_{!}$

where the first and last lines are the definitions, the second line follows
from the Cartesianity of the left square, the third from the commutativity of
the right square, and then fourth from the fact that the right square is
Cartesian.

Because $f=g\circ\Phi$ these morphisms are compatible with the isomorphisms
$\tilde{T}^{\theta}$, namely the following squares are commutative:

${(\Phi_{0})^{\ast}\phi_{g}}$${\phi_{f}\Phi^{\ast}}$${(\Phi_{0})_{!}\phi_{f}}$${\phi_{g}\Phi^{\ast}}$${(\Phi_{0})^{\ast}\phi_{g}^{\theta}}$${\phi_{f}^{\theta}\Phi^{\ast}}$${(\Phi_{0})_{!}\phi_{f}^{\theta}}$${\phi_{g}^{\theta}\Phi_{!}}$$\scriptstyle{(\Phi_{0})^{\ast}\tilde{T}^{\theta}_{g}}$$\scriptstyle{\tilde{T}^{\theta}_{f}\Phi^{\ast}}$$\scriptstyle{(\Phi_{0})_{!}\tilde{T}^{\theta}_{f}}$$\scriptstyle{\tilde{T}^{\theta}_{g}\Phi_{!}}$

in particular these morphisms are compatible with the monodromy.

Consider now $U,U^{\prime},V,V^{\prime}$ algebraic spaces with regular
functions to $\mathbb{C}$ $f,f^{\prime},g,g^{\prime}$ and maps $\Phi:U\to
U^{\prime}$, $\Psi:V\to V^{\prime}$. Considering the Cartesian diagram, where
the horizontal maps are closed embeddings:

${U\times V}$${(\tilde{U\times
V})_{0}}$${\tilde{U}_{0}\times\tilde{V}_{0}}$${U^{\prime}\times
V^{\prime}}$${(\tilde{U^{\prime}\times
V^{\prime}})_{0}}$${\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}$$\scriptstyle{\Phi\times\Psi}$$\scriptstyle{\Phi\times\Psi}$$\scriptstyle{\Phi\times\Psi}$

One has then commutative squares of morphisms from the six-functor formalism:

${(\Phi\times\Psi)^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}}$${R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}(\Phi\times\Psi)^{\ast}}$${(\Phi\times\Psi)_{!}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}}$${R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}(\Phi\times\Psi)_{!}}$${(\Phi\times\Psi)^{\ast}R\Gamma_{(\tilde{U^{\prime}\times
V^{\prime}})_{0}}}$${R\Gamma_{(\tilde{U\times
V})_{0}}(\Phi\times\Psi)^{\ast}}$${(\Phi\times\Psi)_{!}R\Gamma_{(\tilde{U\times
V})_{0}}}$${R\Gamma_{(\tilde{U^{\prime}\times
V^{\prime}})_{0}}(\Phi\times\Psi)_{!}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Consider now the Cartesian diagram:

${U_{0}\times V_{0}}$${(U\times V)_{0}}$${U\times V}$${U^{\prime}_{0}\times
V^{\prime}_{0}}$${(U^{\prime}\times V^{\prime})_{0}}$${U^{\prime}\times
V^{\prime}}$$\scriptstyle{k}$$\scriptstyle{\Phi_{0}\times\Psi_{0}}$$\scriptstyle{q}$$\scriptstyle{(\Phi\times\Psi)_{0}}$$\scriptstyle{\Phi\times\Psi}$$\scriptstyle{k}$$\scriptstyle{q}$

One has then commutative squares of morphisms from the six-functor formalism:

${(\Phi_{0}\times\Psi_{0})^{\ast}(\iota^{\prime}\times\jmath^{\prime})^{\ast}}$${(\iota\times\jmath)^{\ast}(\Phi\times\Psi)^{\ast}}$${(\Phi_{0}\times\Psi_{0})_{!}(\iota\times\jmath)^{\ast}}$${(\iota^{\prime}\times\jmath^{\prime})^{\ast}(\Phi\times\Psi)_{!}}$${k^{\ast}((\Phi\times\Psi)_{0})^{\ast}(q^{\prime})^{\ast}}$${(\iota\times\jmath)^{\ast}(\Phi\times\Psi)^{\ast}}$${k^{\ast}((\Phi\times\Psi)_{0})_{!}q^{\ast}}$${(\iota^{\prime}\times\jmath^{\prime})^{\ast}(\Phi\times\Psi)_{!}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

And one obtains then finally commutative square of morphisms:

${(\Phi_{0}\times\Psi_{0})^{\ast}(\iota^{\prime}\times\jmath^{\prime})^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}}$${(\iota\times\jmath)^{\ast}(\Phi\times\Psi)^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}}$${(\iota\times\jmath)^{\ast}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}(\Phi\times\Psi)^{\ast}}$${k^{\ast}(\Phi\times\Psi)_{0}^{\ast}(q^{\prime})^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}}$${(\iota\times\jmath)^{\ast}(\Phi\times\Psi)^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}}$${(\iota\times\jmath)^{\ast}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}(\Phi\times\Psi)^{\ast}}$${k^{\ast}(\Phi\times\Psi)_{0}^{\ast}(q^{\prime})^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}}$${(\iota\times\jmath)^{\ast}(\Phi\times\Psi)^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}}$${(\iota\times\jmath)^{\ast}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}(\Phi\times\Psi)^{\ast}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

And similarly:

${(\Phi_{0}\times\Psi_{0})_{!}(\iota\times\jmath)^{\ast}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}}$${(\iota^{\prime}\times\jmath^{\prime})^{\ast}(\Phi\times\Psi)_{!}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}}$${(\iota^{\prime}\times\jmath^{\prime})^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}(\Phi\times\Psi)_{!}}$${(k^{\prime})^{\ast}((\Phi\times\Psi)_{0})_{!}q^{\ast}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}}$${(\iota^{\prime}\times\jmath^{\prime})^{\ast}(\Phi\times\Psi)_{!}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}}$${(\iota^{\prime}\times\jmath^{\prime})^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}(\Phi\times\Psi)_{!}}$${(k^{\prime})^{\ast}((\Phi\times\Psi)_{0})_{!}q^{\ast}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}}$${(\iota^{\prime}\times\jmath^{\prime})^{\ast}(\Phi\times\Psi)_{!}R\Gamma_{\tilde{U}_{0}\times\tilde{V}_{0}}}$${(\iota^{\prime}\times\jmath^{\prime})^{\ast}R\Gamma_{\tilde{U^{\prime}}_{0}\times\tilde{V^{\prime}}_{0}}(\Phi\times\Psi)_{!}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Hence the following diagrams are commutative:

${(\Phi_{0}\times\Psi_{0})^{\ast}(\phi_{f^{\prime}}\boxtimes\phi_{g^{\prime}})}$${(\phi_{f}\boxtimes\phi_{g})(\phi\times\Psi)^{\ast}}$${(\Phi_{0}\times\Psi_{0})_{!}(\phi_{f}\boxtimes\phi_{g})}$${k^{\ast}(\phi_{f^{\prime}}\boxtimes\phi_{g^{\prime}})(\phi\times\Psi)_{!}}$${k^{\ast}((\Phi\times\Psi)_{0})^{\ast}\phi_{f^{\prime}\boxplus
g^{\prime}}}$${\phi_{f\boxplus
g}(\Phi\times\Psi)^{\ast}}$${(k^{\prime})^{\ast}((\Phi\times\Psi)_{0})_{!}\phi_{f\boxplus
g}}$${k^{\ast}\phi_{f^{\prime}\boxplus
g^{\prime}}(\Phi\times\Psi)_{!}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

i.e. the morphisms of the proposition are compatible with the Thom-Sebastiani
isomorphism.

Consider now $U,V,W$ algebraic spaces, with three regular maps to
$\mathbb{C}$, and maps $\Phi:U\to V,\Psi:V\to W$ such that
$f=g\circ\Phi,g=h\circ\Psi$. Considering the naturality of the six-functor
formalism in the Cartesian diagram:

${U_{0}}$${U}$${\tilde{U}_{\theta}}$${V_{0}}$${V}$${\tilde{V}_{\theta}}$${W_{0}}$${W}$${\tilde{W}_{\theta}}$${\\{0\\}}$${\mathbb{C}}$${e^{i\theta}\\{v\in\mathbb{C}|\Re(v)\leq
0\\}}$$\scriptstyle{\iota_{U}}$$\scriptstyle{\Phi_{0}}$$\scriptstyle{\Phi}$$\scriptstyle{j_{\theta,U}}$$\scriptstyle{\tilde{\Phi}_{\theta}}$$\scriptstyle{\iota_{V}}$$\scriptstyle{\Psi_{0}}$$\scriptstyle{\Psi}$$\scriptstyle{j_{\theta,V}}$$\scriptstyle{\tilde{\Psi}_{\theta}}$$\scriptstyle{\iota_{W}}$$\scriptstyle{h}$$\scriptstyle{h}$$\scriptstyle{j_{\theta,W}}$$\scriptstyle{h}$

one obtains directly that the following diagrams of morphisms are commutative:

${(\Phi_{0})^{\ast}\phi_{g}\Psi^{\ast}}$${((\Psi\circ\Phi)_{0})^{\ast}\phi_{h}=(\Phi_{0})^{\ast}(\Psi_{0})^{\ast}\phi_{h}}$${\phi_{f}\Phi^{\ast}\Psi^{\ast}=\phi_{f}(\Psi\circ\Phi)^{\ast}}$${(\Psi_{0})_{!}\phi_{g}\Phi_{!}}$${((\Psi\circ\Phi)_{0})_{!}\phi_{f}=(\Psi_{0})_{!}(\Phi_{0})_{!}\phi_{f}}$${\phi_{f}\Psi_{!}\Phi_{!}=\phi_{f}(\Psi\circ\Phi)_{!}}$

hence the morphisms of the proposition are compatible with the composition.

Consider now a Cartesian square of morphisms between algebraic spaces:

${U\times_{W}V}$${V}$${U}$${W}$$\scriptstyle{\Phi^{\prime}}$$\scriptstyle{\Psi^{\prime}}$$\scriptstyle{\Psi}$$\scriptstyle{\Phi}$

and a regular function $f:W\to\mathbb{C}$. Consider now the following
commutative diagram, where all the squares are Cartesian:

${(U\times_{W}V)_{0}}$${U\times_{W}V}$${(\tilde{U\times_{W}V})_{0}}$${U_{0}}$${U}$${\tilde{U}_{0}}$${V_{0}}$${V}$${\tilde{V}_{0}}$${W_{0}}$${W}$${\tilde{W}_{0}}$

Hence from the naturality of the isomorphisms expressing the functoriality and
base change of the six-functor formalism, one obtains that the following
diagram of morphisms is commutative:

${(\Phi^{\prime}_{0})_{!}(\Psi^{\prime}_{0})^{\ast}\phi_{f\circ\Phi}}$${(\Phi^{\prime}_{0})_{!}\phi_{f\circ\Phi\circ\Psi^{\prime}}(\Psi^{\prime})^{\ast}}$${(\Psi_{0})^{\ast}(\Phi_{0})_{!}\phi_{f\circ\Phi}}$${\phi_{f\circ\Psi}(\Phi^{\prime})_{!}(\Psi^{\prime})^{\ast}}$${(\Psi_{0})^{\ast}\phi_{f}\Phi_{!}}$${\phi_{f\circ\Psi}\Psi^{\ast}\Phi_{!}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where the diagonal arrows are the isomorphisms from the base change and the
vertical and horizontal arrows are the morphisms from the proposition. Then
the morphisms of the proposition are compatible with base change.

The functoriality of functor of vanishing cycles for mixed Hodge modules shows
that the morphisms of the proposition are defined at the level of mixed Hodge
modules. Because of the compatibility results of [10, Appendix A], the functor
$rat$ sends the various squares expressing the compatibility conditions in
this proposition to the corresponding squares at the level of perverse
sheaves, which are commutative, hence they commute at the level of mixed Hodge
modules. $\Box$

Now, for $\Phi$ smooth of relative dimension $d$ (resp. $\Psi$ proper),
observing that $\mathbb{D}\Phi^{\ast}[d]\mathbb{D}=\Phi^{\ast}[d]$ (resp.
$\mathbb{D}\Psi_{!}\mathbb{D}=\Psi_{!}$ and then also
$\mathbb{D}(\Phi_{0})^{\ast}[d]\mathbb{D}=(\Phi_{0})^{\ast}[d]$ because
$\Phi_{0}$ is smooth of codimension $d$ (resp.
$\mathbb{D}(\Psi_{0})_{!}\mathbb{D}=(\Psi_{0})_{!}$ because $\Psi_{0}$ is
proper), consider the following diagrams:

${(\Phi_{0})^{\ast}[d]\phi_{g}}$${\phi_{f}\Phi^{\ast}[d]}$${(\Psi_{0})_{!}\phi_{f}}$${\phi_{g}\Psi_{!}}$${\mathbb{D}(\Phi_{0})^{\ast}[d]\phi_{g}\mathbb{D}}$${\mathbb{D}\phi_{f}\Phi^{\ast}[d]\mathbb{D}}$${\mathbb{D}(\Psi_{0})_{!}\phi_{f}\mathbb{D}}$${\mathbb{D}\phi_{g}\Psi_{!}\mathbb{D}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where the vertical arrows comes from the self-duality of the vanishing cycles
functor. It is not hard to show that these squares are commutative, and we
find again the standard fact that for $\Phi$ smooth of relative dimension $d$,
and $\Psi$ proper, there are natural isomorphisms:

$\displaystyle(\Phi_{0})^{\ast}[d]\phi_{g}\simeq$
$\displaystyle\phi_{f}\Phi^{\ast}[d]$ (3.10)
$\displaystyle(\Psi_{0})_{!}\phi_{f}\simeq$ $\displaystyle\phi_{g}\Psi_{!}$

commuting with the monodromy and the duality of the vanishing cycles.

Consider now a $\mathbb{C}^{\ast}$ action on $U$ such that $f$ is
$\mathbb{C}^{\ast}$ invariant, i.e. $U$ is can be considered as an algebraic
space over $\mathbb{A}^{1}_{\mathbb{C}}$. In particular $U$ is locally of
finite type and quasi-separated over a quasi-separated base, hence the
$\mathbb{C}^{\ast}$-action on it is étale-locally linearizable, and then the
$\mathbb{C}^{\ast}$-action on $U^{an}$ (and also on any base change of
$U^{an}$) is analytic-locally linearizable over $\mathbb{C}$, hence the
hyperbolic localization diagram of $U^{an}$ and all its base change exists as
diagrams of analytic spaces. Consider a $\mathbb{C}^{\ast}$-equivariant
constructible complex $A$. The functor $\phi_{f}$ is defined using the
morphisms of the six-functor formalism defined $\mathbb{C}^{\ast}$-equivariant
functions, hence $\phi_{f}(A)$ is $\mathbb{C}^{\ast}$-equivariant. We will use
the idea of [31] to prove the commutation of the hyperbolic localization with
the vanishing cycle:

######  Proposition 3.2.

For $A\in D^{b}_{c}(U)^{T-mon}$, the natural morphism
$(p^{\pm}_{U_{0}})_{!}(\eta^{\pm}_{U_{0}})^{\ast}\phi_{f}\simeq\phi_{f^{0}}(p^{\pm}_{U})_{!}(\eta^{\pm}_{U})^{\ast}(A)$
is an isomorphism, compatible with monodromy and duality, in the sense that
the following diagram is commutative:

${(p^{+}_{U_{0}})_{!}(\eta^{+}_{U_{0}})^{\ast}\phi_{f}\mathbb{D}(A)}$${\phi_{f^{0}}(p^{+}_{U})_{!}(\eta^{+}_{U})^{\ast}\mathbb{D}(A)}$${\mathbb{D}(p^{-}_{U_{0}})_{!}(\eta^{-}_{U_{0}})^{\ast}\phi_{f}\mathbb{D}(A)}$${\mathbb{D}\phi_{f^{0}}(p^{-}_{U})_{!}(\eta^{-}_{U})^{\ast}\mathbb{D}(A)}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where the vertical arrows are given by the isomorphisms of the vanishing
cycles with its dual and the isomorphism of proposition 2.3, and commutes with
Thom-Sebastiani isomorphism and pullback by smooth maps.

Proof: It follows directly from proposition 3.1 that the isomorphism of the
proposition is compatible with monodromy. To show that the natural morphism is
an isomorphism, we need only to show that the diagram of the proposition is
commutative. In fact, the following diagram of $\mathbb{C}^{\ast}$-equivariant
spaces:

${U^{an}_{0}}$${\tilde{U}^{an}_{0}\cap\tilde{U}^{an}_{\pi}}$${\tilde{U}^{an}_{0}}$${\tilde{U}^{an}_{\pi}}$${U^{an}}$$\scriptstyle{q}$$\scriptstyle{\hat{j}_{\pi}}$$\scriptstyle{\hat{j}_{0}}$$\scriptstyle{m}$$\scriptstyle{j_{0}}$$\scriptstyle{j_{\pi}}$

it is obtain by base change from the space $U^{an}\to 0$, hence the hyperbolic
localization diagram of all the spaces in this diagram are also obtained by
base change from the hyperbolic localization diagram. Hence all the squares
are Cartesian in the diagram obtained by superposing the hyperbolic
localization diagrams of those spaces. hence the natural isomorphisms of the
six-functor formalism gives that the following square of isomorphisms is
commutative:

${(p^{+}_{U_{0}})_{!}(\eta^{+}_{U_{0}})^{\ast}\phi_{f}^{\pi}\mathbb{D}}$${\phi_{f^{0}}^{\pi}(p^{+}_{U})_{!}(\eta^{+}_{U})^{\ast}\mathbb{D}}$${\mathbb{D}(p^{-}_{U_{0}})_{!}(\eta^{-}_{U_{0}})^{\ast}\phi_{f}\mathbb{D}(A)}$${\mathbb{D}\phi_{f^{0}}(p^{-}_{U})_{!}(\eta^{-}_{U})^{\ast}\mathbb{D}(A)}$

where the vertical arrow are given by Richarz morphism $S_{U}$ and Massey’s
isomorphism $\mathbb{D}\phi_{f}\simeq\phi_{f}^{\pi}\mathbb{D}$. Because the
isomorphism of the proposition is compatible with the monodromy, hence the
following square is commutative:

${(p^{+}_{U_{0}})_{!}(\eta^{+}_{U_{0}})^{\ast}\phi_{f}\mathbb{D}}$${\phi_{f^{0}}(p^{+}_{U})_{!}(\eta^{+}_{U})^{\ast}\mathbb{D}}$${(p^{+}_{U_{0}})_{!}(\eta^{+}_{U_{0}})^{\ast}\phi_{f}^{\pi}\mathbb{D}}$${\phi_{f}^{\pi}(p^{+}_{U})_{!}(\eta^{+}_{U})^{\ast}\mathbb{D}}$$\scriptstyle{(p^{+}_{U_{0}})_{!}(\eta^{+}_{U_{0}})^{\ast}\tilde{T}^{\pi}_{f}}$$\scriptstyle{\tilde{T}^{\pi}_{f^{0}}}$

and then gluing vertically the two last squares we obtain that the square of
the proposition is commutative.

Consider now $U,V$ two algebraic spaces with $\mathbb{C}^{\ast}$ action,
$\mathbb{C}^{\ast}$-invariants regular functions
$f:U\to\mathbb{C},g:V\to\mathbb{C}$, and a smooth map $\Phi:U\to V$ of
relative dimension $d$. Applying the compatibility of the morphisms of
proposition 3.1 with composition and base change to the morphisms in the
following commutative diagram 2.2, one obtains that the following diagram is
commutative:

${\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}(\Phi^{0}_{0,\pi})^{\ast}[d^{0}_{\pi}](p^{\pm}_{V_{0}})_{!}(\eta^{\pm}_{V_{0}})^{\ast}\phi_{g}}$${\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}(\Phi^{0}_{0,\pi})^{\ast}[d^{0}_{\pi}]\phi_{g^{0}}(p^{\pm}_{V})_{!}(\eta^{\pm}_{V})^{\ast}}$${(p^{\pm}_{U_{0}})_{!}(\eta^{\pm}_{U_{0}})^{\ast}(\Phi_{0})^{\ast}[d]\phi_{g}}$${\phi_{f^{0}}\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}(\Phi^{0}_{0,\pi})^{\ast}[d^{0}_{\pi}](p^{\pm}_{V})_{!}(\eta^{\pm}_{V})^{\ast}}$${(p^{\pm}_{U_{0}})_{!}(\eta^{\pm}_{U_{0}})^{\ast}\phi_{f}\Phi^{\ast}[d]}$${\phi_{f^{0}}(p^{\pm}_{U})_{!}(\eta^{\pm}_{U})^{\ast}\Phi^{\ast}[d]}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

hence this isomorphism is compatible with pullbacks by smooth maps.

Consider two smooth spaces $U,V$ with a $\mathbb{C}^{\ast}$ action. The
product space $U\times V$ is itself provided with a $\mathbb{C}^{\ast}$
action, and functors $X^{0},X^{\pm}$ are obviously compatible with products,
hence the hyperbolic localization diagram of $U\times V$ is the product of the
hyperbolic localization diagrams of $U$ and $V$. In particular:

$\displaystyle(p_{U\times V})_{!}(\eta_{U\times V})^{\ast}$
$\displaystyle=(p_{U}\times
p_{V})_{!}(\eta_{U}\times\eta_{V})^{\ast}:D^{b}_{c}(U\times V)\to
D^{b}_{c}(U^{0}\times V^{0}=(U\times V)^{0})$ (3.11)
$\displaystyle(p_{U_{0}\times V_{0}})_{!}(\eta_{U_{0}\times V_{0}})^{\ast}$
$\displaystyle=(p_{U_{0}}\times
p_{V_{0}})_{!}(\eta_{U_{0}}\times\eta_{V_{0}})^{\ast}:D^{b}_{c}(U_{0}\times
V_{0})\to D^{b}_{c}(U_{0}^{0}\times V_{0}^{0}=(U_{0}\times V_{0})^{0})$

Denote now by $k:U_{0}\times V_{0}\to(U\times V)_{0}$ the natural closed
embedding closed embedding. Using the two commutative diagram of 3.2, one
obtains that the following diagram is commutative:

${(p^{\pm}_{U_{0}}\times
p^{\pm}_{V_{0}})_{!}(\eta^{\pm}_{U_{0}}\times\eta^{\pm}_{V_{0}})^{\ast}(\phi_{f}\boxtimes\phi_{g})}$${(p^{\pm}_{U_{0}}\times
p^{\pm}_{V_{0}})_{!}(\phi_{f^{+}}\boxtimes\phi_{g^{+}})(\eta^{\pm}_{U}\times\eta^{\pm}_{V})^{\ast}}$${(\phi_{f^{0}}\boxtimes\phi_{g^{0}})(p^{\pm}_{U}\times
p^{\pm}_{V})_{!}(\eta^{\pm}_{U}\times\eta^{\pm}_{V})^{\ast}}$${(k^{0})^{\ast}(p^{\pm}_{U_{0}}\times
p^{\pm}_{V_{0}})_{!}(\eta^{\pm}_{U_{0}}\times\eta^{\pm}_{V_{0}})^{\ast}\phi_{f\boxplus
g}}$${(k^{0})^{\ast}(p^{\pm}_{U_{0}}\times
p^{\pm}_{V_{0}})_{!}\phi_{f^{+}\boxplus
g^{+}}(\eta^{\pm}_{U}\times\eta^{\pm}_{V})^{\ast}}$${(k_{0})^{\ast}\phi_{f^{0}\boxplus
g^{0}}(p^{\pm}_{U}\times
p^{\pm}_{V})_{!}(\eta^{\pm}_{U}\times\eta^{\pm}_{V})^{\ast}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

hence the isomorphism is compatible with Thom-Sebastiani isomorphism.

Again the same compatibility results can be lifted at the level of complexes
of mixed Hodge modules.$\Box$

###  3.3. Hyperbolic localization of the perverse sheaf of vanishing cycles

Consider a critical chart $(R,U,f,i)$, i.e. a smooth algebraic space $U$ with
a regular function $f:U\to\mathbb{C}$ with $i:R\to U$ denoting the embedding
of the critical locus. The image of $\phi_{f}$ is supported on $R$, hence
denoting $i_{c}:R_{c}:=R\cap U_{c}\to U_{c}$ the embedding,
$(i_{c})^{\ast}\phi_{f}\simeq(i_{c})^{!}\phi_{f}$ is a functor from ${\rm
Perv}(U)$ to ${\rm Perv}(R_{c})$. One defines in particular the perverse sheaf
on $R$:

(3.12) $\displaystyle\mathcal{PV}_{U,f}:=\bigoplus_{c\in
f(R)}i_{c}^{\ast}\phi_{f-c}\mathcal{IC}_{U}$

with a monodromy operator
$\tau_{U,f}:=\mathcal{PV}_{U,f}\to\mathcal{PV}_{U,f}$ defined by:

(3.13) $\displaystyle\tau_{U,f}=\bigoplus_{c\in
f(R)}\tilde{T}_{f-c}^{2\pi}|_{R_{c}}$

and a polarization operator
$\sigma_{U,f}:\mathcal{PV}_{U,f}\overset{\simeq}{\to}\mathbb{D}\mathcal{PV}_{U,f}$
defined by the composition of isomorphisms:

${\mathcal{PV}_{U,f}=\bigoplus_{c\in
f(R)}i_{c}^{\ast}\phi_{f-c}\mathcal{IC}_{U}}$${\bigoplus_{c\in
f(R)}i_{c}^{\ast}\phi_{f-c}\mathbb{D}\mathcal{IC}_{V}}$${\mathbb{D}\mathcal{PV}_{U,f}=\mathbb{D}\bigoplus_{c\in
f(R)}i_{c}^{\ast}\phi_{f-c}\mathcal{IC}_{U}}$${\bigoplus_{c\in
f(R)}i_{c}^{\ast}\mathbb{D}\phi_{f-c}\mathcal{IC}_{U}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where we have used first the self-duality of the intersection complex,
secondly Massey’s isomorphism, and thirdly the fact that
$(i_{0})^{\ast}\phi_{f}=(i_{0})^{!}\phi_{f}$.

This perverse sheaf is given the structure of strongly polarized a monodromic
mixed Hodge module in [10, sec 2.10], using the polarized mixed Hodge module
on $\mathcal{IC}_{U}$, the strongly polarized monodromic mixed Hodge module
obtained by applying $\phi^{H}_{f}$, and then using $i^{\ast}$ at the level of
mixed Hodge modules. The functor $rat$ sends then the monodromy automorphism
to $\tau_{U,f}$, and the strong polarization to $\sigma_{U,f}$.

######  Proposition 3.3.

For a $\mathbb{C}^{\ast}$-equivariant critical chart $(U,R,f,i)$, there are
natural isomorphisms in the abelian category $\bigoplus_{\pi\in\Pi}{\rm
MMHM}(R^{0}_{\pi})[-{\rm Ind}_{\pi}]$:

(3.14)
$\displaystyle\beta^{\pm}_{U,f}:(p_{R}^{\pm})_{!}(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f}\to\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}}$

compatible with the strong polarization, i.e. such that the following square
is commutative:

${(p_{R}^{+})_{!}(\eta_{R}^{+})^{\ast}\mathcal{PV}_{U,f}}$${\bigoplus_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}}}$${\mathbb{D}(p_{R}^{-})_{!}(\eta_{R}^{-})^{\ast}\mathcal{PV}_{U,f}}$${\bigoplus_{\pi\in\Pi}\mathbb{L}^{-{\rm
Ind}_{\pi}/2}\mathbb{D}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}}}$$\scriptstyle{\beta^{+}_{U,f}}$$\scriptstyle{B_{R}^{-1}\circ\sigma_{U,f}}$$\scriptstyle{\bigoplus_{\pi\in\Pi}\sigma_{U^{0}_{\pi},f^{0}_{\pi}}}$$\scriptstyle{\beta_{U,f}}$

Proof: Consider the following commutative diagram:

${U_{c}}$${U^{\pm}_{c}}$${U^{0}_{c}}$${(p^{\pm}_{U_{c}})^{-1}(R^{0}_{c})}$${R^{0}_{c}}$${R_{c}}$${R_{c}^{\pm}}$$\scriptstyle{\eta^{\pm}_{U_{c}}}$$\scriptstyle{p^{\pm}_{U_{c}}}$$\scriptstyle{\bar{i}}$$\scriptstyle{\bar{p}}$$\scriptstyle{i^{0}_{c}}$$\scriptstyle{i_{c}}$$\scriptstyle{p^{\pm}_{R}}$$\scriptstyle{\eta^{\pm}_{R}}$$\scriptstyle{\hat{i}}$

where $\hat{i}$ and $\bar{i}$ are the obvious inclusions, and
$\bar{p}:=p^{\pm}_{U_{c}}|_{(p^{\pm}_{U_{c}})^{-1}(R^{0}_{c})}$. The upper
right square is Cartesian. One can define the following isomorphisms:

$\displaystyle(p^{\pm}_{R_{c}})_{!}(\eta^{\pm}_{R_{c}})^{\ast}(i_{c})^{\ast}$
$\displaystyle\simeq\bar{p}_{!}\hat{i}_{!}\hat{i}^{\ast}\bar{i}^{\ast}(\eta^{\pm}_{U_{c}})^{\ast}$
$\displaystyle\simeq\bar{p}_{!}\bar{i}^{\ast}(\eta_{U_{c}})^{\ast}$ (3.15)
$\displaystyle\simeq(i^{0}_{c})^{\ast}(p^{\pm}_{U_{c}})_{!}(\eta^{\pm}_{U_{c}})^{\ast}$

where we have used the fact that $\hat{i}_{!}\hat{i}^{\ast}={\rm Id}$ because
$\hat{i}$ is a closed embedding in the second line, and the base change
theorem in the upper right square in the last line. Moreover, because these
isomorphism, and also the isomorphisms $S_{R},S_{U_{c}}$ are built using the
natural morphisms of the six-functor formalism in the above diagram, the
following diagram is commutative, for $A\in
D^{b}_{c}(U_{c})_{R_{c}}^{\mathbb{C}^{\ast}-eq}$:

${(p^{+}_{R_{c}})_{!}(\eta_{R_{c}}^{+})^{\ast}i_{c}^{\ast}\mathbb{D}(A)}$${(i_{c}^{0})^{\ast}(p^{+}_{U_{c}})_{!}(\eta^{+}_{U_{c}})^{\ast}\mathbb{D}(A)}$${(p^{+}_{R_{c}})_{!}(\eta_{R_{c}}^{+})^{\ast}\mathbb{D}i_{c}^{\ast}(A)}$${(i_{c}^{0})^{\ast}\mathbb{D}(p^{-}_{U_{c}})_{!}(\eta^{-}_{U_{c}})^{\ast}(A)}$${\mathbb{D}(p^{-}_{R_{c}})_{!}(\eta_{R_{c}}^{-})^{\ast}i_{c}^{\ast}(A)}$${\mathbb{D}(i_{c}^{0})^{\ast}(p^{-}_{U_{c}})_{!}(\eta^{-}_{U_{c}})^{\ast}(A)}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Then one obtains isomorphisms:

(3.16)
$\displaystyle\beta^{\pm}_{U,f}:(p^{\pm}_{R})_{!}(\eta^{\pm}_{R})^{\ast}\mathcal{PV}_{U,f}\simeq\bigoplus_{\pi\in\Pi}\mathbb{L}^{{}^{\pm}{\rm
Ind}_{\pi}/2}{PV}_{U^{0}_{\pi},f^{0}_{\pi}}$

by the following composition of isomorphisms:

${(p_{R}^{\pm})_{!}(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f}:=(p_{R}^{\pm})_{!}(\eta_{R}^{\pm})^{\ast}\bigoplus_{c\in
f(R)}i_{c}^{\ast}\phi_{f-c}\mathcal{IC}_{U}}$${\bigoplus_{c\in
f(R)}(i_{c}^{0})^{\ast}(p^{\pm}_{U_{c}})_{!}(\eta^{\pm}_{U_{c}})^{\ast}\phi_{f-c}\mathcal{IC}_{U}}$${\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}{PV}_{U^{0}_{\pi},f^{0}_{\pi}}:=\bigoplus_{c\in
f(R)}i_{c}^{\ast}\phi_{f-c}\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\mathcal{IC}_{U^{0}_{\pi}}}$${\bigoplus_{c\in
f(R)}i_{c}^{\ast}\phi_{f-c}(p_{U}^{\pm})_{!}(\eta_{U}^{\pm})^{\ast}\mathcal{IC}_{U}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where the first isomorphism is (3.3), the second from proposition 3.2 where
one has remaked that $\mathcal{IC}_{U}$ is $\mathbb{C}^{\ast}$-equivariant,
and the third from corollary 2.6.

Using the commutativity of the diagram 3.2, one obtains that the following
square is commutative:

${(p_{U_{0}}^{\pm})_{!}(\eta_{U_{0}}^{\pm})^{\ast}\phi_{f}\mathcal{IC}_{U}}$${\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\phi_{f^{0}_{\pi}}\mathcal{IC}_{U^{0}_{\pi}}}$${(p_{U_{0}}^{\pm})_{!}(\eta_{U_{0}}^{\pm})^{\ast}\phi_{f}\mathcal{IC}_{U}}$${\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\phi_{f^{0}_{\pi}}\mathcal{IC}_{U^{0}_{\pi}}}$$\scriptstyle{\simeq}$$\scriptstyle{(p_{U_{0}}^{\pm})_{!}(\eta_{U_{0}}^{\pm})^{\ast}(\tilde{T}_{f}^{2\pi})}$$\scriptstyle{\bigoplus_{\pi\in\Pi}\tilde{T}^{2\pi}_{f^{0}_{\pi}}}$$\scriptstyle{\simeq}$

the compatibility with the monodromy results then by applying $(i_{0})^{\ast}$
to this diagram, applying the functorial isomorphism (3.3), and finally
writing the definition of $\tau_{U,f}$ by noticing that
$(-1)^{\dim(U)}=(-1)^{\pm{\rm Ind}_{\pi}+\dim(U^{0}_{\pi})}$.

The proof of the compatibility with $\sigma_{U,f}$ is slightly more technical.
In fact $\beta^{\pm}_{U,f}$ and $\sigma_{U,f}$ are both defined by composing
three isomorphisms expressing the commutation of the hyperbolic localization
and Verdier duality with $i_{c}^{\ast}$, with $\phi_{f}$ and with
$\mathcal{IC}_{U}$, and then one should divide the square expressing the
compatibility into nine sub-squares:

${11}$${12}$${13}$${14}$${21}$${22}$${23}$${24}$${31}$${32}$${33}$${34}$${41}$${42}$${43}$${44}$

Where the objects are:

$\displaystyle
11:(p_{R}^{+})_{!}(\eta_{R}^{+})^{\ast}(i_{0})^{\ast}\phi_{f}\mathcal{IC}_{U}$
$\displaystyle
12:(i^{0}_{0})^{\ast}(p^{+}_{U_{0}})_{!}(\eta^{+}_{U_{0}})^{\ast}\phi_{f}\mathcal{IC}_{U}$
$\displaystyle
13:(i^{0}_{0})^{\ast}\phi_{f^{0}}(p^{+}_{U})_{!}(\eta^{+}_{U})^{\ast}\mathcal{IC}_{U}$
$\displaystyle
14:(i^{0}_{0})^{\ast}\phi_{f^{0}}\bigoplus_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}}$ $\displaystyle
21:(p_{R}^{+})_{!}(\eta_{R}^{+})^{\ast}(i_{0})^{\ast}\phi_{f}\mathbb{D}\mathcal{IC}_{U}$
$\displaystyle
22:(i^{0}_{0})^{\ast}(p^{+}_{U_{0}})_{!}(\eta^{+}_{U_{0}})^{\ast}\phi_{f}\mathbb{D}\mathcal{IC}_{U}$
$\displaystyle
23:(i^{0}_{0})^{\ast}\phi_{f^{0}}(p^{+}_{U})_{!}(\eta^{+}_{U})^{\ast}\mathbb{D}\mathcal{IC}_{U}=(i^{0}_{0})^{\ast}\phi_{f^{0}}\mathbb{D}(p^{-}_{U})_{!}(\eta^{-}_{U})^{\ast}\mathcal{IC}_{U}$
$\displaystyle
24:(i^{0}_{0})^{\ast}\phi_{f^{0}}\bigoplus_{\pi\in\Pi}\mathbb{D}\mathbb{L}^{-{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}}$ $\displaystyle
31:(p_{R}^{+})_{!}(\eta_{R}^{+})^{\ast}(i_{0})^{\ast}\mathbb{D}\phi_{f}\mathcal{IC}_{U}$
$\displaystyle
32:(i^{0}_{0})^{\ast}(p^{+}_{U_{0}})_{!}(\eta^{+}_{U_{0}})^{\ast}\mathbb{D}\phi_{f}\mathcal{IC}_{U}=(i^{0}_{0})^{\ast}\mathbb{D}(p^{-}_{U_{0}})_{!}(\eta^{-}_{U_{0}})^{\ast}\phi_{f}\mathcal{IC}_{U}$
$\displaystyle
33:(i^{0}_{0})^{\ast}\phi_{f^{0}}(p^{+}_{U})_{!}(\eta^{+}_{U})^{\ast}\mathbb{D}\mathcal{IC}_{U}=(i^{0}_{0})^{\ast}\mathbb{D}\phi_{f^{0}}(p^{-}_{U})_{!}(\eta^{-}_{U})^{\ast}\mathcal{IC}_{U}$
$\displaystyle
34:(i^{0}_{0})^{\ast}\mathbb{D}\phi_{f^{0}}\bigoplus_{\pi\in\Pi}\mathbb{L}^{-{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}}$ $\displaystyle
41:(p_{R}^{+})_{!}(\eta_{R}^{+})^{\ast}\mathbb{D}(i_{0})^{\ast}\phi_{f}\mathcal{IC}_{U}=\mathbb{D}(p_{R}^{-})_{!}(\eta_{R}^{-})^{\ast}(i_{0})^{\ast}\phi_{f}\mathcal{IC}_{U}$
$\displaystyle
42:\mathbb{D}(i^{0}_{0})^{\ast}(p^{+}_{U_{0}})_{!}(\eta^{+}_{U_{0}})^{\ast}\phi_{f}\mathcal{IC}_{U}=(i^{0}_{0})^{\ast}\mathbb{D}(p^{-}_{U_{0}})_{!}(\eta^{-}_{U_{0}})^{\ast}\phi_{f}\mathcal{IC}_{U}$
$\displaystyle
43:\mathbb{D}(i^{0}_{0})^{\ast}\phi_{f^{0}}(p^{+}_{U})_{!}(\eta^{+}_{U})^{\ast}\mathcal{IC}_{U}=(i^{0}_{0})^{\ast}\mathbb{D}\phi_{f^{0}}(p^{-}_{U})_{!}(\eta^{-}_{U})^{\ast}\mathcal{IC}_{U}$
(3.17) $\displaystyle
44:\mathbb{D}(i^{0}_{0})^{\ast}\phi_{f^{0}}\bigoplus_{\pi\in\Pi}\mathbb{L}^{-{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}}$

The down left square commutes because it is the diagram 3.3 with
$A=\phi_{f}\mathcal{IC}_{U}$. The central square commutes because 3.2
commutes, and the upper right square commutes because it is
$(i^{0}_{0})^{\ast}\phi_{f^{0}}$ applied to the commutative diagram in
corollary 2.6. The squares which are not off the down-left/up right diagonal
are commutative because the isomorphisms constructed are functorial. Then the
whole diagram is commutative, and, summing over $c\in f(R)$, one obtains the
commutativity of the diagram expressing the compatibility with the
polarization.

Again the same compatibility results can be directly lifted to the derived
category of monodromic mixed Hodge modules.$\Box$

###  3.4. Compatibility with smooth maps

Consider two critical charts $(R,U,f,i),(S,V,g,j)$ and $\Phi:U\to V$ a smooth
map of relative dimension $d$ with $f=g\circ\Phi$. Joyce defines then in [7]
the isomorphism

(3.18)
$\displaystyle\Xi_{\Phi}:(\Phi|_{R})^{\ast}[d]\mathcal{PV}_{V,g}\simeq\mathcal{PV}_{U,f}$

by the following composition of isomorphisms:

${(\Phi|_{R})^{\ast}[d]\mathcal{PV}_{V,g}=(\Phi|_{R})^{\ast}[d]\bigoplus_{c\in
g(S)}j_{c}^{\ast}\phi_{g-c}\mathcal{IC}_{V}}$${\bigoplus_{c\in
f(R)}i_{c}^{\ast}(\Phi_{c})^{\ast}[d]\phi_{g-c}\mathcal{IC}_{V}}$${\mathcal{PV}_{U,f}=\bigoplus_{c\in
f(R)}i_{c}^{\ast}\phi_{f-c}\mathcal{IC}_{U}}$${\bigoplus_{c\in
f(R)}i_{c}^{\ast}\phi_{f-c}\Phi^{\ast}[d]\mathcal{IC}_{V}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

identifying $(\Phi|_{R})^{\ast}[d](\sigma_{V,g})$ with $\sigma_{U,f}$ and
$(\Phi|_{R})^{\ast}[d](\tau_{V,g})$ with $\tau_{U,f}$ according to proposition
4.3 of [7].

######  Proposition 3.4.

The isomorphisms $\beta^{\pm}_{U,f}$ commute with pullback of smooth maps of
$\mathbb{C}^{\ast}$-equivariant critical charts, i.e. for $\Phi:U\to V$ a
$\mathbb{C}^{\ast}$ equivariant and smooth map of relative dimension $d$ and
critical charts $(R,U,f,i)$ and $(S,V,g,j)$ with $f=g\circ\Phi$ one has the
commutative diagram:

${\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}(\Phi^{0}_{\pi}|_{R^{0}})^{\ast}[d^{0}_{\pi}](p^{\pm}_{S})_{!}(\eta^{\pm}_{S})^{\ast}\mathcal{PV}_{V,g}}$${\bigoplus_{\pi,\pi^{\prime}\in\Pi}\mathbb{L}^{(\pm{\rm
Ind}_{\pi^{\prime}})/2}(\Phi^{0}_{\pi}|_{R^{0}})^{\ast}[d^{0}_{\pi}]\mathcal{PV}_{V^{0}_{\pi^{\prime}-\pi},g^{0}_{\pi^{\prime}-\pi}}}$${(p^{\pm}_{R})_{!}(\eta^{\pm}_{R})^{\ast}(\Phi|_{R})^{\ast}[d]\mathcal{PV}_{V,g}}$${(p^{\pm}_{R})_{!}(\eta^{\pm}_{R})^{\ast}\mathcal{PV}_{U,f}}$${\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi^{\prime}}/2}\mathcal{PV}_{U^{0}_{\pi^{\prime}},f^{0}_{\pi^{\prime}}}}$$\scriptstyle{\beta^{\pm}_{V,g}}$$\scriptstyle{\Xi_{\Phi^{0}}}$$\scriptstyle{\Xi_{\Phi}}$$\scriptstyle{\simeq}$$\scriptstyle{\beta^{\pm}_{U,f}}$

In particular, one obtains that $\beta^{\pm}_{U,f}$ commute with the
restriction on an étale neighborhood.

Proof: Consider now the following diagram, where we denote
$G=\phi_{g}\mathcal{IC}_{V}$:

${(p_{R})_{!}(\eta_{R})^{\ast}(\Phi|_{R})^{\ast}(j_{0})^{\ast}G=(\Phi|_{R^{0}})^{\ast}(p_{S})_{!}(\eta_{S})^{\ast}(j_{0})^{\ast}G}$${(\Phi|_{R^{0}})^{\ast}(j^{0}_{0})^{\ast}(p_{V_{0}})_{!}(\eta_{V_{0}})^{\ast}G}$${(p_{R})_{!}(\eta_{R})^{\ast}(i_{0})^{\ast}(\Phi_{0})^{\ast}G}$${(i^{0}_{0})^{\ast}(p_{U_{0}})_{!}(\eta_{U_{0}})^{\ast}(\Phi_{0})^{\ast}G=(i^{0}_{0})^{\ast}(\Phi^{0}_{0})^{\ast}(p_{V_{0}})_{!}(\eta_{V_{0}})^{\ast}}$

where the horizontal isomorphisms are those of (3.3) applied to $V$ and $U$:
it is a diagram of isomorphisms between objects in an abelian category, and
the isomorphisms are built from the natural isomorphisms of the six-functor
formalism in the diagram:

${V_{0}}$${V_{0}^{\pm}}$${V_{0}^{0}}$${U_{0}}$${U_{0}^{\pm}}$${U_{0}^{0}}$${p^{-1}_{V_{0}}}$${S^{0}}$${p^{-1}_{U_{0}}}$${R^{0}}$${S}$${S^{\pm}}$${R}$${R^{\pm}}$

hence this diagram commutes. Finally, consider the diagram:

${11}$${12}$${13}$${14}$${21}$${22}$${23}$${24}$${31}$${32}$${33}$${34}$${41}$${42}$${43}$${44}$

Where the objects are:

$\displaystyle
11:(p_{R})_{!}(\eta_{R})^{\ast}(\Phi|_{R})^{\ast}[d](j_{0})^{\ast}\phi_{g}\mathcal{IC}_{V}=(\Phi|_{R^{0}})^{\ast}[d](p_{S})_{!}(\eta_{S})^{\ast}(j_{0})^{\ast}\phi_{g}\mathcal{IC}_{V}$
$\displaystyle
12:(\Phi|_{R^{0}})^{\ast}[d](j^{0}_{0})^{\ast}(p_{V_{0}})_{!}(\eta_{V_{0}})^{\ast}\phi_{g}\mathcal{IC}_{V}$
$\displaystyle
13:(\Phi|_{R^{0}})^{\ast}[d](j^{0}_{0})^{\ast}\phi_{g^{0}}(p_{V})_{!}(\eta_{V})^{\ast}\mathcal{IC}_{V}$
$\displaystyle
14:(\Phi|_{R^{0}})^{\ast}[d](j^{0}_{0})^{\ast}\phi_{g^{0}}\bigoplus_{n\in\mathbb{Z}}\mathbb{L}^{n/2}\mathcal{PV}_{V^{0}_{n},g^{0}_{n}}$
$\displaystyle
21:(p_{R})_{!}(\eta_{R})^{\ast}(i_{0})^{\ast}(\Phi_{0})^{\ast}[d]\phi_{g}\mathcal{IC}_{V}$
$\displaystyle
22:(i^{0}_{0})^{\ast}(p_{U_{0}})_{!}(\eta_{U_{0}})^{\ast}(\Phi_{0})^{\ast}[d]\phi_{g}\mathcal{IC}_{V}=(i^{0}_{0})^{\ast}(\Phi^{0}_{0})^{\ast}[d](p_{V_{0}})_{!}(\eta_{V_{0}})^{\ast}\phi_{g}\mathcal{IC}_{V}$
$\displaystyle
23:(i^{0}_{0})^{\ast}(\Phi^{0}_{0})^{\ast}[d]\phi_{g^{0}}(p_{V})_{!}(\eta_{V})^{\ast}\mathcal{IC}_{V}$
$\displaystyle
24:(i^{0}_{0})^{\ast}(\Phi^{0}_{0})^{\ast}[d]\phi_{g^{0}}\bigoplus_{n\in\mathbb{Z}}\mathbb{L}^{n/2}\mathcal{PV}_{V^{0}_{n},g^{0}_{n}}$
$\displaystyle
31:(p_{R})_{!}(\eta_{R})^{\ast}(i_{0})^{\ast}\phi_{f}\Phi^{\ast}[d]\mathcal{IC}_{V}$
$\displaystyle
32:(i^{0}_{0})^{\ast}(p_{U_{0}})_{!}(\eta_{U_{0}})^{\ast}\phi_{f}\Phi^{\ast}[d]\mathcal{IC}_{V}$
$\displaystyle
33:(i^{0}_{0})^{\ast}\phi_{f^{0}}(p_{U})_{!}(\eta_{U})^{\ast}\Phi^{\ast}[d]\mathcal{IC}_{V}=(i^{0}_{0})^{\ast}\phi_{f^{0}}(\Phi^{0})^{\ast}[d](p_{V})_{!}(\eta_{V})^{\ast}\mathcal{IC}_{V}$
$\displaystyle
34:(i^{0}_{0})^{\ast}\phi_{f^{0}}(\Phi^{0})^{\ast}[d]\bigoplus_{n\in\mathbb{Z}}\mathbb{L}^{n/2}\mathcal{PV}_{V^{0}_{n},g^{0}_{n}}$
$\displaystyle
41:(p_{R})_{!}(\eta_{R})^{\ast}(i_{0})^{\ast}\phi_{f}\mathcal{IC}_{U}$
$\displaystyle
42:(i^{0}_{0})^{\ast}(p_{U_{0}})_{!}(\eta_{U_{0}})^{\ast}\phi_{f}\mathcal{IC}_{U}$
$\displaystyle
43:(i^{0}_{0})^{\ast}\phi_{f^{0}}(p_{U})_{!}(\eta_{U})^{\ast}\mathcal{IC}_{U}$
(3.19) $\displaystyle
44:(i^{0}_{0})^{\ast}\phi_{f^{0}}\bigoplus_{n\in\mathbb{Z}}\mathbb{L}^{n/2}\mathcal{PV}_{U^{0}_{n},f^{0}_{n}}$

the three diagonal squares of the above diagram are respectively the
commutative diagrams 3.4, 3.2 and the commutative diagram expressing the fact
that the isomorphism giving the commutation of the hyperbolic localization and
pullback by smooth maps in proposition 2.5 is compatible with composition of
smooth maps, and the off diagonal squares commute because the functoriality of
the isomorphism. Then the whole diagram is commutative. According to the
definition of $\Xi_{\Phi},\Xi_{\Phi^{0}}$ and $\beta_{U,f},\beta_{V,g}$, the
diagram 3.4 is the direct sum of the above diagram with $f$ replaced by $f-c$
for $c\in f(R)$, hence it is also commutative. $\Box$

###  3.5. Compatibility with Thom-Sebastiani isomorphism

Consider two critical charts $(U,R,f,i)$ and $(S,V,g,j)$. Notice that because
$U$ and $V$ are smooth ${\rm Crit}(f\boxplus g)={\rm Crit}(f)\times{\rm
Crit}(g)R\times S$. Consider the sequence of closed embeddings:

${R\times S}$${U_{c}\times V_{d}}$${(U\times
V)_{c+d}}$$\scriptstyle{i_{c}\times j_{d}}$$\scriptstyle{k_{c,d}}$

In [10], Joyce defined the Thom-Sebastiani isomorphism:

(3.20) $\displaystyle\mathcal{TS}_{U,f,V,g}:\mathcal{PV}_{U\times V,f\boxplus
g}\simeq\mathcal{PV}_{U,f}\boxtimes\mathcal{PV}_{S,g}$

By the following sequence of isomorphisms:

$\displaystyle\mathcal{PV}_{U\times V,f\boxplus g}=$
$\displaystyle\bigoplus_{e\in f\boxplus g(R\times S)}((i\times
j)_{e})^{\ast}\phi_{f\boxplus g-e}\mathcal{IC}_{U\times V}$
$\displaystyle\simeq$ $\displaystyle\bigoplus_{c\in f(R),d\in
g(S)}(i_{c}\times
j_{d})^{\ast}k_{c,d}^{\ast}\phi_{(f-c)\boxplus(g-d)}\mathcal{IC}_{U\times V}$
$\displaystyle\simeq$ $\displaystyle\bigoplus_{c\in f(R),d\in
g(S)}(i_{c}\times
j_{d})^{\ast}(\phi_{f-c}\boxtimes\phi_{g-d})\mathcal{IC}_{U\times V}$
$\displaystyle\simeq$ $\displaystyle(\bigoplus_{c\in
f(R)}(i_{c})^{\ast}\phi_{f-c}\mathcal{IC}_{U})\boxtimes(\bigoplus_{d\in
g(S)}(j_{d})^{\ast}\phi_{g-d}\mathcal{IC}_{V})$ (3.21) $\displaystyle=$
$\displaystyle\mathcal{PV}_{U,f}\boxtimes\mathcal{PV}_{S,g}$

where the third line is the Thom-Sebastiani isomorphism of [24].

######  Proposition 3.5.

The isomorphism $\beta_{U,f}$ commutes with Thom-Sebastiani isomorphism,
namely for $\mathbb{C}^{\ast}$-equivariant critical chart $(U,R,f,i)$ and
$(S,V,g,j)$, one has the commutative diagram:

$(p_{R}^{\pm}\times
p_{S}^{\pm})|_{!}(\eta_{R}^{\pm}\times\eta_{S}^{\pm})^{\ast}(\mathcal{PV}_{U\times
V,f\boxplus g})$ $\bigoplus_{\pi,\pi^{\prime}\in\Pi}\mathbb{L}^{\pm({\rm
Ind}_{\pi}+{\rm Ind}_{\pi^{\prime}})/2}\mathcal{PV}_{U^{0}_{\pi}\times
V^{0}_{\pi^{\prime}},f^{0}_{\pi}\boxplus g^{0}_{\pi^{\prime}}}$
$((p_{R}^{\pm})_{!}(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f})\boxtimes((p_{S}^{\pm})_{!}(\eta_{S}^{\pm})^{\ast}\mathcal{PV}_{S,g})$
$\bigoplus_{\pi\in\mathbb{Z}}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}})\boxtimes\bigoplus_{\pi^{\prime}\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi^{\prime}}/2}\mathcal{PV}_{V^{0}_{\pi^{\prime}},g^{0}_{\pi^{\prime}}})$
$\scriptstyle{\beta^{\pm}_{U\times V}}$ $\scriptstyle(p_{R}^{\pm}\times
p_{S}^{\pm})|_{!}(\eta_{R}^{\pm}\times\eta_{S}^{\pm})^{\ast}\mathcal{TS}_{U,f,V,g}$
$\scriptstyle\bigoplus_{\pi,\pi^{\prime}\in\Pi}\mathbb{L}^{\pm({\rm
Ind}_{\pi}+{\rm
Ind}_{\pi^{\prime}})/2}\mathcal{TS}_{U^{0}_{\pi},f^{0}_{\pi},V^{0}_{\pi^{\prime}},g^{0}_{\pi^{\prime}}}$
$\scriptstyle{\beta_{U}\times\beta_{V}}$

Proof: Consider the following diagram:

${11}$${12}$${13}$${14}$${21}$${22}$${23}$${24}$

Where the objects are:

$\displaystyle 11:(p_{R}^{\pm}\times
P_{S}^{\pm})_{!}(\eta_{R}^{\pm}\times\eta_{S}^{\pm})^{\ast}(i_{0}\times
j_{0})^{\ast}k^{\ast}\phi_{f\boxplus g}\mathcal{IC}_{U\times V}$
$\displaystyle 12:(i^{0}_{0}\times j^{0}_{0})^{\ast}(p^{\pm}_{U_{0}}\times
p^{\pm}_{V_{0}})_{!}(\eta^{\pm}_{U_{0}}\eta^{\pm}_{V_{0}})^{\ast}k^{\ast}\phi_{f\boxplus
g}\mathcal{IC}_{U\times V}=(i^{0}_{0}\times
j^{0}_{0})^{\ast}(k^{0})^{\ast}(p^{\pm}_{(U\times
V)_{0}})_{!}(\eta^{\pm}_{(U\times V)_{0}})^{\ast}\phi_{f\boxplus
g}\mathcal{IC}_{U\times V}$ $\displaystyle 13:(i^{0}_{0}\times
j^{0}_{0})^{\ast}(k^{0})^{\ast}\phi_{f^{0}\boxplus g^{0}}(p^{\pm}_{U}\times
p^{\pm}_{V})_{!}(\eta^{\pm}_{U}\times\eta^{\pm}_{V})^{\ast}\mathcal{IC}_{U\times
V}$ $\displaystyle 14:(i^{0}_{0}\times
j^{0}_{0})^{\ast}(k^{0})^{\ast}\phi_{f^{0}\boxplus
g^{0}}\bigoplus_{\pi,\pi^{\prime}\in\Pi}\mathbb{L}^{({\rm Ind}_{\pi}+{\rm
Ind}_{\pi^{\prime}})/2}\mathcal{IC}_{U^{0}_{\pi}\times V^{0}_{\pi^{\prime}}}$
$\displaystyle 21:(p_{R}^{\pm}\times
P_{S}^{\pm})_{!}(\eta_{R}^{\pm}\times\eta_{S}^{\pm})^{\ast}(i_{0}\times
j_{0})^{\ast}(\phi_{f}\boxtimes\phi_{g})\mathcal{IC}_{U\times V}$
$\displaystyle 22:(i^{0}_{0}\times j^{0}_{0})^{\ast}(p^{\pm}_{U_{0}}\times
p^{\pm}_{V_{0}})_{!}(\eta^{\pm}_{U_{0}}\eta^{\pm}_{V_{0}})^{\ast}(\phi_{f}\boxtimes\phi_{g})\mathcal{IC}_{U\times
V}$ $\displaystyle 23:(i^{0}_{0}\times
j^{0}_{0})^{\ast}(\phi_{f^{0}}\boxtimes\phi_{g^{0}})(p^{\pm}_{U}\times
p^{\pm}_{V})_{!}(\eta^{\pm}_{U}\times\eta^{\pm}_{V})^{\ast}\mathcal{IC}_{U\times
V}$ (3.22) $\displaystyle 24:(i^{0}_{0}\times
j^{0}_{0})^{\ast}(\phi_{f^{0}}\boxtimes\phi_{g^{0}})\bigoplus_{\pi,\pi^{\prime}\in\Pi}\mathbb{L}^{({\rm
Ind}_{\pi}+{\rm Ind}_{\pi^{\prime}})/2}\mathcal{IC}_{U^{0}_{\pi}\times
V^{0}_{\pi^{\prime}}}$

the left and the right squares commutes from the functoriality of the
corresponding isomorphisms, and the central square commutes because it is the
diagram 3.2, which commutes, hence the whole diagram commutes. Identifying
$\mathcal{IC}_{U\times V}$ with $\mathcal{IC}_{U}\boxtimes\mathcal{IC}_{V}$
and $\mathcal{IC}_{U^{0}_{\pi}\times V^{0}_{\pi^{\prime}}}$ with
$\mathcal{IC}_{U^{0}_{\pi}}\boxtimes\mathcal{IC}_{V^{0}_{\pi^{\prime}}}$, and
summing over $c\in f(R),d\in g(S)$, one obtains the commutative square of the
proposition. $\Box$

As explained in example 2.14 of [10], for $q$ a non-degenerate quadratic form
on an $n$-dimensional vector space $E$, one has ${\rm Crit}(q)=\\{0\\}$, and
then:

(3.23)
$\displaystyle\mathcal{PV}_{E,q}=H^{n-1}(MF_{q}(0),\mathbb{Q})\otimes\mathbb{Q}_{\\{0\\}}\simeq\mathbb{Q}_{\\{0\\}}$

where $MF_{q}(0)$ denotes the Milnor fiber of $q$ at $0$, which is
$T^{\ast}S^{n-1}$, and the second isomorphism comes from the orientation of
$S^{n-1}$ coming from an orientation of $E$. Consider now $E$ with a linear
$\mathbb{C}^{\ast}$-action and a $\mathbb{C}^{\ast}$-invariant non-degenerate
quadratic form $q$. We can decompose $E=E^{0}\oplus E^{+}\oplus E^{-}$
according to the $\mathbb{C}^{\ast}$-weights, and the non-degenerate invariant
quadratic form $q$ gives a natural isomorphism $E^{-}=(E^{+})^{\vee}$, hence a
natural isomorphism $K_{E}=K_{E^{0}}$, i.e. a natural bijection between
orientations of $E$ and orientations of $E^{0}$. Denoting $q^{0}:=q|_{E^{0}}$,
one obtains directly the commutativity of the following square:

${\mathcal{PV}_{E,q}}$${\mathcal{PV}_{E^{0},q^{0}}}$${\mathbb{Q}_{\\{0\\}}}$${\mathbb{Q}_{\\{0\\}}}$$\scriptstyle{\beta^{\pm}_{E,q}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where the vertical arrows comes from consistent orientations of $V$ and
$V^{0}$.

##  4\. Białynicki-Birula decomposition on a d-critical algebraic space

###  4.1. D-critical algebraic space

We recall here the notions and results of [21] and [10] about d-critical
structures and the Donaldson-Thomas sheaf. Joyce and his collaborators have
developed the theory of d-critical schemes and build the perverse sheaf on it
using the Zariski topology, mainly in order to define also motivic Donaldson-
Thomas invariants, which can be glued in the Zariski topology, but not in the
étale topology. But because coherent sheaves and perverse sheaves glue in the
étale topology, and the perverse sheaf of vanishing cycles transforms
naturally under étale maps, the same formalism can be developed in the étale
topology. We will then work in the étale topology, and consider d-critical
algebraic spaces.

In [21, Theo 2.1], Joyce constructs sheaves $\mathcal{S}_{X}$ and
$\mathcal{S}^{0}_{X}$ on any algebraic space $X$. Given an étale map $R\to X$
and a closed embedding $i:R\hookrightarrow U$ into a smooth scheme $U$, denote
by $I_{R,U}$ the sheaf of ideal in $i^{-1}(\mathcal{O}_{U})$ of functions on
$U$ near $i(R)$ which vanishes on $i(R)$. There is then an exact sequence of
sheaves on $R$ defining locally $\mathcal{S}_{X}$:

(4.1) $\displaystyle
0\to\mathcal{S}_{X}|_{R}\overset{\iota_{R,U}}{\to}\frac{i^{-1}(\mathcal{O}_{U})}{I^{2}_{R,U}}\overset{d}{\to}\frac{i^{-1}(T^{\ast}U)}{I_{R,U}.i^{-1}(T^{\ast}U)}$

One has a decomposition $\mathcal{S}_{X}=\mathcal{S}^{0}_{X}\oplus\mathbb{C}$,
and $\mathcal{S}^{0}_{X}\subset\mathcal{S}_{X}$ is the kernel of the
composition:

(4.2) $\displaystyle\mathcal{S}_{X}\to\mathcal{O}_{X}\to\mathcal{O}_{X^{red}}$

with $X^{red}$ the reduced subspace of $X$. This construction is functorial,
i.e. for $\Phi:X\to Y$ a morphism of scheme there is a sheaf morphism
$\Phi^{\star}:\mathcal{S}^{0}_{Y}\to\mathcal{S}^{0}_{X}$.

A d-critical structure on $X$ is a section $s$ of $\mathcal{S}^{0}_{X}$ such
that for each $x\in X$ there exist an étale neighborhood $R$ of $x$, and
embedding $i:R\hookrightarrow U$ into a smooth algebraic space $U$, and a
regular function $f:U\to\mathbb{C}$ such that $i(R)={\rm Crit}(f)$ and
$\iota_{R,U}(s|_{R})=i^{-1}(f)+I_{R,U}^{2}$. Informally, the data of $s$
precise the functional $f$ of critical charts up to second order terms, and
one has $f|_{R^{red}}=0$. We deal then with critical charts $(R,U,f,i)$:
considering an étale map $U^{\prime}\to U$, one can consider a subchart
$(R^{\prime}:=R\times_{U}U^{\prime},U^{\prime},f^{\prime}:=f|_{U^{\prime}},i^{\prime}:=i|_{R^{\prime}})\to(R,U,f,i)$.

One can also consider embeddings of charts
$\Phi:(R,U,f,i)\hookrightarrow(S,V,g,j)$ for $R\to S\to X$ étale maps, i.e. a
locally closed embedding $\Phi:U\to V$ such that $\Phi\circ i=j|_{R}:R\to V$
and $f=g\circ\Phi:U\to\mathbb{C}$. According to [21, Theo 2.20], one can
compare two critical charts using embeddings: namely, for $(R,U,f,i)$ and
$(S,V,g,j)$ two critical charts and $x\in R\cap S$, there exist subcharts
$(R^{\prime},U^{\prime},f^{\prime},i^{\prime})\subset(R,U,f,i)$ and
$(S^{\prime},V^{\prime},g^{\prime},j^{\prime})\subset(S,V,g,j)$ such that
$x\in R^{\prime}\cap S^{\prime}$, a critical chart $(T,W,h,k)$ and embeddings
$\Phi:(R^{\prime},U^{\prime},f^{\prime},i^{\prime})\hookrightarrow(T,W,h,k)$
and
$\Psi(S^{\prime},V^{\prime},g^{\prime},j^{\prime})\hookrightarrow(T,W,h,k)$.

For a $-1$-shifted symplectic structure, on a d-critical chart $(R,U,f,i)$,
the tangent-obstruction complex $\mathbb{L}_{X}|_{R^{red}}$ is quasi-
isomorphic with $0\to TU\to T^{\ast}U\to 0$, and then
$det(\mathbb{L}_{X})|_{R^{red}}=i^{\ast}(K_{U}^{\otimes 2})|_{R^{red}}$, i.e.
the sheaves $i^{\ast}(K_{U}^{\otimes 2})|_{R^{red}}$ glue in a sheaf on
$X^{red}$ For a general d-critical scheme and an embedding
$\Phi:(R,U,f,i)\to(S,V,g,j)$, from definition 2.26 of [21], one has a natural
isomorphism:

(4.3) $\displaystyle J_{\Phi}:i^{\ast}(K_{U}^{\otimes^{2}})|_{R^{red}}\to
j^{\ast}(K^{\otimes^{2}}_{V})|_{R^{red}}$

Using the fact that two maps can be locally embedded in a single map, it is
shown in [21, Theo 2.28] that these sheaves glue together into a single sheaf
$K_{X,s}$ on $X^{red}$ called the canonical sheaf, with natural local
isomorphisms:

$\displaystyle\iota_{R,U,f,i}:K_{X,s}|_{R^{red}}\to
i^{\ast}(K_{U}^{\otimes^{2}})|_{R^{red}}$ (4.4) $\displaystyle
J_{\Phi}\circ\iota_{R,U,f,i}=\iota_{S,V,g,j}$

###  4.2. Gluing the Donaldson-Thomas sheaves

Given a d-critical scheme, one could try naively to define a perverse sheaf
and strongly polarized monodromic mixed Hodge modules modeled locally on
$\mathcal{PV}_{U,f}$ for each critical charts $(R,U,f,i)$, gluing these
sheaves and strongly polarized monodromic mixed Hodge modules by constructing
isomorphisms on intersections of critical charts, satisfying the cocycle
conditions. This is complicated by the orientations issue seen above. The
construction of $\mathcal{PV}_{U,f}$ is natural with respect to étale
restriction of maps, and étale locally two intersecting critical charts are
related by stabilization i.e. embedding of the form
$\Phi:(R,U,f,i)\hookrightarrow(S,U\times E,g\boxplus q,j\times 0)$, with $E$ a
vector space and $q$ a non-degenerate quadratic form. As seen above, the
descent data to define the orientation $K_{X,s}^{1/2}$ is equivalent to give a
natural orientation on $E$ for any such stabilization. We can then consider
the chain of isomorphisms in ${\rm Perv}(X)$ (or $MMHM(X)$):

(4.5) $\displaystyle\Theta(\Phi):\mathcal{PV}_{U\times E,f\boxplus
q}\simeq\mathcal{PV}_{U,f}\boxtimes\mathcal{PV}_{E,q}\simeq\mathcal{PV}_{U,f}$

Where the first isomorphism is the Thom-Sebastiani isomorphism
$\mathcal{TS}_{U,f,E,q}$, and the second comes from the natural orientation of
$E$. The technical work of [10] is to check that
$(\mathcal{PV}_{U,f},\Theta(\Phi))$ defines a descent data, namely that the
$\Theta(\Phi)$ glue on intersections of critical charts to define comparison
isomorphisms, and that the cocycle relations are verified.

We are here interested in building an isomorphism between perverse sheaves, so
we have to work one categorical level below: namely, we must define these
isomorphisms locally, and check that this isomorphism commutes with the gluing
isomorphisms. We will then use only the above presentation
$(\mathcal{PV}_{U,f},\Theta(\Phi))$ for $P_{X,s}$, and will not use the
equivalent, but more technical, presentation of [10].

###  4.3. Białynicki-Birula decomposition

An action $\mu:T\times X\to X$ of a one dimensional torus $\mathbb{C}^{\ast}$
on a d-critical scheme $X$ is said to leave invariant the d-critical structure
$s$ (resp. the orientation $K_{X,s}^{1/2}$) if $\mu(\gamma)^{\star}(s)=s$
(resp. $\mu(\gamma)^{\ast}K_{X,s}^{1/2}=K_{X,s}^{1/2}$) for $\gamma\in T$. In
particular, the d-critical structure of the classical truncation of a
$(-1)$-shifted symplectic scheme with a $\mathbb{C}^{\ast}$-action leaving the
$(-1)$-shifted symplectic structure invariant is
$\mathbb{C}^{\ast}$-invariant. If the action is étale locally linearizable,
then translating [21, Prop 2.43, 2.44] from the Zariski to the étale topology,
we can then work with $\mathbb{C}^{\ast}$-equivariant critical charts. Namely,
we can cover $X$ in the étale topology by charts $(R,U,f,i)$ such that $U$ has
a $\mathbb{C}^{\ast}$-action for which $i$ is equivariant and $f$ is
invariant. Moreover, considering $(R,U,f,i)$ and $(S,V,g,j)$ two
$\mathbb{C}^{\ast}$-equivariant critical charts, and $x\in R\cap S$, one has
étale restrictions $(R^{\prime},U^{\prime},f^{\prime},i^{\prime})\to(R,U,f,i)$
and $(S^{\prime},V^{\prime},g^{\prime},j^{\prime})\to(S,V,g,j)$ with $x\in
R^{\prime}\cap S^{\prime}$, and $\mathbb{C}^{\ast}$-equivariant critical chart
$(T,W,h,k)$ and $\mathbb{C}^{\ast}$-equivariant embeddings
$\Phi:(R^{\prime},U^{\prime},f^{\prime},i^{\prime})\hookrightarrow(T,W,h,k)$
and
$\Psi(S^{\prime},V^{\prime},g^{\prime},j^{\prime})\hookrightarrow(T,W,h,k)$.

As explained in [21, Cor 2.45], considering the closed embedding
$\xi_{X}:X^{0}\hookrightarrow X$, $X^{0}$ admits a natural d-critical
structure $(X^{0},s^{0}:=\xi_{X}^{\star}(s))$. Indeed, for $(R,U,f,i)$ a
$\mathbb{C}^{\ast}$-equivariant d-critical chart, denoting by $R^{0}$, $U^{0}$
the $\mathbb{C}^{\ast}$-fixed locus of $R$,$U$, and $f^{0}=f|_{U^{0}}$,
$i^{0}=i|_{R^{0}}$, $(R^{0},U^{0},f^{0},i^{0})$ is a d-critical chart of
$X^{0}$ such that $s^{0}_{R^{0},U^{0}}=f^{0}+I^{2}_{R^{0},U^{0}}$, and then
$X^{0}$ is covered by d-critical charts.

Consider now the orientation issue. Suppose first that $(X,s)$ is the
classical truncation of a $-1$-shifted symplectic space with a
$\mathbb{C}^{\ast}$-action. Then the tangent-obstruction complex
$\mathbb{L}_{X}|_{X^{0}}$ splits as a direct sum:

(4.6)
$\displaystyle\mathbb{L}_{X}|_{X^{0}}=\mathbb{L}_{X^{0}}\oplus\mathbb{L}^{+}_{X^{0}}\oplus\mathbb{L}^{-}_{X^{0}}$

Here $\mathbb{L}^{+}_{X^{0}}$ (resp $\mathbb{L}^{-}_{X^{0}}$) denotes the part
of contracting (resp repelling) weight under the $\mathbb{C}^{\ast}$-action.
The $-1$-shifted symplectic structure provides then a canonical isomorphism
between $\mathbb{L}^{+}_{X^{0}}$ and $(\mathbb{L}^{-}_{X^{0}})^{\vee}[1]$,
hence, taking:

$\displaystyle det(\mathbb{L}_{X})|_{X^{0}}=det(\mathbb{L}_{X^{0}})\otimes
det(\mathbb{L}^{+}_{X^{0}})^{2}$ (4.7) $\displaystyle\implies$ $\displaystyle
K_{X^{0},s^{0}}=K_{X,s}\otimes det(\mathbb{L}^{+}_{X^{0}})^{-2}$

Hence, given an orientation $K_{X,s}^{1/2}$ on $X$, we can define canonically
an orientation on $X^{0}$:

(4.8) $\displaystyle K_{X^{0},s^{0}}^{1/2}:=K_{X,s}^{1/2}\otimes
det(\mathbb{L}^{+}_{X^{0}})^{-1}$

Such construction is still possible if $(X,s)$ is a d critical space with
$\mathbb{C}^{\star}$ action, not necessarily coming from a $-1$-shifted
symplectic space:

######  Lemma 4.1.

If $(X,s)$ is a d critical space with $\mathbb{C}^{\star}$ action, there is a
natural bijection between orientations of $(X,s)$ and orientations of
$(X^{0},s^{0})$.

Proof: We can define an orientation $K_{X,s}^{1/2}$ (resp
$K_{X^{0},s^{0}}^{1/2}$) by descent: we define $(K_{X,s}^{1/2})_{R}:=K_{U}$
(resp $(K_{X^{0},s^{0}}^{1/2})_{R^{0}}:=K_{U^{0}}$) for each critical chart
$(R,U,f,i)$ (resp $(R^{0},U^{0},f^{0},i^{0})$, and gluing isomorphisms on
double intersections which satisfies cocycle on triple intersections. Using
[21, Prop 2.43], we can use $\mathbb{C}^{\ast}$-equivariant critical charts,
and, using [21, Prop 2.44] and Lemma 5.1, it suffice to define gluing
isomorphisms for $\mathbb{C}^{\ast}$-equivariant embeddings of charts of the
form $(R,U,f,i)\hookrightarrow(R,U\times E,f\boxplus q,i\times 0)$, where $E$
is a finite dimensional vector space with linear $\mathbb{C}^{\ast}$ action
and $q$ is a $\mathbb{C}^{\ast}$-invariant non-degenerate quadratic form. As
seen above (3.5), an orientation on $E$, which defines a descent isomorphism
for $K^{1/2}_{X,s}$ for the embedding $(R,U,f,i)\hookrightarrow(R,U\times
E,f\boxplus q,i\times 0)$, provides then naturally an orientation of $V^{0}$,
which is a descent isomorphism for $K_{X^{0},s^{0}}^{1/2}$ and the embedding
$(R^{0},U^{0},f^{0},i^{0})\hookrightarrow(R^{0},U^{0}\times
E^{0},f^{0}\boxplus q^{0},i^{0}\times 0)$. The isomorphisms for
$K_{X^{0},s^{0}}^{1/2}$ glue on triple intersections if and only if those for
$K_{X,s}^{1/2}$ glue also. $\Box$

Consider the decomposition of $X^{0}$ into connected components
$X^{0}=\bigsqcup_{\pi\in\Pi}X^{0}_{\pi}$: one can further consider the
oriented d-critical schemes
$(X^{0}_{\pi},s^{0}_{\pi},K^{1/2}_{X^{0}_{\pi},s^{0}_{\pi}})$. Denote by ${\rm
Ind}_{\pi}$ the number of contracting weight in the tangent-obstruction
complex of $X$ at $X^{0}_{\pi}$. In a critical chart $(R,U,f,i)$, the tangent-
obstruction complex is given by $0\to TU\to T^{\ast}U\to 0$, and then,
denoting by $d_{+},d_{0},d_{-}$ respectively the number of contracting,
invariant and repelling weights in $TU$ at $i(X^{0}_{\pi}\cap R)$, one has
${\rm Ind}_{\pi}=d_{+}-d_{-}$, i.e. this definition is consistent with the
previous definition on a critical chart.

######  Theorem 4.2.

For $X$ an oriented d-critical algebraic space $X$ with an étale linearizable
$\mathbb{C}^{\ast}$ action leaving the d-critical structure and the
orientation invariant, one has natural isomorphisms of perverse sheaves and
monodromic mixed Hodge modules on $X^{0}$ (with its orientation defined in
Lemma 4.1):

(4.9)
$\displaystyle\beta^{\pm}_{X,s}:(p_{X}^{\pm})_{!}(\eta_{X}^{\pm})^{\ast}P_{X,s}\overset{\simeq}{\to}\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}P_{X^{0}_{\pi},s^{0}_{\pi}}$

which are compatible with polarization, in the sense that the following
diagram is commutative:

${(p_{X}^{+})_{!}(\eta_{X}^{+})^{\ast}P_{X,s}}$${\bigoplus_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}P_{X^{0}_{\pi},s^{0}_{\pi}}}$${\mathbb{D}(p_{X}^{-})_{!}(\eta_{X}^{-})^{\ast}P_{X,s}}$${\mathbb{D}\bigoplus_{\pi\in\Pi}\mathbb{L}^{-{\rm
Ind}_{\pi}/2}P_{X^{0}_{\pi},s^{0}_{\pi}}}$$\scriptstyle{\beta^{+}_{X,s}}$$\scriptstyle{S_{X}^{-1}\circ(p_{X}^{+})_{!}(\eta_{X}^{+})^{\ast}(\Sigma_{X,s})}$$\scriptstyle{\bigoplus_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}\Sigma_{X^{0}_{\pi},s^{0}_{\pi}}}$$\scriptstyle{\mathbb{D}\beta^{-}_{X,s}}$

Proof: We can cover $X^{0}$ by critical charts of the form
$(R^{0},U^{0},f^{0},i^{0})$, which are the $\mathbb{C}^{\ast}$-fixed part of
$\mathbb{C}^{\ast}$-equivariant charts $(R,U,f,i)$, and the relate them by the
transitive action of stabilization, i.e. embedding of the form
$\Phi^{0}:(R^{0},U^{0},f^{0},i^{0})\hookrightarrow(R^{0},U^{0}\times
E^{0},f^{0}\boxplus q^{0},i^{0}\times 0)$, which are the
$\mathbb{C}^{\ast}$-fixed part of $\mathbb{C}^{\ast}$-equivariant embedding
$\Phi:(R,U,f,i)\hookrightarrow(R,U\times E,f\boxplus q,i\times 0)$. Hence
${\rm Perv}(X^{0})$ (or $MMHM(X^{0})$) can be defined by descent datum with
these charts and embeddings. Namely, $P_{X^{0},s^{0}}$ is defined as said
before by the descent datum $(\mathcal{PV}_{U,f},\Theta(\Phi^{0}))$. Because
hyperbolic localization commutes with étale restriction,
$(p_{X}^{\pm})_{!}(\eta_{X}^{\pm})^{\ast}P_{X,s}$ can be defined by the
descent datum
$((p_{R}^{\pm})_{!}(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f},(p_{R}^{\pm})_{!}(\eta_{R}^{\pm})^{\ast}\Theta(\Phi))$.

On a chart $(R^{0},U^{0},f^{0},i^{0})$, we consider the isomorphism built in
Proposition 3.3:

(4.10)
$\displaystyle\beta^{\pm}_{U,f}:(p_{R}^{\pm})_{!}(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f}\to\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}}$

We have to check he compatibility of these isomorphisms with stabilization.
Consider as above
$\Phi^{0}:(R^{0},U^{0},f^{0},i^{0})\hookrightarrow(R^{0},U^{0}\times
E^{0},f^{0}\boxplus q^{0},i^{0}\times 0)$, and consider the natural
orientation on $E$ and $E^{0}$ defined by $K_{X,s}^{1/2}$ and
$K_{X^{0},s^{0}}^{1/2}$. consider now the following diagram:

$(p_{R}^{\pm})|_{!}(\eta_{R}^{\pm})^{\ast}(\mathcal{PV}_{U\times E,f\boxplus
q})$ $\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm({\rm
Ind}_{\pi}}\mathcal{PV}_{U^{0}_{\pi}\times E^{0},f^{0}_{\pi}\boxplus q^{0}}$
$((p_{R}^{\pm})_{!}(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f})\boxtimes(\mathcal{PV}_{E,q})$
$\bigoplus_{\pi\in\mathbb{Z}}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}})\boxtimes\mathcal{PV}_{E^{0},q^{0}})$
${(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f})}$${\bigoplus_{\pi\in\mathbb{Z}}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}})}$$\scriptstyle{\beta^{\pm}_{U\times
E}}$
$\scriptstyle(p_{R}^{\pm})|_{!}(\eta_{R}^{\pm})^{\ast}\mathcal{TS}_{U,f,E,q}$
$\scriptstyle\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm({\rm
Ind}_{\pi})/2}\mathcal{TS}_{U^{0}_{\pi},f^{0}_{\pi},E^{0},q^{0}}$
$\scriptstyle{\beta^{\pm}_{U}\times\beta^{\pm}_{E}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\beta^{\pm}_{U}}$

The above square is the commutative square of Proposition 3.5 expressing the
commutation of hyperbolic localization with Thom-Sebastiani isomorphism, and
the square below is the tensor product of the commutative square 3.5 with
$(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f})$. Hence the above diagram is
commutative, and is equal to:

$(p_{R}^{\pm})|_{!}(\eta_{R}^{\pm})^{\ast}(\mathcal{PV}_{U\times E,f\boxplus
q})$ $\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm({\rm
Ind}_{\pi}}\mathcal{PV}_{U^{0}_{\pi}\times E^{0},f^{0}_{\pi}\boxplus q^{0}}$
${(\eta_{R}^{\pm})^{\ast}\mathcal{PV}_{U,f})}$${\bigoplus_{\pi\in\mathbb{Z}}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}\mathcal{PV}_{U^{0}_{\pi},f^{0}_{\pi}})}$$\scriptstyle{\beta^{\pm}_{U\times
E}}$ $\scriptstyle(p_{R}^{\pm})|_{!}(\eta_{R}^{\pm})^{\ast}\Theta(\Phi)$
$\scriptstyle\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm({\rm
Ind}_{\pi})/2}\Theta(\Phi^{0}_{\pi})$ $\scriptstyle{\beta^{\pm}_{U}}$

Hence the $\beta^{\pm}_{U}$ are compatible with stabilization, and then by
descent glue to give an isomorphism on $X^{0}$. for each
$\mathbb{C}^{\ast}$-equivariant critical chart $(R,U,f,i)$ compatible with the
monodromy and the duality. From Proposition 3.3, the $\beta^{\pm}_{U}$ are
compatible with polarization and monodromy, hence by descent $\beta^{\pm}_{X}$
are also compatible (the commutativity of the corresponding squares can be
checked locally). $\Box$

######  Corollary 4.3.

We define the cohomological DT invariant $[X]^{vir}:=[H_{c}(X,P_{X,s})]$ as
the class of the hypercohomology with compact support of $P_{X,s}$ in the
Grothendieck group of monodromic mixed Hodge structures. In the context of
Theorem 4.2, one has:

(4.11) $\displaystyle[X]^{vir}=\sum_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}[X^{0}_{\pi}]^{vir}+[X-\eta(X^{+})]^{vir}$

Proof: $(X-\eta(X^{+})\cup\bigcup_{\pi\in\Pi}X^{+}_{\pi}$ provides a
stratification of $X$ by locally closed subvariety, hence one has:

$\displaystyle[H_{c}(X,P_{X,s})]^{vir}$
$\displaystyle=[H_{c}(X^{+},\eta^{\ast}P_{X,s})]+[H_{c}(X-\eta(X^{+}),P_{X,s})]$
$\displaystyle=[H_{c}(X^{0},p_{!}\eta^{\ast}P_{X,s})]+[H_{c}(X-\eta^{\ast}(X^{+}),P_{X,s})]$
$\displaystyle=\bigoplus_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}[H_{c}(X^{0},P_{X^{0},s^{0}})]+[H_{c}(X-\eta^{\ast}(X^{+}),P_{X,s})]$
(4.12) $\displaystyle=\sum_{\pi\in\Pi}\mathbb{L}^{{\rm
Ind}_{\pi}/2}[X^{0}_{\pi}]^{vir}+[X-\eta(X^{+})]^{vir}$

Here the first line holds because the class in the Grothendieck of monodromic
mixed Hodge structures is a motivic invariant, the second line follows from
the six functor formalism by viewing the cohomology with compact support as
the proper pushforward to a point, the third line from the main result of this
paper, and the last line by the definition of the cohomological DT invariants.
$\Box$

###  4.4. Compatibility with smooth pullbacks

As mentioned in the subsection 1, the result of this subsection, the
compatibility of the localization isomorphism with smooth pullbacks, is useful
if one want to extends our result to DT theory on Artin stacks, following the
approach of [7]. This was done in an earlier version of the paper, but removed
here for clarity, but we have kept this subsection for future reference.

Consider a smooth map of d-critical algebraic spaces $\phi:(X,s)\to(Y,t)$ of
relative dimension $d$ (i.e. a smooth map such that $\phi^{\star}(t)=s$). [7,
Cor 3.8] shows that:

(4.13) $\displaystyle
K_{X,s}=K_{Y,t}\otimes(\Lambda^{top}T^{\ast}_{X/Y})^{\otimes 2}|_{X^{red}}$

A smooth map of oriented d-critical scheme
$\phi:(X,s,K_{X,s}^{1/2})\to(Y,t,K_{Yt}^{1/2})$ is then the data a smooth map
$\phi:(X,s)\to(Y,t)$ of oriented d-critical scheme together with the data of
an isomorphism:

(4.14) $\displaystyle
K^{1/2}_{X,s}=\phi^{\ast}(K^{1/2}_{Y,t})\otimes(\Lambda^{top}T^{\ast}_{X/Y})|_{X^{red}}.$

which is a square root of the above isomorphism. Given such a map, [7, Prop
4.5] built an isomorphism:

(4.15) $\displaystyle\Delta_{\phi}:\phi^{\ast}[d]P_{Y,t}\simeq P_{X,s}$

Consider now a $\mathbb{C}^{\ast}$-equivariant smooth map of oriented
d-critical schemes $\phi:(X,s,K_{X,s}^{1/2})\to(Y,t,K_{Yt}^{1/2})$ of relative
dimension $d$. From Lemma 5.2, $X$ is covered by
$\mathbb{C}^{\ast}$-equivariant critical charts $(R,U,f,i)$ such that there is
a $\mathbb{C}^{\ast}$-equivariant critical chart and $(S,V,g,j)$ of $Y$ with a
smooth map $\Phi:U\to V$ of relative dimension $d$ such that $f=g\circ\Phi$
and $\Phi\circ i=j\circ\phi$. Hence sheaves and perverse sheaves on $X^{0}$
can be defined by descent with a definition on each critical chart
$(R^{0},U^{0},f^{0},i^{0})$ coming from a chart of this form, and comparison
on charts coming from simultaneous stabilization of $(S,V,g,j)$ and
$(R,U,f,i)$.

######  Lemma 4.4.

If $\phi:(X,s,K_{X,s}^{1/2})\to(Y,s,K_{Y,t}^{1/2})$ is a smooth
$\mathbb{C}^{\ast}$ morphism of oriented d-critical loci, then
$\phi^{0}:(X^{0},s^{0},K_{X^{0},s^{0}}^{1/2})\to(Y^{0},t^{0},K_{Y^{0},t^{0}}^{1/2})$
is aloso a smooth morphism of oriented d-critical loci.

Proof: The induced morphism $\phi^{0}:X^{0}\to Y^{0}$ is smooth of dimension
$d$. One has:

$\displaystyle(\phi^{0})^{\star}(t^{0})$
$\displaystyle=(\phi^{0})^{\star}\xi_{Y}^{\star}(t)$
$\displaystyle=\xi_{X}^{\star}\phi^{\star}(t)$
$\displaystyle=\xi_{X}^{\star}(s)$ (4.16) $\displaystyle=s^{0}$

where the first and the last line are the definition of $t^{0}$ and $s^{0}$,
the second line follows from the functoriality of ⋆ and the fact that
$\xi_{Y}\phi^{0}=\phi\xi_{X}$, and the third line follows from the fact that
$\phi$ is a morphism of d-critical structure. Then $\phi^{0}$ is a morphism of
d-critical algebraic space.

We will now prove that:

(4.17) $\displaystyle
K_{X^{0},s^{0}}^{1/2}=(\phi^{0})^{\ast}(K_{Y^{0},t^{0}}^{1/2})\otimes(\Lambda^{top}T^{\ast}_{X^{0}/Y^{0}})|_{(X^{0})^{red}}$

We prove this equality by descent: this equality holds by the definitions
$K_{X^{0},s^{0}}^{1/2}|_{R^{0}}=K_{U^{0}}$ and
$K_{Y^{0},t^{0}}^{1/2}=K_{V^{0}}$ for each pair of critical charts of $X,Y$ of
the form described above. Now, considering a stabilization for a pair of
critical charts of $X,Y$, the compatibility of the descent isomorphisms on $X$
with (4.14) implies the compatibility of the descent isomorphisms on $X^{0}$
with (4.17), which proves (4.17). $\Box$

We can in particular consider the isomorphism:

(4.18) $\displaystyle\Delta_{\phi^{0}}:(\phi^{0})^{\ast}[d]P_{Y^{0},t^{0}}\to
P_{X^{0},s^{0}}$

######  Proposition 4.5.

The hyperbolic localization isomorphism commutes with smooth pullbacks, namely
for $\phi:(X,s,K_{X,s}^{1/2})\to(Y,t,K_{Y,t}^{1/2})$ a
$\mathbb{C}^{\ast}$-equivariant smooth map of oriented critical algebraic
space of relative dimension $d$, the following diagram of isomorphisms
commutes:

${(\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}(\phi^{0}_{\pi})^{\ast}[d^{0}_{\pi}](p^{\pm}_{Y})_{!}(\eta^{\pm}_{Y})^{\ast}P_{Y,t})|_{R^{0}}}$${\bigoplus_{\pi,\pi^{\prime}\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi^{\prime}}/2}(\phi^{0}_{\pi})^{\ast}[d^{0}_{\pi}]P_{Y^{0}_{\pi^{\prime}-\pi},t^{0}_{\pi^{\prime}-\pi}}}$${((p^{\pm}_{X})_{!}(\eta^{\pm}_{X})^{\ast}\phi^{\ast}[d]P_{Y,t}))|_{R^{0}}}$${((p^{\pm}_{X})_{!}(\eta^{\pm}_{X})^{\ast}P_{X,s})|_{R^{0}}}$${\bigoplus_{\pi^{\prime}\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi^{\prime}}/2}P_{X^{0}_{\pi^{\prime}},s^{0}_{\pi^{\prime}}}}$$\scriptstyle{\bigoplus_{\pi\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi}/2}(\phi^{0}_{\pi})^{\ast}{[}d^{0}_{\pi}{]}\beta^{\pm}_{Y,t}}$$\scriptstyle{\bigoplus_{\pi,\pi^{\prime}\in\Pi}\mathbb{L}^{\pm{\rm
Ind}_{\pi^{\prime}}/2}\Delta_{\phi^{0}_{\pi}}}$$\scriptstyle{\Delta_{\phi}}$$\scriptstyle{\simeq}$$\scriptstyle{\beta^{\pm}_{X,s}}$

Proof: It suffices to show that this square commutes étale locally on $X^{0}$.
According to lemma 5.2, we can then choose $\mathbb{C}^{\ast}$-equivariant
critical charts $(R,U,f,i)$ and $(S,V,g,j)$ of $X$ and $Y$ containing $x$ and
$y$ with a smooth map $\Phi:U\to V$ of relative dimension $d$ such that
$f=g\circ\Phi$ and $\Phi\circ i=j\circ\phi$. On such charts, the commutation
of the proposition is given by Proposition 3.4. $\Box$

##  5\. Technical lemmas

###  5.1. Trivialization of torus-equivariant embeddings

######  Lemma 5.1.

Let $\Phi:(R,U,f,i)\to(S,V,g,j)$ be a $\mathbb{C}^{\ast}$-equivariant
embedding of $\mathbb{C}^{\ast}$-equivariant critical charts. For $x\in
i(R^{0})$, there are smooth $\mathbb{C}^{\ast}$-equivariant algebraic space
$U^{\prime},V^{\prime}$, $x^{\prime}\in U^{\prime}$,
$\mathbb{C}^{\ast}$-equivariant morphisms $\iota:U^{\prime}\to U$ with
$\iota(x^{\prime})=x$, $\jmath:V^{\prime}\to V$, $\Phi^{\prime}:U^{\prime}\to
V^{\prime}$, $\alpha:V^{\prime}\to U$ and $\beta:V^{\prime}\to E$ with $E$ a
vector space with $\mathbb{C}^{\ast}$ action, with $\iota$, $\jmath$ and
$\alpha\otimes\beta$ being étale, such that the following square commutes:

${U}$${U^{\prime}}$${U}$${V}$${V^{\prime}}$${U\times
E}$$\scriptstyle{\Phi}$$\scriptstyle{\iota}$$\scriptstyle{\iota}$$\scriptstyle{\Phi^{\prime}}$$\scriptstyle{{\rm
Id}_{U}\times 0}$$\scriptstyle{\jmath}$$\scriptstyle{\alpha\times\beta}$

and $g\circ\jmath=f\circ q\circ\alpha:V^{\prime}\to\mathbb{C}$, with $q$ a
non-degenerate $\mathbb{C}^{\ast}$-invariant quadratic form on $E$.

Proof: We follow here closely the proof of [21, Prop 2.22, 2.23], adapting it
to the equivariant case: we take care to consider at each step only étale
cover with $\mathbb{C}^{\ast}$ action, and to keep
$\mathbb{C}^{\ast}$-equivariant functions and coordinates. The map $\Phi:U\to
V$ is a $\mathbb{C}^{\ast}$-equivariant embedding of smooth schemes of
respective dimension $\dim(U)=m,\dim(V)=m+n$. We can then find
$\mathbb{C}^{\ast}$-equivariant étale coordinates
$(\dot{y}_{1},...,\dot{y}_{m},\dot{z}_{1},...,\dot{z}_{n})$ on a
$\mathbb{C}^{\ast}$-equivariant Zariski open neighborhood $\dot{V}$ of $j(x)$
in $V$ such that $j(x)=(0,...,0)$ and $\Phi(U)\cap\dot{V}$ is the locus
$\dot{z}_{1}=...=\dot{z}_{n}=0$ in $\dot{V}$. Set $\dot{U}=\Phi^{-1}(\dot{V})$
and $\dot{x}_{a}=\dot{y}_{a}\circ\Phi|_{\dot{U}}$ for $a=1,...,m$. Then
$\dot{U}$ is an open neighborhood of $i(x)$ in $U$ , and
$(\dot{x}_{1},...,\dot{x}_{m})$ are étale coordinates on $\dot{U}$ with
$i(x)=(0,...,0)$. Then the ideal $I_{R,U}=I_{(df)}$ is on $\dot{U}$ the ideal
generated by the $\mathbb{C}^{\ast}$-equivariant functions $\frac{\partial
f}{\partial\dot{x}_{a}}$ for $a=1,...,m$, and the ideal $I_{S,V}=I_{(dg)}$ is
on $\dot{V}$ the ideal generated by the $\mathbb{C}^{\ast}$-equivariant
functions $\frac{\partial g}{\partial\dot{y}_{a}}$ for $a=1,...,m$ and
$\frac{\partial g}{\partial\dot{z}_{b}}$ for $b=1,...,n$. Since $\Phi$ maps
$U$ to $\dot{z}_{1}=...=\dot{z}_{n}=0$ and $i(R)$ to $j(R)\subset j(S)$, we
have $I_{(df)}\simeq I_{(dg)}|_{\dot{z}_{1}=...=\dot{z}_{n}=0}$ , that is:

$\displaystyle(\frac{\partial
f}{\partial\dot{x}_{a}}(\dot{y}_{1},...,\dot{y}_{m}):a=1,...,m)=($
$\displaystyle\frac{\partial
g}{\partial\dot{y}_{a}}(\dot{y}_{1},...,\dot{y}_{m},0,...,0):a=1,...,m,$ (5.1)
$\displaystyle\frac{\partial
g}{\partial\dot{z}_{b}}(\dot{y}_{1},...,\dot{y}_{m},0,...,0):b=1,...,n)$

this holds provided each $\frac{\partial
g}{\partial\dot{z}_{b}}(\dot{y}_{1},...,\dot{y}_{m},0,...,0)$ lies in
$(\frac{\partial
g}{\partial\dot{y}_{a}}(\dot{y}_{1},...,\dot{y}_{m},0,...,0):a=1,...,m$. Thus,
making $\dot{U}$, $\dot{V}$ smaller if necessary, we can suppose there exist
étale locally $\mathbb{C}^{\ast}$-equivariant functions
$A_{ab}(\dot{y}_{1},...,\dot{y}_{m})$ on $\dot{U}$ for $a=1,...,m,b=1,...,n$
such that for each b:

(5.2) $\displaystyle\frac{\partial
g}{\partial\dot{z}_{b}}(\dot{y}_{1},...,\dot{y}_{m},0,...,0)=\sum_{a=1}^{m}A_{ab}(\dot{y}_{1},...,\dot{y}_{m})\frac{\partial
g}{\partial\dot{y}_{a}}(\dot{y}_{1},...,\dot{y}_{m},0,...,0)$

Defining the $\mathbb{C}^{\ast}$-equivariant functions
$\tilde{y}_{a}=\dot{y}_{a}-\sum_{a=1}^{m}A_{ab}(\dot{y}_{1},...,\dot{y}_{m})\dot{z}_{b}$,
and $\tilde{z}_{b}=\dot{z}_{b}$, they give $\mathbb{C}^{\ast}$-equivariant
étale coordinates on a $\mathbb{C}^{\ast}$-invariant neighborhood $\tilde{V}$
of $j(x)$. We also define the $\mathbb{C}^{\ast}$-invariant subspace
$\tilde{U}=\Phi^{-1}(\tilde{V})$, and define on it the
$\mathbb{C}^{\ast}$-equivariant étale coordinates
$\tilde{x}=\dot{x}|_{\tilde{U}}$: then
$\tilde{y}_{a}\circ\Phi|_{\tilde{U}}=\tilde{x}_{a}$ and
$\tilde{z}_{b}\circ\Phi|_{\tilde{U}}=0$. Then:

$\displaystyle\frac{\partial
g}{\partial\tilde{z}_{b}}(\tilde{y}_{1},...,\tilde{y}_{m},0,...,0)=$
$\displaystyle\sum_{a=1}^{m}\frac{\partial
g}{\partial\dot{y}_{a}}(\dot{y}_{1},...,\dot{y}_{m},0,...,0).\frac{\partial\dot{y}_{a}}{\partial\tilde{z}_{b}}+\sum_{c=1}^{n}\frac{\partial
g}{\partial\dot{z}_{c}}(\dot{y}_{1},...,\dot{y}_{m},0,...,0).\frac{\partial\dot{z}_{c}}{\partial\tilde{z}_{b}}$
$\displaystyle=$ $\displaystyle\sum_{a=1}^{m}\frac{\partial
g}{\partial\dot{y}_{a}}(\dot{y}_{1},...,\dot{y}_{m},0,...,0)(-A_{ab}(\dot{y}_{1},...,\dot{y}_{m}))$
$\displaystyle+\sum_{c=1}^{n}(\sum_{a=1}^{n}A_{ac}(\dot{y}_{1},...,\dot{y}_{m})).\frac{\partial