# Hyperbolic localization of the Donaldson-Thomas sheaf

Pierre Descombes

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Pierre Descombes. Hyperbolic localization of the Donaldson-Thomas sheaf. 2023. hal-03976357

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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 scheme or stack with a $C^*$ action leaving the $-1$-shifted symplectic form invariant. This includes the moduli stack of sheaves or complexes of sheaves on a Calabi-Yau threefold with a $C^*$-invariant Calabi-Yau form, or the intersection of two $C^*$-invariant Lagrangians in a symplectic space with a $C^*$-invariant symplectic form. In this case we express the restriction of the Donaldson-Thomas perverse sheaf defined by Joyce et al. to the attracting variety as a sum of cohomological shifts of the DT perverse sheaves on the $C^*$ fixed components. This result can be seen as a $-1$-shifted version of the Białynicki-Birula decomposition for smooth schemes.

Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>2</td><td>Classical Białynicki-Birula decomposition</td><td>9</td></tr><tr><td>2.1</td><td>Hyperbolic localization</td><td>9</td></tr><tr><td>2.2</td><td>Compatibility with smooth morphisms</td><td>12</td></tr><tr><td>3</td><td>Critical Białynicki-Birula decomposition</td><td>15</td></tr><tr><td>3.1</td><td>The functor of vanishing cycles</td><td>15</td></tr><tr><td>3.2</td><td>Functoriality of the vanishing cycles</td><td>17</td></tr><tr><td>3.3</td><td>Hyperbolic localization of the perverse sheaf of vanishing cycles</td><td>23</td></tr><tr><td>3.4</td><td>Compatibility with smooth maps</td><td>27</td></tr><tr><td>3.5</td><td>Compatibility with Thom-Sebastiani isomorphism</td><td>29</td></tr><tr><td>4</td><td>Białynicki-Birula decomposition on a d-critical algebraic space</td><td>31</td></tr><tr><td>4.1</td><td>D-critical algebraic space</td><td>31</td></tr><tr><td>4.2</td><td>Gluing the Donaldson-Thomas sheaves</td><td>32</td></tr><tr><td>4.3</td><td>Białynicki-Birula decomposition</td><td>34</td></tr><tr><td>4.4</td><td>Compatibility with smooth pullbacks</td><td>41</td></tr></table>
cases 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):

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

An example where this procedure can be carried out completely is given by framed invariants of toric quivers, as studied in [1]. When $X$ is projective, or more generally when each point of $X$ is in the attracting locus of a fixed component (one says that the $\mathbb{C}^*$-action is circle-compact), we obtain a formula similar to the localization formula for K-theoretic invariants:

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

## The Donaldson-Thomas perverse sheave

Donaldson-Thomas theory was first designed to count sheaves on Calabi-Yau threefolds. In [2], 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 Serre Duality on the Ext spaces of the sheaves. In [3], 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 number weighted at each point by the Milnor number. In [4] and [5], Kontsevich and Soibelman have 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 [6], [7], [8] and [9], Joyce and collaborators have developed a rigorous cohomological Donaldson-Thomas theory, using the language of $[-1]$ shifted symplectic structures in derived geometry introduced in [10]. 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 [10], it was shown that moduli stack of complexes on a Calabi-Yau 3-fold, and intersections of two Lagrangians in a symplectic space, carry naturally a $[-1]$-shifted symplectic structure.

In [7] and [9], it was shown that $[-1]$-shifted derived schemes (resp. stacks) can be written 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, $f: U \to \mathbb{C}$ a regular map, and $i: R \to U$ a closed embedding such that $i(R) = \mathrm{Crit}(f)$, and there is a way to compare intersecting critical charts. More formally, such a scheme (resp. stack) carries a d-critical structure $s$, a notion that we will introduce below. In [8], Joyce and collaborators has constructed the Donaldson-Thomas perverse sheaf $P_{X,s}$ carrying a monodromic mixed Hodge module for a d-critical scheme or stack $(X, s)$, with extra data called orientation $K_X^{1/2}$. Its restriction to a critical $(R, U, f, i)$ is given by:

$$P_{X,s}|_{R} := \mathcal{P}\mathcal{V}_{U,f} \otimes_{\mathbb{Z}/2\mathbb{Z}} Q_{R,U,f,i} \quad (1.5)$$
where $PV_{U,f}$ is the perverse sheaf (or monodromic mixed Hodge module) of vanishing cycles defined by applying the vanishing cycles functor $\phi_f$ of $f$ to $\mathcal{I}C_U$, the intersection cohomology complex of $U$ with its natural mixed Hodge module structure, and then restricting to $R$; and $Q_{R,U,f,i}$ is a $\mathbb{Z}/2\mathbb{Z}$-principal bundle giving the difference between the orientation of $U$ and the canonical orientation given by $K_{X,s}^{1/2}$. The numerical Donaldson-Thomas invariant defined in [2] is then, as expected, the Euler number of the cohomology of this perverse sheaf. We will use the formalism of d-critical structures in the present article, but we shall not use the formalism of derived geometry and shifted symplectic structures.

## Hyperbolic localization

The aim of this work is to provide a way to compute the cohomological Donaldson-Thomas invariants by localization, namely, given a d-critical scheme (or stack) $X$ with a $\mathbb{C}^*$-action and a $\mathbb{C}^*$-invariant d-critical structure, express the invariants of $X$ in terms of the invariants of the $\mathbb{C}^*$-fixed scheme (or stack) $X^0$. Graber and Pandharipande have proven a torus localization formula for numerical Donaldson-Thomas invariants in [11]. An analogue formula was derived in [12] using the alternative definition as weighted Euler numbers: the numerical Donaldson-Thomas invariants of $X$ are the sum of those of each components of $X^0$ weighted by a sign given by the parity of the dimension of the normal space to the component. Hence numerical Donaldson-Thomas invariants localize under torus action exactly like the Euler numbers of smooth spaces.

In [13], Białynicki-Birula proved that, on a smooth scheme $X$ with $\mathbb{C}^*$ action, the attracting subset of $X$ (i.e. the subset of points $x \in X$ such that $\lim_{t \to 0} t \cdot x$ exists) admits a sort 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 $\mathbb{C}^*$-weights in the tangent space of $X^0$. Hence one can compute the cohomology of the attracting subset of $X$ in terms of the cohomology of $X^0$. In [14] and [15], Braden and Drinfeld gave a functorial reformulation of the Białynicki-Birula decomposition, which can be extended to singular algebraic spaces or stacks. For any algebraic space (resp. stack) $X$ with $\mathbb{C}^*$-action one can define the $\mathbb{C}^*$-fixed space (resp. stack) $X^0$ and also the contracting (resp. repelling) space (resp. stack) $X^\pm$ using the functor of points in an obvious way. There is then the so-called hyperbolic localization diagram:

$$X \xleftarrow{\eta_X^\pm} X^\pm \xrightarrow{p_X^\pm} X^0$$

where $\eta_X^\pm$ is the inclusion of the attracting (resp. repelling) subset, and $p_X^+$ (resp. $p_X^-$) sends $x$ to $\lim_{t \to 0} x$ (resp. $\lim_{t \to \infty} x$). This diagram is easy to describe when $X$ is an affine scheme: therefore it suffices to find $\mathbb{C}^*$-equivariant covering of $X$ by affine spaces. The existence of such covering is not obvious at all, and is far too restrictive in the Zariski topology. Fortunately, a deep result of [16] and [17] shows that quasi-separated algebraic spaces locally of finite type have a $\mathbb{C}^*$-equivariant étale cover by affine spaces. For an algebraic stack $\mathcal{X}$ with a $\mathbb{C}^*$-action, consider $\mathcal{Y} := [\mathcal{X}/T]$; for $x \in \mathcal{X}$, considering the corresponding point $y \in \mathcal{Y}$, there is an exact sequence:

$$1 \longrightarrow G_x \longrightarrow G_y \longrightarrow T_x \longrightarrow 1$$

where $G_x, G_y$ are the stabilizer groups of $x, y$ and $T_x \subset T$ is the subgroup fixing $x$. Now according to [16], a quasi-separated algebraic stack locally of finite type with affine stabilizers, such that the above exact sequence is split for all $x \in \mathcal{X}$, has a smooth $\mathbb{C}^*$-equivariant
presentation from an affine scheme with $\mathbb{C}^*$-action. Under those assumptions, $X^0$ and $X^\pm$ are then algebraic spaces (resp. stacks). The functors $(p_X^\pm)!(\eta_X^\pm)^*$ are called hyperbolic localization functors. From [15] there is a natural morphism build from morphisms of the six-functor formalism in the derived category of constructible sheaves, or the derived category of monodromic mixed Hodge modules:

$$S_X : \mathbb{D}(p_X^-)!(\eta_X^-)^*\mathbb{D} \to (p_X^+)_!(\eta_X^+)^* \quad (1.6)$$

This is by [15] and [18] an isomorphism when those functors are applied to $\mathbb{C}^*$-equivariant complexes, i.e. complexes equivariant under the action of $\mathbb{C}^*$, in a sense which we will explain below.

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

$$ \begin{gathered} (p_X^\pm)_!(\eta_X^\pm)^*\mathcal{Q}_U = \bigoplus_{\pi \in \Pi} \mathbb{L}^{d_\pi^\pm} \mathcal{Q}_{U_\pi^0} \\ \Rightarrow (p_X^\pm)_!(\eta_X^\pm)^*\mathcal{I}\mathcal{C}_U = \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_\pi/2} \mathcal{I}\mathcal{C}_{U_\pi^0} \end{gathered} \quad (1.7) $$

where $\mathcal{Q}_Y$ and $\mathcal{I}\mathcal{C}_Y$ 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 $\mathbb{C}^*$-equivariant oriented d-critical scheme or stack $X$ satisfying the assumptions under which the hyperbolic localization functors were defined, we will prove in section 4.3.1 that $X^0$ carries 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 (1.7) holds for the Donaldson-Thomas perverse sheaf (resp. monodromic mixed Hodge module) on a d-critical oriented algebraic space (theorem 4.1) or stack (theorem 5.3):

**Theorem 1.1.** For $X$ a quasi-separated locally of finite type algebraic space (resp. a quasi-separated locally of finite type algebraic stack with affine stabilizers such that the exact sequence 1 is split for each $x \in X$) with an action of a one dimensional torus $\mathbb{C}^*$ and $\mathbb{C}^*$-equivariant d-critical structure $s$ and orientation, there are natural isomorphisms of perverse sheaves and monodromic mixed Hodge modules:

$$ \beta_{X,s}^{\pm} : (p_X^{\pm})_!(\eta_X^{\pm})^* P_{X,s} \xrightarrow{\sim} \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} P_{X_{\pi}^0, s_{\pi}^0} \quad (1.8) $$

where $\mathrm{Ind}_\pi$ denotes the signed number of contracting weights in the $\mathbb{C}^*$ action on the tangent obstruction complex of $X$ at $X_\pi^0$.
## Sketch of the proof

The main ingredient of the proof of this theorem is the commutation of the hyperbolic localization and vanishing cycles functor proven in [18]. On a critical chart $(R, U, f, i)$, denoting $(R^0, U^0, f^0, i^0)$, combining this with the classical Bialynicki-Birula decomposition of $U$ (1.7), one obtains isomorphisms in the derived category of constructible sheaves with monodromy, or of monodromic mixed Hodge modules:

$$ \beta_{U,f}^{\pm} : (p_{R}^{\pm})!(\eta_{R}^{\pm})^{*}\mathcal{P}\mathcal{V}_{U,f} \to \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} \mathcal{P}\mathcal{V}_{U_{\pi}, f_{\pi}^{0}} \quad (1.9) $$

This is the content of proposition 3.3. We must show that these isomorphisms behave well with respect to the pullback by étale (resp. smooth) maps of critical locus: this is the content of proposition 3.4. We must also show that they behave well with respect to the Thom-Sebastiani isomorphism from [19]:

$$ \mathcal{T}\mathcal{S}_{U,f,V,g} : \mathcal{P}\mathcal{V}_{U \times V, f \boxplus g} \simeq \mathcal{P}\mathcal{V}_{U,f} \boxtimes \mathcal{P}\mathcal{V}_{S,g} \quad (1.10) $$

This is the content of proposition 3.5.

Now, consider a d-critical oriented algebraic space $(X, s)$. The isomorphism $\beta_{X,s}^{\pm}$ is an isomorphism in the abelian category $\bigoplus_{\pi} \mathrm{Perv}(X_{\pi}^{0})[-\mathrm{Ind}_{\pi}]$ of shifted perverse sheaves, or the abelian category $\bigoplus_{\pi} \mathrm{MMHM}(X_{\pi}^{0})[-\mathrm{Ind}_{\pi}]$ of monodromic mixed Hodge modules, hence to define them it suffices to define them on each critical chart, and to show that they agree on intersections of critical charts, the compatibility with monodromy and self-duality can be checked on critical charts. Paying special care to orientation issues, in subsection 4.3.2 we define the isomorphisms:

$$ \beta_{R}^{\pm} : ((p_{X}^{\pm})!(\eta_{X}^{\pm})^{*}P_{X,s})|_{R^{0}} \to \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} P_{X_{\pi}^{0}, s_{\pi}^{0}}|_{R^{0}} \quad (1.11) $$

Notice that the tangent obstruction complex of $X$ on $R$ is quasi-isomorphic with $0 \to T_U \to T_U^* \to 0$, hence $\mathrm{Ind}_\pi$ is the signed number of contracting weights in the tangent-obstruction complex of $X$ at $X_\pi^0$.

For d-critical algebraic spaces, one can glue the Donaldson-Thomas perverse sheaf on intersecting critical charts by embedding them étale-locally into a single critical chart. To show that the isomorphisms $\beta_R^\pm$ glue into a single isomorphism, one must show that they are compatible with embedding. Etale-locally, an embedding of critical charts is of the form $(U, f) \hookrightarrow (U \times \mathbb{C}^n, f \oplus q)$ where $q$ is a quadratic form on $\mathbb{C}^n$, and one glues the perverse sheaves using the isomorphism:

$$ \mathcal{P}\mathcal{V}_{U \times \mathbb{C}^n, f \boxplus q} \xrightarrow{\mathcal{T}\mathcal{S}_{U,f,\mathbb{C}^n,q}} \mathcal{P}\mathcal{V}_{U,f} \boxtimes \mathcal{P}\mathcal{V}_{\mathbb{C}^n, q} \xrightarrow{\simeq} \mathcal{P}\mathcal{V}_{U,f} $$

where we have used the isomorphism $\mathcal{P}\mathcal{V}_{\mathbb{C}^n, q} \simeq \mathbb{Q}$, using the canonical orientation of $\mathbb{C}^n$ (this choice is at the root of orientation issues in cohomological DT theory). The proof of the compatibility of the isomorphisms $\beta^\pm$ with embeddings is given in subsection 4.3.3, using the compatibility with étale pullback and Thom-Sebastiani isomorphism proven above, and dealing with orientation issues.

For a d-critical oriented algebraic stack $(\mathcal{X}, s)$, an isomorphism $\beta_{\mathcal{X},s}^{\pm}$ in the abelian category $\bigoplus_{\pi} \mathrm{Perv}(\mathcal{X}_{\pi}^{0})[-\mathrm{Ind}_{\pi}]$ of shifted perverse sheaves can be defined by giving its pullback $\beta_{Y,y^*(s)}^{\pm}$
by smooth 1-morphisms $y: Y \to \mathcal{X}$ from d-critical oriented algebraic space $(Y, y^*(s))$, and
by checking that for each smooth 2-morphism $(\phi, \eta)$ between 1-morphisms $y: Y \to \mathcal{X}$,
$z: Z \to \mathcal{X}$, with $\phi: Y \to Z$ a smooth morphism of d-critical oriented algebraic spaces, one
has:

$$ \phi^*(\beta_{Z,z^*(s)}^{\pm}) = \beta_{Y,y^*(s)}^{\pm} \qquad (1.12) $$

This is done in proposition 4.2.

In this paper, we use the functorial properties of the six-functor formalism in the derived categories $D_c^b(Y)$ of bounded constructible complex over various algebraic spaces (or stacks) $Y$, or, more formally, an $(\infty, 1)$-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. Because the final diagrams lie in the abelian heart $\bigoplus_{\pi} \mathrm{Perv}(\mathcal{X}_{\pi}^0)[-\mathrm{Ind}_{\pi}]$ of $D_c^b(X^0)$, which is a classical category, they truly commute. Moreover, without specification, all the algebraic spaces are assumed to be quasi-separated and locally of finite type, and all algebraic stacks are assumed to be quasi-separated, locally of finite type, and with affine stabilizers.

For $X$ an algebraic space, the category $\mathrm{MHM}(X)$ of mixed Hodge modules is an abelian category with a faithful and exact functor $\mathrm{rat}: \mathrm{MHM}(X) \to \mathrm{Perv}(X)$. In particular, because $\mathrm{rat}$ is faithful, the commutativity of a diagram can be checked at the level of perverse sheaves. Moreover, because a mixed Hodge module $M$ is the data of filtrations on the $\mathcal{D}$-module associated to the complexification of the perverse sheaf $\mathrm{rat}(M)$ 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 projects under $\mathrm{rat}$ on the six-functor formalism on $D_c^b(X)$, the derived category of constructible complexes. Furthermore, because $\mathrm{rat}$ is exact, a morphism between complexes of perverse sheaves is an isomorphism in the derived category if the underlying morphism of constructible complexes is an isomorphism in the derived category. Hence the extension of our results to monodromic mixed Hodge modules is straightforward: our morphisms being defined easily 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 consists in checking 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 [8], 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 an holomorphic function. These constructions can also be extended to d-critical oriented analytic stacks using an analytic version of the constructions of [9]. Unfortunately we are missing crucial results on $\mathbb{C}^*$-action on analytic spaces or stacks. When an analytic space (resp. stack) with $\mathbb{C}^*$-action has a $\mathbb{C}^*$-equivariant analytic (resp. smooth) cover by an affine analytic space, we show that the fixed and attracting/repelling variety are algebraic spaces (resp. stack), hence the hyperbolic localization functors, and the morphism $S_X$, are defined. However, we do not know under which generality such a $\mathbb{C}^*$-equivariant cover exists, and, according to
David Rydh (private communication), the main techniques of [16] does not apply directly to the analytic setting. Moreover, we need an analytic version of [18, Theo B] saying that $S_X$ is an isomorphism when applied to $C^*$-equivariant complexes. Furthermore, according to [8, Remark 2.20], the six-functor formalism is not fully defined for mixed Hodge modules in the analytic case, hence it could 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, the whole results of this article should extend directly to the analytic case.

## 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 section 8.4 of [20]. It was applied in section 6 of [21] and in the section 8.3 [22], where it was used in a specific example of framed representations of quiver with potential. In [18], Timo Richarz proved the commutativity of the hyperbolic localization with the vanishing cycles functor in greater generality. We have used this result to establish the formula (1.3) for any critical locus of a potential in [1], and the extension of this result to d-critical structures was suggested to us by Richard Thomas.

In [23], the authors defined a Donaldson-Thomas motive, gluing the motive of vanishing cycles defined using arc spaces. Motives glue in the Zariski topology and not in the étale topology, hence to prove a localization formula one must a priori restrict to d-critical schemes admitting Zariski $C^*$-equivariant critical charts. According to [6, Prop 2.43], the existence of such charts is equivalent to the existence of a $C^*$-equivariant Zariski cover (such a $C^*$-action is then called 'good'), which holds for normal spaces from Sumihiro's theorem [24], but not in greater generality (see example 2.46 of [6]). Davesh Maulik has proven a formula similar to (1.4) for motivic DT invariants on d-critical schemes with a good circle-compact $C^*$-action, as explained in section 5.3 of [9], in an unpublished preprint (private communication). A generalization of this result for non-Archimedean geometry was proven after in [25, Theo 7.17]. For non circle-compact actions one can compute the DT motive of the attracting variety, which is by definition circle-compact, by this way.

A toric localization similar to (1.4) exists also for K-theoretic DT invariants as defined in [26]. The K-theoretic DT invariants are a refinement of numerical DT invariants defined for projective moduli spaces with symmetric obstruction theories, which was developed parallel to the motivic and cohomological refinement of Kontsevich-Soibelman and Joyce and collaborators. One expect in general that they correspond to the $\chi_y$ genus of the Hodge polynomial of cohomological DT theory, hence in particular one replace $\mathbb{L}^{1/2}$ by $-y$ in K-theoretic formulas. When the moduli space $X$ is non compact, but has a $C^*$-action with compact fixed components, it was suggested in [26] to use the equation (1.4) to define the $K$ theoretic invariants of $X$. However, this definition depends on the choice of the $C^*$-action (this choice is called a choice of 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 $C^*$-action.

This dependency on the slope was studied explicitly in [27] for the moduli space of framed representations of a toric quiver, and this was related to the ambiguity in the refined topological vertex of [28]. 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, hence the space of slopes is $\mathbb{P}_{\mathbb{R}}^1$. The fixed points can be described as molten crystals
from [29]. In [27, 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 quite strange at first sight, because inside a given chamber the cohomological weight of a given molten crystal does changes on many walls, but the final result does not change: those walls are 'invisible'. In [1] it was established that the attracting variety is the subspace of representations where the cycles with repelling weights are nilpotent, hence the attracting variety changes exactly on the walls defined in [27], i.e. (1.3) give an explanation of this wall and chamber structure. Moreover, using a nilpotent/invertible decomposition for unframed representation and a wall-crossing relation between framed and unframed invariants, [1] 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 simple to compute and has a universal closed formula for all toric quivers. Notice that in this case the moduli space is the critical locus of the potential of the quiver, hence one does not need all the subtleties of the gluing, i.e. one needs only proposition 3.3.

## Acknowledgments

I thank Boris Pioline and Olivier Schiffmann, my PHD advisors, and David Rydh, Benjamin Hennion, Davesh Maulik, 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 $\mathbb{C}^*$ over $S$.

**Definition 2.1.** An algebraic space $X$ over an algebraic space $S$ with a relative $\mathbb{C}^*$ over $S$ is said to be étale locally (resp. analytical locally) linearizable if there is a $\mathbb{C}^*$-equivariant étale (resp. analytical) covering family $\{U_i \to X\}_i$ where the $U_i$ are affine $S$-algebraic spaces with $\mathbb{C}^*$-action.

According to [17, 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 [18] to the complex analytic case. Consider the following functors on the category of $S$-algebraic spaces:

$$ \begin{align} X^0 : Y &\mapsto \operatorname{Hom}_S^T(Y, X) \notag \\ X^+ : Y &\mapsto \operatorname{Hom}_S^T((\mathcal{A}_Y^1)^+, X) \notag \\ X^- : Y &\mapsto \operatorname{Hom}_S^T((\mathcal{A}_Y^1)^-, X) \tag{2.1} \end{align} $$

where the superscript $\mathbb{C}^*$ denotes the $\mathbb{C}^*$-equivariant morphism, and $T, (\mathcal{A}_Y^1)^+, (\mathcal{A}_Y^1)^-$ has the trivial, resp. usual, resp. opposite $\mathbb{C}^*$-action.
The structure morphism $(\mathcal{A}_S^1)^\pm \to S$ is $\mathbb{C}^*$-equivariant and defines then a morphism:

$$ \zeta^\pm : X^0 \to X^\pm \qquad (2.2) $$

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

$$ p^\pm : X^\pm \to X^0 \qquad (2.3) $$

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

$$ \eta^\pm : X^\pm \to X \qquad (2.4) $$

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 [18]:

<figure><img src="image_2.png" /></figure>

**Proposition 2.2.** For $X$ an $S$ algebraic space with an étale (resp. analytical)-locally linearizable $\mathbb{C}^*$-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 [18, 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 [18, 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 $\mathbb{C}^*$-invariant open subset of $X$ in the analytic topology. One can adapt [18, 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:

<figure><img src="image_3.png" /></figure>

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

<figure><img src="image_4.png" /></figure>

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 $C^*$-invariant open subset of $X$ in the analytic topology, one can adapt [18, 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 $C^*$-equivariant morphisms:

<figure><img src="image_5.png" /></figure>

where $Y \to (A_Y^1)^\pm$ is the zero section. The open subset $f^{-1}(U)$ contains $Y$ and is $C^*$ invariant in $A_Y^1$, hence is the whole $A_Y^1$, i.e. $\phi \in U^\pm(Y)$. Hence $U^\pm = U^0 \times_{X_0} X^\pm$ as functors, and taking a $C^*$-equivariant analytic covering family $\{U_i \to X\}_i$ where the $U_i$ are S-affine, one obtains a Cartesian commutative diagram:

<figure><img src="image_6.png" /></figure>

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. □

There is a natural transformation $\mathbb{D}(\eta^-)_!(p^-)^*\mathbb{D} \simeq (\eta^-)_*(p^-)^! \to (\eta^+)_!(p^+)^*$ built using the six-functor formalism between derived functors $D_c^b(X) \to D_c^b(X^0)$, as explained in [18, Section 2.2], and we can define the same similar transformation at the level of mixed Hodge modules. We change slightly the terminology of [18], 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 $A \in D_c^b(X)$ is naively $C^*$-equivariant if there exists an isomorphism $a^*A \simeq p^*A$ in $D_c^b(T \otimes_S S)$. Let us define $D_c^b(X)^{C^*-eq}$ to be the full subcategory strongly generated by naively $C^*$-equivariant complexes, i.e. generated by a finite iteration of taking the cone of a morphism in $D_c^b(X)$. The objects in $D_c^b(X)^{C^*-eq}$ are called $C^*$-equivariant. The result [18, Theo B] can then be rephrased as:

**Proposition 2.3.** For $X$ an algebraic space with $C^*$ action, the natural morphism of constructible complexes (or complexes of mixed Hodge modules) $S_X(A) : \mathbb{D}(\eta_X^-)_!(p_X^-)^*\mathbb{D}(A) \to (\eta_X^+)_!(p_X^+)^*(A)$ is an isomorphism when $A$ is a $C^*$-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 $\mathbb{C}^*$-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 commute in the following diagram:

<figure><img src="image_7.png" /></figure>

**Proposition 2.4.** For a smooth $\mathbb{C}^*$-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 [18, 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 $\mathbb{C}^*$-equivariant fibration $\phi: Y \times_S \mathbb{A}_S^d \to Y$ where the fiber has a linear $\mathbb{C}^*$ action. Then $\phi^0: Y^0 \times_S (\mathbb{A}_S^d)^0 \to Y^0$ and $\phi^{\pm}: Y^{\pm} \times_S (\mathbb{A}_S^d)^{\pm} \to Y^{\pm}$ are smooth, and:

$$ (Y \times_S \mathbb{A}_S^d)^{\pm} = Y^{\pm} \times_S (\mathbb{A}_S^d)^{\pm} \\ \to (Y \times_S \mathbb{A}_S^d)^0 \times_{Y^0} Y^{\pm} = Y^{\pm} \times_S (\mathbb{A}_S^d)^0 \qquad (2.5) $$

is affine.

Suppose now that $\phi$ is smooth. For $x \in X^0$, we can restrict to a $\mathbb{C}^*$-invariant étale (resp. open in the analytic topology) neighborhood of $\phi(x)$ in which $Y$ is affine, and further restricts to a $\mathbb{C}^*$-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 [6, Prop 2.43], because $\mathbb{C}^*$ is a torus, an affine $\mathbb{C}^*$-equivariant space can be written as $\text{Spec}(R[x_1, ..., x_n]/(h_1, ..., h_r))$ where the $x_i$ are $\mathbb{C}^*$-equivariant coordinates, and the $h_i$ are $\mathbb{C}^*$-equivariant polynomial. Then the $\mathbb{C}^*$-equivariant smooth map $\phi: X \to Y$ is étale locally given by the dual of the map of rings:

$$ \tilde{R}[x_1, ..., x_n]/(h_1, ..., h_r) \to \tilde{R}[y_1, ..., y_m]/(k_1, ..., k_s) \qquad (2.6) $$

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:

$$ R \to R[y_1, ..., y_m]/(l_1, ..., l_{r+s}) \qquad (2.7) $$

where $R := \tilde{R}[x_1, ..., x_n]/(h_1, ..., h_r)$ and the $l_i$ are the $\mathbb{C}^*$-equivariant polynomials (resp. holomorphic functions) $k_1, ..., k_r, x_1 - \phi'(x_1), ..., x_n - \phi'(x_n)$. 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 $(\frac{\partial l_i}{\partial y_j}|_x)_{0 \le i, j \le c}$ is invertible at $x$, hence in the neighborhood of $x$ it is still invertible, i.e. $\tilde{R}[y_{c+1}, ..., y_m] \to \tilde{R}[y_1, ..., y_m]/(l_1, ..., l_{r+s})$ is étale (resp. a local homeomorphism) and $\tilde{R}[y_1, ..., y_m]/(l_1, ..., l_c) \to$
$R[y_1, ..., y_m]/(l_1, ..., l_{r+s})$ is a local homeomorphism, hence $\phi$ can be étale locally written as a composition of $\mathbb{C}^*$-equivariant morphisms:

$$R \to R[y_{c+1}, ..., y_m] \to S[y_1, ..., y_m]/(l_1, ..., l_{r+s}) \quad (2.8)$$

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 $X \to Y \mathbb{C}^*$-equivariant and étale (resp. open embeddings in the analytic topology) and for $X \to Y$ affine fibrations with $\mathbb{C}^*$-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 [30, 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 $\mathcal{B} = \bigoplus_{i \ge 0} \mathcal{B}_i$ (resp. $\mathcal{C} = \bigoplus_{i \ge 0} \mathcal{C}_i$). 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 $\kappa(z)$, 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 $r \le \dim(q^{-1}(z)) = \dim(\mathrm{Spec}(\mathcal{C} \otimes \kappa(z)))$. Since $q$ is flat, we have:

$$\mathcal{C} \otimes \kappa(z) = (\mathcal{B} \otimes_{\mathcal{B}_0} \mathcal{C}_0[F_1, ..., F-r]/\ker(p)) \otimes \kappa(z) = \frac{\kappa(z)[F_1, ..., F_r]}{\ker(p) \otimes \kappa(z)} \quad (2.9)$$

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 $X^\pm = \mathbb{A}^r \times_{X^0 \times_{Y^0} Y^\pm} X^0 \times_{Y^0} Y^\pm$, 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. $\square$

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

**Proposition 2.5.** For a smooth $\mathbb{C}^*$-equivariant morphism $\phi: X \to Y$ of relative dimension $d$ there are natural isomorphisms in the six-functor formalism for constructible complexes:

$$(p_X^{\pm})!(\eta_X^{\pm})^* \phi^*[d] \simeq \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_\pi/2} (\phi_\pi^0)^*[d_\pi^0](p_Y^{\pm})!(\eta_Y^{\pm})^* \quad (2.10)$$

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:

$$(p_X^{\pm})!(\eta_X^{\pm})^* \phi^*[d] \simeq (\phi^0)^*[d](p_Y^{\pm})!(\eta_Y^{\pm})^* \quad (2.11)$$
Proof: Consider the following commutative diagram:

<figure><img src="image_8.png" /></figure>

One has the following sequence of isomorphisms:

$$ (p_X^{\pm})!(\eta_X^{\pm})^*\phi^*[d] \simeq (p_X^{\pm})!(\phi^{\pm})^*[d](\eta_Y^{\pm})^* \\ \simeq (\tilde{p}_X^{\pm})!f_1f^*(\tilde{\phi}^{\pm})^*[d](\eta_Y^{\pm})^* \qquad (2.12) $$

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 a decomposition as a disjoint union of those diagrams:

<figure><img src="image_9.png" /></figure>

hence one obtains:

$$ (\tilde{p}_{X,\pi}^{\pm})!(f_{\pi})!(f_{\pi})^*(\tilde{\phi}_{\pi}^{\pm})^*[d](\eta_{Y}^{\pm})^* \simeq (\tilde{p}_{X,\pi}^{\pm})![-2d_{\pi}^{\pm}](\tilde{\phi}_{\pi}^{\pm})^*[d_{\pi}^{+} + d_{\pi}^{0} + d_{\pi}^{-}](\eta_{Y}^{\pm})^* \\ \simeq \mathbb{L}^{\pm\mathrm{Ind}_{\pi}/2}\phi_{\pi}^{0}[d_{\pi}^{0}](p_{Y}^{\pm})!(\eta_{Y}^{\pm})^* \qquad (2.13) $$

where we have used the fact that $f_{\pi}$ is an affine fibration with $d_{\pi}^{\pm}$-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:

$$ (p_X^{\pm})!(\eta_X^{\pm})^*\phi^*[d] \simeq \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} (\phi^0)^*[d_{\pi}^0](p_Y^{\pm})!(\eta_Y^{\pm})^* \qquad (2.14) $$

<figure><img src="image_10.png" /></figure>

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

$$ \mathcal{I}C_U = \phi^*[d]\mathcal{I}C_S = \phi^![-d]\mathcal{I}C_S \qquad (2.15) $$

in particular $\mathcal{I}C_S$ is $\mathbb{C}^*$-equivariant, hence $\mathcal{I}C_U$ is also $\mathbb{C}^*$-equivariant, and self-Verdier dual, hence the natural arrow $\mathcal{D}(p_U^*)(\eta_U^*)^*\mathcal{I}C_U \to (p_X^+)(\eta_X^+)^*\mathcal{I}C_U$ is an isomorphism. Moreover, there is then an open and closed decomposition $U^0 = \bigsqcup_{\pi \in \Pi} U_{\pi}^0$, and:

$$ \mathcal{I}C_{U_{\pi}^{0}} = (\phi_{\pi}^{0})^{*}[d_{\pi}^{0}]\mathcal{I}C_{S} = (\phi_{\pi}^{0})^{!}[-d_{\pi}^{0}]\mathcal{I}C_{S} \qquad (2.16) $$

hence an absolute version of the last proposition gives:
**Corollary 2.6.** For a smooth $\mathbb{C}^*$-equivariant S-algebraic (resp. analytic) space U there are natural isomorphisms in the six-functor formalism such that the following diagram commutes:

<figure><img src="image_11.png" /></figure>

# 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 $\mathbb{C}^*$-actions will 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 [8]. Consider the following commutative diagram of complex analytic spaces, where $U^{an}$ is the analytification of $U$, and the square are Cartesian:

<figure><img src="image_12.png" /></figure>

We define the functors:

$$ \phi_f^\theta := \iota^*(j_\theta)!(j_\theta)^1 : D(X) \to D(X_0) \qquad (3.1) $$

There are natural isomorphisms $\tilde{T}_f^\theta: \phi_f^0 \to \phi_f^\theta$ induced by rotating $\mathbb{C}$ counterclockwise by an angle $\theta$ around the origin. One defines the vanishing cycles functor $\phi_f := \phi_f^0$, and the natural isomorphism $\tilde{T}_f^{2\pi} : \phi_f \to \phi_f$ 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 [8], i.e. it defines a functor $\phi_f: D_c^b(U) \to D_c^b(U_0)$, and it even defines functors $\phi_f: \mathrm{Perv}(U) \to \mathrm{Perv}(U_0)$.

Massey defines in [31] a natural isomorphism giving the commutation of the vanishing cycles functor (considered as a functor $\phi_f: D_c^b(U) \to D_c^b(U_0)$) with the Verdier duality
Dφ<sub>f</sub> ≃ φ<sub>f</sub>D. Consider the Cartesian diagram of closed subspaces:

<figure><img src="image_13.png" /></figure>

Then Massey defines the following sequence of isomorphisms:

$$
\begin{align*}
\mathbb{D}\phi_f &\simeq \mathbb{D}q^*m^*(j_0)!(j_0)! \\
&\simeq \mathbb{D}q^!m^*(j_0)!(j_0)! \\
&\simeq \mathbb{D}q!(\hat{j}_0)!^*(j_\pi)^* \\
&\simeq q^*(\hat{j}_0)^*(j_\pi)!\mathbb{D} \\
&\simeq q^*m^*(j_\pi)!(j_\pi)!\mathbb{D} \\
&\simeq \phi_f^\pi\mathbb{D} \tag{3.2}
\end{align*}
$$

where in the first and last line we have used the fact that $\iota = m \circ q$, in the second line
[31, Lem 2.1] which proves that the natural morphism $q^!m^*(j_0)!(j_0)! \to q^*m^*(j_0)!(j_0)!$ is an
isomorphism and in the third and fifth line the isomorphisms built in [31, Lem 2.2] using the
six-functor formalism in the Cartesian square of closed subspace of $X$, Using the fact that
$U^{an} = \tilde{U}_0^{an} \cup \tilde{U}_\pi^{an}$. The Massey isomorphism is then defined by the composition:

$$
\mathbb{D}\phi_f \xrightarrow{\approx} \phi_f^{\pi} \mathbb{D} \xrightarrow{(\tilde{T}_f^{\pi})^{-1}} \phi_f \mathbb{D}
$$

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

For two algebraic spaces X, Y with projections π₁ : X × Y → X, π₂ : X × Y → Y, for
F ∈ D<sup>b</sup><sub>c</sub>(X), G ∈ D<sup>b</sup><sub>c</sub>(Y), one denotes:

$$
F \boxtimes G := \pi_1^*(F) \otimes \pi_2^*G \tag{3.3}
$$

For $f: X \to \mathbb{C}$, $g: Y \to \mathbb{C}$, denote by $f \boxtimes g := f \circ \pi_1 + g \circ \pi_2: X \times Y \to \mathbb{C}$. For $j: Y \to X$ a closed embedding, denote $R\Gamma_Y := j!j^! : D_c^b(X) \to D_c^b(X)$.

Consider two algebraic spaces *U*, *V* with a regular function *f*: *U* → *C*, *g*: *V* → *C*, denote by *k*: *U*₀ × *V*₀ → (*U* × *V*)₀ the embedding into the zero locus of *f* ⊕ *g*. Massey defines in [19] a natural Thom-Sebastiani isomorphism:

$$
\phi_f \boxtimes \phi_g \simeq k^* \phi_{f \boxtimes g} \qquad (3.4)
$$

denote by $\iota: U_0 \to U, j: V_0 \to V, q: (U \times V)_0 \to U \times V$ the closed embeddings of the zero set of $f, g, f \boxtimes g$. One has the closed embedding $\tilde{U}_0 \times \tilde{V}_0 \to (U \times V)_0$, and [19, lem 1.2] shows that the natural arrow of the six-functor formalism $R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} \to R\Gamma_{(U \times V)_0}$ is an isomorphism. The
Thom-Sebastiani isomorphism is then defined by the composition of natural isomorphisms of the six-functor formalism:

$$
\begin{align*}
\phi_f \boxtimes \phi_g &= \iota^* R \Gamma_{\tilde{U}_0} \boxtimes j^* R \Gamma_{\tilde{V}_0} \\
&\simeq (\iota \times j)^* R \Gamma_{\tilde{U}_0 \times \tilde{V}_0} \\
&\xrightarrow{\sim} k^* q^* R \Gamma_{U \tilde{\times} V_0} \\
&= k^* \phi_{f \boxtimes g} \tag{3.5}
\end{align*}
$$

it is shown in [19] 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 [8, sec 2.10]. There is a vanishing cycle functor:

$$
\phi_f^H : \mathrm{MHM}(U) \to \mathrm{MMHM}(U_0) \qquad (3.6)
$$

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 [8, appendix A] that *rat* sends them to the corresponding isomorphisms of perverse sheaves. When a mixed Hodge module $M \in \mathrm{MHM}(U)$ is polarized, i.e. is provided with an isomorphism $\sigma : \mathbb{D}M \simeq M$, the self-duality isomorphism of $\phi_f^H$ provides them $\phi_f^H M$ with a strong polarization, i.e. an isomorphism with its dual commuting with monodromy.

3.2 Functionality 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:*

$$
\begin{align}
(\Phi_0)^* \phi_g^\theta &\to \phi_f^\theta \Phi^* \notag \\
(\Phi_0)_! \phi_f^\theta &\to \phi_g^\theta \Phi_! \tag{3.7}
\end{align}
$$

They are compatible with monodromy and Thom-Sebastiani isomorphism, and which are com-
patible with composition and base change of Φ.

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

<figure><img src="image_14.png" /></figure>
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:

$$ (\Phi_0)^* \phi_g^\theta = \Phi_0^*(\iota_V)^*(j_{\theta,V})!(j_{\theta,V})! \\ \simeq (\iota_U)^* \Phi^*(j_{\theta,V})!(j_{\theta,V})! \\ \simeq (\iota_U)^*(j_{\theta,U})! \tilde{\Phi}^*(j_{\theta,V})! \\ \to (\iota_U)^*(j_{\theta,U})!(j_{\theta,U})! \Phi^* \\ = \phi_f^\theta \Phi^* \qquad (3.8) $$

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

$$ (\Phi_0)! \phi_f^\theta = (\Phi_0)!(\iota_U)^*(j_{\theta,U})!(j_{\theta,U})! \\ \simeq (\iota_V)^* \Phi!(j_{\theta,U})!(j_{\theta,U})! \\ \simeq (\iota_V)^*(j_{\theta,V,U})! \\ \to (\iota_V)^*(j_{\theta,V})!(j_{\theta,V})! \Phi! \\ = \phi_g^\theta \Phi! \qquad (3.9) $$

where the first and last lines are the definitions, the second line follows from the Cartesianianity 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:

$$ \begin{array}{ccc} (\Phi_0)^* \phi_g & \longrightarrow & \phi_f \Phi^* \\ \Big\downarrow^{(\Phi_0)^* \tilde{T}_g^\theta} & \Big\downarrow^{\tilde{T}_f^\theta \Phi^*} & \Big\downarrow^{(\Phi_0)! \tilde{T}_f^\theta} \\ (\Phi_0)^* \phi_g^\theta & \longrightarrow & \phi_f^\theta \Phi^* \end{array} \qquad \begin{array}{ccc} (\Phi_0)! \phi_f & \longrightarrow & \phi_g \Phi^* \\ \Big\downarrow^{(\Phi_0)! \tilde{T}_g^\theta} & \Big\downarrow^{\tilde{T}_f^\theta \Phi^*} & \Big\downarrow^{(\Phi_0)! \phi_f^\theta} \\ (\Phi_0)! \phi_f^\theta & \longrightarrow & \phi_g^\theta \Phi^* \end{array} $$

in particular these morphisms are compatible with the monodromy.

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

$$ \begin{array}{ccc} U \times V & \leftarrow & (U \tilde{\times} V)_0 \leftarrow \tilde{U}_0 \times \tilde{V}_0 \\ \Big\downarrow^{\Phi \times \Psi} & & \Big\downarrow^{\Phi \times \Psi} \\ U' \times V' & \leftarrow & (U' \tilde{\times} V')_0 \leftarrow \tilde{U}'_0 \times \tilde{V}'_0 \end{array} $$

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

$$ \begin{array}{ccc} (\Phi \times \Psi)^* R \Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} & \to & R \Gamma_{\tilde{U}_0 \times \tilde{V}_0} (\Phi \times \Psi)^* \\ \Big\downarrow^{\simeq} & & \Big\downarrow^{\simeq} \\ (\Phi \times \Psi)^* R \Gamma_{(U' \tilde{\times} V')_0} & \to & R \Gamma_{(U \tilde{\times} V)_0} (\Phi \times \Psi)^* \end{array} \qquad \begin{array}{ccc} (\Phi \times \Psi)_! R \Gamma_{\tilde{U}_0 \times \tilde{V}_0} & \to & R \Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} (\Phi \times \Psi)_! \\ \Big\downarrow^{\simeq} & & \Big\downarrow^{\simeq} \\ (\Phi \times \Psi)_! R \Gamma_{(U' \tilde{\times} V')_0} & \to & R \Gamma_{(U \tilde{\times} V)_0} (\Phi \times \Psi)_! \end{array} $$
Consider now the Cartesian diagram:

$$ \begin{array}{ccc} U_0 \times V_0 & \xrightarrow{k} & (U \times V)_0 \\ & \searrow & \searrow \\ & \Phi_0 \times \Psi_0 & (\Phi \times \Psi)_0 \\ & \searrow & \searrow \\ U'_0 \times V'_0 & \xrightarrow{k} & (U' \times V')_0 \end{array} \xrightarrow{q} U \times V $$

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

$$ \begin{array}{ccc} (\Phi_0 \times \Psi_0)^*(\iota' \times \jmath')^* & \to & (\iota \times \jmath)^*(\Phi \times \Psi)^* \\ \searrow & \approx & \searrow \\ k^*((\Phi \times \Psi)_0)^*(q')^* & \to & (\iota \times \jmath)^*(\Phi \times \Psi)^* \end{array} \qquad \begin{array}{ccc} (\Phi_0 \times \Psi_0)!(\iota \times \jmath)^* & \to & (\iota' \times \jmath')^*(\Phi \times \Psi)_! \\ \searrow & \approx & \searrow \\ k^*((\Phi \times \Psi)_0)!q^* & \to & (\iota' \times \jmath')^*(\Phi \times \Psi)_! \end{array} $$

And one obtains then finally commutative square of morphisms:

$$ \begin{array}{ccc} (\Phi_0 \times \Psi_0)^*(\iota' \times \jmath')^* R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} & \cong & (\iota \times \jmath)^*(\Phi \times \Psi)^* R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} \\ \searrow & \approx & \searrow \\ k^*(\Phi \times \Psi)_0^*(q')^* R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} & \cong & (\iota \times \jmath)^*(\Phi \times \Psi)^* R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} \\ \searrow & \approx & \searrow \\ k^*(\Phi \times \Psi)_0^*(q')^* R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} & \cong & (\iota \times \jmath)^*(\Phi \times \Psi)^* R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} \end{array} \qquad \begin{array}{ccc} (\Phi_0 \times \Psi_0)!(\iota \times \jmath)^* R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} & \cong & (\iota' \times \jmath')^*(\Phi \times \Psi)_! R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} \\ \searrow & \approx & \searrow \\ (k')^*((\Phi \times \Psi)_0)!q^* R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} & \cong & (\iota' \times \jmath')^*(\Phi \times \Psi)_! R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} \\ \searrow & \approx & \searrow \\ (k')^*((\Phi \times \Psi)_0)!q^* R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} & \cong & (\iota' \times \jmath')^*(\Phi \times \Psi)_! R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} \end{array} $$

And similarly:

$$ \begin{array}{ccc} (\Phi_0 \times \Psi_0)!(\iota \times \jmath)^* R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} & \cong & (\iota' \times \jmath')^*(\Phi \times \Psi)_! R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} \\ \searrow & \approx & \searrow \\ (k')^*((\Phi \times \Psi)_0)!q^* R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} & \cong & (\iota' \times \jmath')^*(\Phi \times \Psi)_! R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} \\ \searrow & \approx & \searrow \\ (k')^*((\Phi \times \Psi)_0)!q^* R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} & \cong & (\iota' \times \jmath')^*(\Phi \times \Psi)_! R\Gamma_{\tilde{U}_0 \times \tilde{V}_0} \end{array} \qquad \begin{array}{ccc} (\Phi_0 \times \Psi_0)!(\iota \times \jmath)^* R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} & \cong & (\iota' \times \jmath')^*(\Phi \times \Psi)_! R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} \\ \searrow & \approx & \searrow \\ (k')^*((\Phi \times \Psi)_0)!q^* R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} & \cong & (\iota' \times \jmath')^*(\Phi \times \Psi)_! R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} \\ \searrow & \approx & \searrow \\ (k')^*((\Phi \times \Psi)_0)!q^* R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} & \cong & (\iota' \times \jmath')^*(\Phi \times \Psi)_! R\Gamma_{\tilde{U}'_0 \times \tilde{V}'_0} \end{array} $$

Hence the following diagrams are commutative:

$$ \begin{array}{ccc} (\Phi_0 \times \Psi_0)^*(\phi_{f'} \boxtimes \phi_{g'}) & \to & (\phi_f \boxtimes \phi_g)(\phi \times \Psi)^* \\ \searrow & \approx & \searrow \\ k^*((\Phi \times \Psi)_0)^* \phi_{f' \boxplus g'} & \to & \phi_{f \boxplus g}(\Phi \times \Psi)^* \end{array} \qquad \begin{array}{ccc} (\Phi_0 \times \Psi_0)!(\phi_f \boxtimes \phi_g) & \to & k^*(\phi_{f'} \boxtimes \phi_{g'})(\phi \times \Psi)_! \\ \searrow & \approx & \searrow \\ (k')^*((\Phi \times \Psi)_0)!\phi_{f \boxplus g} & \to & k^*\phi_{f' \boxplus g'}(\Phi \times \Psi)_! \end{array} $$

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:

<figure><img src="image_15.png" /></figure>

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

<figure><img src="image_16.png" /></figure>

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

Consider now a Cartesian square of morphisms between algebraic spaces:

<figure><img src="image_17.png" /></figure>

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

<figure><img src="image_18.png" /></figure>
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:

<figure><img src="image_19.png" /></figure>

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 [8, 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. □

Now, for $\Phi$ smooth of relative dimension $d$ (resp. $\Psi$ proper), observing that $\mathbb{D}\Phi^*[d]\mathbb{D} = \Phi^*[d]$ (resp. $\mathbb{D}\Psi_!\mathbb{D} = \Psi_!$ and then also $\mathbb{D}(\Phi_0)^*[d]\mathbb{D} = (\Phi_0)^*[d]$ 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:

<figure><img src="image_20.png" /></figure>

where the vertical arrows come 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:

$$ \begin{aligned} (\Phi_0)^*[d]\phi_g &\simeq \phi_f\Phi^*[d] \\ (\Psi_0)_!\phi_f &\simeq \phi_g\Psi_! \end{aligned} \qquad (3.10) $$

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

Consider now a $\mathbb{C}^*$ action on $U$ such that $f$ is $\mathbb{C}^*$ invariant, i.e. $U$ is can be considered as an algebraic space over $A_{\mathbb{C}}^1$. In particular $U$ is locally of finite type and quasi-separated over a quasi-separated base, hence the $\mathbb{C}^*$-action on it is étale-locally linearizable, and then the $\mathbb{C}^*$-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 $\mathbb{C}^*$-equivariant constructible complex $A$. The functor $\phi_f$ is defined using the morphisms of the six-functor formalism defined $\mathbb{C}^*$-equivariant functions, hence $\phi_f(A)$ is $\mathbb{C}^*$-equivariant. We will use the idea of [18] to prove the commutation of the hyperbolic localization with the vanishing cycle:
**Proposition 3.2.** For $A \in D_c^b(U)^{T-mon}$, the natural morphism $(p_{U_0}^{\pm})!(\eta_{U_0}^{\pm})^*\phi_f \simeq \phi_{f^0}(p_U^{\pm})!(\eta_U^{\pm})^*(A)$ is an isomorphism, compatible with monodromy and duality, in the sense that the following diagram is commutative:

$$ \begin{array}{ccc} (p_{U_0}^{+})!(\eta_{U_0}^{+})^{*}\phi_f \mathbb{D}(A) & \xrightarrow{\simeq} & \phi_{f^0}(p_{U}^{+})!(\eta_{U}^{+})^{*}\mathbb{D}(A) \\ \xrightarrow{\simeq} & & \xrightarrow{\simeq} \\ \mathbb{D}(p_{U_0}^{-})!(\eta_{U_0}^{-})^{*}\phi_f \mathbb{D}(A) & \leftarrow\limits_{\simeq} & \mathbb{D}\phi_{f^0}(p_{U}^{-})!(\eta_{U}^{-})^{*}\mathbb{D}(A) \end{array} $$

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 $\mathbb{C}^*$-equivariant spaces:

<figure><img src="image_21.png" /></figure>

it is obtained by base change from the space $U^{an} \to 0$, hence the hyperbolic localisation 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:

$$ \begin{array}{ccc} (p_{U_0}^{+})!(\eta_{U_0}^{+})^{*}\phi_f \mathbb{D} & \longrightarrow & \phi_{f^0}(p_{U}^{+})!(\eta_{U}^{+})^{*}\mathbb{D} \\ \xrightarrow{\uparrow} & & \xrightarrow{\uparrow} \\ \mathbb{D}(p_{U_0}^{-})!(\eta_{U_0}^{-})^{*}\phi_f \mathbb{D}(A) & \leftarrow\limits & \mathbb{D}\phi_{f^0}(p_{U}^{-})!(\eta_{U}^{-})^{*}\mathbb{D}(A) \end{array} $$

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:

$$ \begin{array}{ccc} (p_{U_0}^{+})!(\eta_{U_0}^{+})^{*}\phi_f \mathbb{D} & \longrightarrow & \phi_{f^0}(p_{U}^{+})!(\eta_{U}^{+})^{*}\mathbb{D} \\ \xrightarrow{\downarrow} & & \xrightarrow{\downarrow} \\ (p_{U_0}^{+})!(\eta_{U_0}^{+})^{*}\phi_f^{\pi} \mathbb{D} & \longrightarrow & \phi_f^{\pi}(p_{U}^{+})!(\eta_{U}^{+})^{*}\mathbb{D} \end{array} $$

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 $\mathbb{C}^*$ action, $\mathbb{C}^*$-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:

<figure><img src="image_22.png" /></figure>

hence this isomorphism is compatible with pullbacks by smooth maps.

Consider two smooth spaces $U, V$ with a $\mathbb{C}^*$ action. The product space $U \times V$ is itself provided with a $\mathbb{C}^*$ 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:

$$ (p_{U \times V})!(\eta_{U \times V})^* = (p_U \times p_V)!(\eta_U \times \eta_V)^* : D_c^b(U \times V) \to D_c^b(U^0 \times V^0 = (U \times V)^0) \\ (p_{U_0 \times V_0})!(\eta_{U_0 \times V_0})^* = (p_{U_0} \times p_{V_0})!(\eta_{U_0} \times \eta_{V_0})^* : D_c^b(U_0 \times V_0) \to D_c^b(U_0^0 \times V_0^0 = (U_0 \times V_0)^0) \qquad (3.11) $$

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:

<figure><img src="image_23.png" /></figure>

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. $\square$

## 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, $(i_c)^*\phi_f \simeq (i_c)^!\phi_f$ is a functor from $\text{Perv}(U)$ to $\text{Perv}(R_c)$. One defines in particular the perverse sheaf on $R$:

$$ \mathcal{P}\mathcal{V}_{U,f} := \bigoplus_{c \in f(R)} i_c^* \phi_{f-c} \mathcal{I}\mathcal{C}_U \qquad (3.12) $$
where we have used the fact that $\hat{i}\hat{i}^* = \text{Id}$ 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 $A \in D_c^b(U_c)_{R_c}^{C^*-eq}$:

<figure><img src="image_24.png" /></figure>

Then one obtains isomorphisms:

$$ \beta_{U,f}^{\pm} : (p_R^{\pm})!(\eta_R^{\pm})^* \mathcal{P}V_{U,f} \simeq \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} PV_{U_{\pi}, f_{\pi}^0} \quad (3.16) $$

by the following composition of isomorphisms:

$$ \begin{aligned} (p_R^{\pm})!(\eta_R^{\pm})^* \mathcal{P}V_{U,f} &:= (p_R^{\pm})!(\eta_R^{\pm})^* \bigoplus_{c \in f(R)} i_c^* \phi_{f-c} \mathcal{I}C_U \xrightarrow{\simeq} \bigoplus_{c \in f(R)} (i_c^0)^* (p_{\bar{U}_c}^{\pm})!(\eta_{\bar{U}_c}^{\pm})^* \phi_{f-c} \mathcal{I}C_U \\ \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} PV_{U_{\pi}, f_{\pi}^0} &:= \bigoplus_{c \in f(R)} i_c^* \phi_{f-c} \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} \mathcal{I}C_{U_{\pi}}^0 \xleftarrow{\simeq} \bigoplus_{c \in f(R)} i_c^* \phi_{f-c} (p_U^{\pm})!(\eta_U^{\pm})^* \mathcal{I}C_U \end{aligned} $$

where the first isomorphism is (3.15), the second from proposition 3.2 where one has remaked that $\mathcal{I}C_U$ is $\mathbb{C}^*$-equivariant, and the third from corollary 2.6.

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

$$ \begin{aligned} (p_{U_0}^{\pm})!(\eta_{U_0}^{\pm})^* \phi_f \mathcal{I}C_U &\xrightarrow{\simeq} \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} \phi_{f_{\pi}^0} \mathcal{I}C_{U_{\pi}^0} \\ \xleftarrow{\downarrow} (p_{\bar{U}_0}^{\pm})!(\eta_{\bar{U}_0}^{\pm})^* (\tilde{T}_f^{2\pi}) &\xleftarrow{\bigoplus_{\pi \in \Pi}} \tilde{T}_{f_{\pi}^0}^{2\pi} \\ (p_{U_0}^{\pm})!(\eta_{U_0}^{\pm})^* \phi_f \mathcal{I}C_U &\xrightarrow{\simeq} \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} \phi_{f_{\pi}^0} \mathcal{I}C_{U_{\pi}^0} \end{aligned} $$

the compatibility with the monodromy results then by applying $(i_0)^*$ to this diagram, applying the functorial isomorphism (3.15), and finally writing the definition of $\tau_{U,f}$ by noticing that $(-1)^{\dim(U)} = (-1)^{\pm \mathrm{Ind}_{\pi} + \dim(U_{\pi}^0)}$.

The proof of the compatibility with $\sigma_{U,f}$ is slightly more technical. In fact $\beta_{U,f}^{\pm}$ and $\sigma_{U,f}$ are both defined by composing three isomorphisms expressing the commutation of the hyperbolic localization and Verdier duality with $i_c^*$, with $\phi_f$ and with $\mathcal{I}C_U$, and then one should divide
the square expressing the compatibility into nine sub-squares:

<figure><img src="image_25.png" /></figure>

Where the objects are:

11 : $(p_R^+)! (\eta_R^+)^* (i_0)^* \phi_f \mathcal{I}C_U$

12 : $(i_0^0)^* (p_{U_0}^+)! (\eta_{U_0}^+)^* \phi_f \mathcal{I}C_U$

13 : $(i_0^0)^* \phi_{f^0} (p_U^+)! (\eta_U^+)^* \mathcal{I}C_U$

14 : $(i_0^0)^* \phi_{f^0} \bigoplus_{\pi \in \Pi} \mathbb{L}^{\mathrm{Ind}_{\pi}/2} \mathcal{P}V_{U_{\pi}, f_{\pi}^0}$

21 : $(p_R^+)! (\eta_R^+)^* (i_0)^* \phi_f \mathcal{D}IC_U$

22 : $(i_0^0)^* (p_{U_0}^+)! (\eta_{U_0}^+)^* \phi_f \mathcal{D}IC_U$

23 : $(i_0^0)^* \phi_{f^0} (p_U^+)! (\eta_U^+)^* \mathcal{D}IC_U = (i_0^0)^* \phi_{f^0} \mathcal{D}(p_U^-)! (\eta_U^-)^* \mathcal{I}C_U$

24 : $(i_0^0)^* \phi_{f^0} \bigoplus_{\pi \in \Pi} \mathcal{D} \mathbb{L}^{-\mathrm{Ind}_{\pi}/2} \mathcal{P}V_{U_{\pi}, f_{\pi}^0}$

31 : $(p_R^+)! (\eta_R^+)^* (i_0)^* \mathcal{D} \phi_f \mathcal{I}C_U$

32 : $(i_0^0)^* (p_{U_0}^+)! (\eta_{U_0}^+)^* \mathcal{D} \phi_f \mathcal{I}C_U = (i_0^0)^* \mathcal{D}(p_{U_0}^-)! (\eta_{U_0}^-)^* \phi_f \mathcal{I}C_U$

33 : $(i_0^0)^* \phi_{f^0} (p_U^+)! (\eta_U^+)^* \mathcal{D}IC_U = (i_0^0)^* \mathcal{D} \phi_{f^0} (p_U^-)! (\eta_U^-)^* \mathcal{I}C_U$

34 : $(i_0^0)^* \mathcal{D} \phi_{f^0} \bigoplus_{\pi \in \Pi} \mathbb{L}^{-\mathrm{Ind}_{\pi}/2} \mathcal{P}V_{U_{\pi}, f_{\pi}^0}$

41 : $(p_R^+)! (\eta_R^+)^* \mathcal{D}(i_0)^* \phi_f \mathcal{I}C_U = \mathcal{D}(p_R^-)! (\eta_R^-)^* (i_0)^* \phi_f \mathcal{I}C_U$

42 : $\mathcal{D}(i_0^0)^* (p_{U_0}^+)! (\eta_{U_0}^+)^* \phi_f \mathcal{I}C_U = (i_0^0)^* \mathcal{D}(p_{U_0}^-)! (\eta_{U_0}^-)^* \phi_f \mathcal{I}C_U$

43 : $\mathcal{D}(i_0^0)^* \phi_{f^0} (p_U^+)! (\eta_U^+)^* \mathcal{I}C_U = (i_0^0)^* \mathcal{D} \phi_{f^0} (p_U^-)! (\eta_U^-)^* \mathcal{I}C_U$

44 : $\mathcal{D}(i_0^0)^* \phi_{f^0} \bigoplus_{\pi \in \Pi} \mathbb{L}^{-\mathrm{Ind}_{\pi}/2} \mathcal{P}V_{U_{\pi}, f_{\pi}^0}$ \qquad (3.17)

The down left square commutes because it is the diagram 3.3 with $A = \phi_f \mathcal{I}C_U$. The central square commutes because 3.2 commutes, and the upper right square commutes because it is $(i_0^0)^* \phi_{f^0}$ 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. $\square$
the natural isomorphisms of the six-functor formalism in the diagram:

<figure><img src="image_26.png" /></figure>

hence this diagram commutes. Finally, consider the diagram:

<figure><img src="image_27.png" /></figure>
Where the objects are:

$$11 : (p_R)!(\eta_R)^*(\Phi|_R)^*[d](j_0)^*\phi_g \mathcal{I}C_V = (\Phi|_{R^0})^*[d](p_S)!(\eta_S)^*(j_0)^*\phi_g \mathcal{I}C_V$$

$$12 : (\Phi|_{R^0})^*[d](j_0^0)^*(p_{V_0})!(\eta_{V_0})^*\phi_g \mathcal{I}C_V$$

$$13 : (\Phi|_{R^0})^*[d](j_0^0)^*\phi_{g^0}(p_V)!(\eta_V)^*\mathcal{I}C_V$$

$$14 : (\Phi|_{R^0})^*[d](j_0^0)^*\phi_{g^0} \bigoplus_{n \in \mathbb{Z}} \mathbb{L}^{n/2} \mathcal{P}V_{V_n^0, g_n^0}$$

$$21 : (p_R)!(\eta_R)^*(i_0)^*(\Phi_0)^*[d]\phi_g \mathcal{I}C_V$$

$$22 : (i_0^0)^*(p_{U_0})!(\eta_{U_0})^*(\Phi_0)^*[d]\phi_g \mathcal{I}C_V = (i_0^0)^*(\Phi_0^0)^*[d](p_{V_0})!(\eta_{V_0})^*\phi_g \mathcal{I}C_V$$

$$23 : (i_0^0)^*(\Phi_0^0)^*[d]\phi_{g^0}(p_V)!(\eta_V)^*\mathcal{I}C_V$$

$$24 : (i_0^0)^*(\Phi_0^0)^*[d]\phi_{g^0} \bigoplus_{n \in \mathbb{Z}} \mathbb{L}^{n/2} \mathcal{P}V_{V_n^0, g_n^0}$$

$$31 : (p_R)!(\eta_R)^*(i_0)^*\phi_f \Phi^*[d] \mathcal{I}C_V$$

$$32 : (i_0^0)^*(p_{U_0})!(\eta_{U_0})^*\phi_f \Phi^*[d] \mathcal{I}C_V$$

$$33 : (i_0^0)^*\phi_{f^0}(p_U)!(\eta_U)^*\Phi^*[d] \mathcal{I}C_V = (i_0^0)^*\phi_{f^0}(\Phi^0)^*[d](p_V)!(\eta_V)^*\mathcal{I}C_V$$

$$34 : (i_0^0)^*\phi_{f^0}(\Phi^0)^*[d] \bigoplus_{n \in \mathbb{Z}} \mathbb{L}^{n/2} \mathcal{P}V_{V_n^0, g_n^0}$$

$$41 : (p_R)!(\eta_R)^*(i_0)^*\phi_f \mathcal{I}C_U$$

$$42 : (i_0^0)^*(p_{U_0})!(\eta_{U_0})^*\phi_f \mathcal{I}C_U$$

$$43 : (i_0^0)^*\phi_{f^0}(p_U)!(\eta_U)^*\mathcal{I}C_U$$

$$44 : (i_0^0)^*\phi_{f^0} \bigoplus_{n \in \mathbb{Z}} \mathbb{L}^{n/2} \mathcal{P}V_{U_n^0, f_n^0} \qquad (3.19)$$

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. $\square$

## 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 $\mathrm{Crit}(f \boxplus g) = \mathrm{Crit}(f) \times \mathrm{Crit}(g)R \times S$. Consider the sequence of closed embeddings:

$$R \times S \xrightarrow{i_c \times j_d} U_c \times V_d \xrightarrow{k_{c,d}} (U \times V)_{c+d}$$

In [8], Joyce defined the Thom-Sebastiani isomorphism:

$$\mathcal{T}S_{U,f,V,g} : \mathcal{P}V_{U \times V, f \boxplus g} \simeq \mathcal{P}V_{U,f} \boxtimes \mathcal{P}V_{S,g} \qquad (3.20)$$
By the following sequence of isomorphisms:

$$
\begin{align*}
\mathcal{P}\mathcal{V}_{U \times V, f \boxplus g} &= \bigoplus_{e \in f \boxplus g(R \times S)} ((i \times j)_e)^* \phi_{f \boxplus g-e} \mathcal{I}\mathcal{C}_{U \times V} \\
&\approx \bigoplus_{c \in f(R), d \in g(S)} (i_c \times j_d)^* k_{c,d}^* \phi_{(f-c) \boxplus (g-d)} \mathcal{I}\mathcal{C}_{U \times V} \\
&\approx \bigoplus_{c \in f(R), d \in g(S)} (i_c \times j_d)^* (\phi_{f-c} \boxtimes \phi_{g-d}) \mathcal{I}\mathcal{C}_{U \times V} \\
&\approx (\bigoplus_{c \in f(R)} (i_c)^* \phi_{f-c} \mathcal{I}\mathcal{C}_U) \boxtimes (\bigoplus_{d \in g(S)} (j_d)^* \phi_{g-d} \mathcal{I}\mathcal{C}_V) \\
&= \mathcal{P}\mathcal{V}_{U,f} \boxtimes \mathcal{P}\mathcal{V}_{S,g} \tag{3.21}
\end{align*}
$$

where the third line is the Thom-Sebastiani isomorphism of [19].

**Proposition 3.5.** The isomorphism $\beta_{U,f}$ commutes with Thom-Sebastiani isomorphism, namely for $C^*$-equivariant critical chart $(U, R, f, i)$ and $(S, V, g, j)$, one has the commutative diagram:

<figure><img src="image_28.png" /></figure>

Proof: Consider the following diagram:

<figure><img src="image_29.png" /></figure>

Where the objects are:

$$
11 : (p_R^\pm \times P_S^\pm)! (\eta_R^\pm \times \eta_S^\pm)^* (i_0 \times j_0)^* k^* \phi_{f \boxplus g} \mathcal{I}\mathcal{C}_{U \times V}
$$

$$
12 : (i_0^0 \times j_0^0)^* (p_{U_0}^\pm \times p_{V_0}^\pm)! (\eta_{U_0}^\pm \eta_{V_0}^\pm)^* k^* \phi_{f \boxplus g} \mathcal{I}\mathcal{C}_{U \times V} = (i_0^0 \times j_0^0)^* (k^0)^* (p_{(U \times V)_0}^\pm)^* (\eta_{(U \times V)_0}^\pm)^* \phi_{f \boxplus g} \mathcal{I}\mathcal{C}_{U \times V}
$$

$$
13 : (i_0^0 \times j_0^0)^* (k^0)^* \phi_{f^0 \boxplus g^0} (p_U^\pm \times p_V^\pm)! (\eta_U^\pm \times \eta_V^\pm)^* \mathcal{I}\mathcal{C}_{U \times V}
$$

$$
14 : (i_0^0 \times j_0^0)^* (k^0)^* \phi_{f^0 \boxplus g^0} \bigoplus_{\pi, \pi' \in \Pi} \mathbb{L}^{(\mathrm{Ind}_{\pi} + \mathrm{Ind}_{\pi'})/2} \mathcal{I}\mathcal{C}_{U_{\pi}^0 \times V_{\pi'}^0}
$$

$$
21 : (p_R^\pm \times P_S^\pm)! (\eta_R^\pm \times \eta_S^\pm)^* (i_0 \times j_0)^* (\phi_f \boxtimes \phi_g) \mathcal{I}\mathcal{C}_{U \times V}
$$

$$
22 : (i_0^0 \times j_0^0)^* (p_{U_0}^\pm \times p_{V_0}^\pm)! (\eta_{U_0}^\pm \eta_{V_0}^\pm)^* (\phi_f \boxtimes \phi_g) \mathcal{I}\mathcal{C}_{U \times V}
$$

$$
23 : (i_0^0 \times j_0^0)^* (\phi_{f^0} \boxtimes \phi_{g^0}) (p_U^\pm \times p_V^\pm)! (\eta_U^\pm \times \eta_V^\pm)^* \mathcal{I}\mathcal{C}_{U \times V}
$$

$$
24 : (i_0^0 \times j_0^0)^* (\phi_{f^0} \boxtimes \phi_{g^0}) \bigoplus_{\pi, \pi' \in \Pi} \mathbb{L}^{(\mathrm{Ind}_{\pi} + \mathrm{Ind}_{\pi'})/2} \mathcal{I}\mathcal{C}_{U_{\pi}^0 \times V_{\pi'}^0} \quad (3.22)
$$

the left and the right squares commutes from the functoriality of the corresponding isomor-
phisms, and the central square commutes because it is the diagram 3.2, which commutes,
hence the whole diagram commutes. Identifying $\mathcal{I}C_{U \times V}$ with $\mathcal{I}C_U \boxtimes \mathcal{I}C_V$ and $\mathcal{I}C_{U_0 \times V_0'}$ with $\mathcal{I}C_{U_0} \boxtimes \mathcal{I}C_{V_0'}$, and summing over $c \in f(R), d \in g(S)$, one obtains the commutative square of the proposition. $\square$

As explained in example 2.14 of [8], for $q$ a non-degenerate quadratic form on $\mathbb{C}^n$, one has $\mathrm{Crit}(q) = \{0\}$, and then:

$$ \mathcal{P}V_{\mathbb{C}^n, q} \simeq H^{n-1}(MF_q(0), \mathbb{Q}) \otimes \mathbb{Q}_{\{0\}} \simeq \mathbb{Q}_{\{0\}} \qquad (3.23) $$

where $MF_q(0)$ denotes the Milnor fiber of $q$ at 0, which is $T^*S^{n-1}$, and the second isomorphism comes from the orientation of $S^{n-1}$ coming from the canonical orientation of $\mathbb{C}$. Consider now $\mathbb{C}^2$ with a $\mathbb{C}^*$-action giving opposite weights to the coordinates $x, y$, and the quadratic $\mathbb{C}^*$-invariant form $q(x, y) = xy$. Then the naturality of the isomorphisms $\beta^\pm$ gives that the following triangle of perverse sheaves on $\{0\}$ is commutative:

<figure><img src="image_30.png" /></figure>

Consider now $\mathbb{C}^m$ with a linear $\mathbb{C}^*$-action and a $\mathbb{C}^*$-invariant quadratic form $q$. Up to changing the coordinates $q$ can be written: $q^0 + \sum_{k=1}^r x_k y_k$ with $q^0$ a quadratic form on $\mathbb{C}^{m-2r} = (\mathbb{C}^m)^0$ and $x_k, y_k$ $\mathbb{C}^*$-equivariant coordinates with opposite weight. Hence using the commutativity of the above triangle and the compatibility of $\beta^\pm$ with Thom-Sebastiani isomorphism, one obtains the commutativity of the following triangle:

<figure><img src="image_31.png" /></figure>

where the vertical arrows comes from the natural orientations on $\mathbb{C}^m$ and $(\mathbb{C}^m)^0$.

# 4 Białyński-Birula decomposition on a d-critical algebraic space

## 4.1 D-critical algebraic space

We recall here the notions and results of [6] and [8] about d-critical structures and the Donaldson-Thomas sheaf. Joyce and its 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 [6, Theo 2.1], Joyce constructs sheaves $S_X$ and $S_X^0$ 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 $i^{-1}(O_U)$ of functions on $U$ near $i(R)$ which vanishes on $i(R)$. There is then an exact sequence of sheaves on $R$ defining locally $S_X$:

$$0 \to S_X|_R \xrightarrow{\iota_{R,U}} \frac{i^{-1}(O_U)}{I_{R,U}^2} \to \frac{i^{-1}(T^*U)}{I_{R,U} \cdot i^{-1}(T^*U)} \quad (4.1)$$

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

$$S_X \to O_X \to O_{X^{red}} \quad (4.2)$$

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 $\Phi^*: S_Y^0 \to S_X^0$.

A d-critical structure on $X$ is a section $s$ of $S_X^0$ 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 $i(R) = \text{Crit}(f)$ 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 $U' \to U$, one can consider a subchart $(R' := R \times_U U', U', f' := f|_{U'}, i' := i|_{R'}) \to (R, U, f, i)$.

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 [6, 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 $(R', U', f', i') \subset (R, U, f, i)$ and $(S', V', g', j') \subset (S, V, g, j)$ such that $x \in R' \cap S'$, a critical chart $(T, W, h, k)$ and embeddings $\Phi: (R', U', f', i') \hookrightarrow (T, W, h, k)$ and $\Psi(S', V', g', j') \hookrightarrow (T, W, h, k)$.

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 $0 \to TU \to T^*U \to 0$, and then $\mathbb{L}_X|_{R^{red}} = i^*(K_U^{\otimes 2})|_{R^{red}}$, i.e. the sheaves $i^*(K_U^{\otimes 2})|_{R^{red}}$ 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 [6], one has a natural isomorphism:

$$J_{\Phi}: i^*(K_U^{\otimes 2})|_{R^{red}} \to j^*(K_V^{\otimes 2})|_{R^{red}} \quad (4.3)$$

Using the fact that two maps can be locally embedded in a single map, it is shown in [6, 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:

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

## 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. Because two critical charts are locally embedded into a single critical chart, it suffices then to compare $PV_{U,f}$ and $(\Phi|_R)^*PV_{V,g}$ for $\Phi: (R, U, f, i) \hookrightarrow (S, V, g, j)$ an embedding.

For an étale morphism $\iota: U' \to U$, giving a critical chart $(R', U', f, \iota)$, $\iota$ is in particular smooth of dimension 0, i.e. there is a natural isomorphism:

$$ \Xi_{\iota}: \iota|_{R'}^*PV_{U,f} \to PV_{U',f \circ \iota} \quad (4.5) $$

which is exactly the isomorphism $PV_{\iota}^{-1}$ in definition 2.15 of [8] (we have adopted here the notation of [9]). Moreover, for $(R, U, f, i)$ and $(S, V, g, j)$ two critical charts, considering the critical chart $(R \times S, U \times V, f \boxtimes g, i \times j$, recall the Thom-Sebastiani isomorphism:

$$ \mathcal{T}S_{U,f,V,g}: PV_{U \times V, f \boxtimes g} \to PV_{U,f} \boxtimes PV_{V,g} \quad (4.6) $$

Consider an embedding of critical charts $\Phi: (R, U, f, i) \hookrightarrow (S, V, g, j)$. From [8, Theo 5.1], an embedding of critical chart is étale locally of the form $(R, U, f, i) \hookrightarrow (R \times \{0\}, U \times \mathbb{C}^m, f \boxtimes q, i \times 0)$ with $q$ a quadratic form. This property is slightly less precise than the result of [8, Theo 5.1], where $q$ is required to be of the form $q(z_1, ..., z_m) = z_1^2 + ... + z_n^2$, but we will need to work in this generality in the $\mathbb{C}^*$-equivariant case. It means that for $x \in U$ there is there is smooth algebraic spaces $U', V'$, $x' \in U'$, morphisms $\iota: U' \to U$ with $\iota(x') = x$, $j: V' \to V$, $\Phi': U' \to V'$, $\alpha: V' \to U$ and $\beta: V' \to \mathbb{C}^m$, with $\iota, j$ and $\alpha \otimes \beta$ being étale, such that the following square commutes:

<figure><img src="image_32.png" /></figure>

and $g \circ j = (f \circ \iota) \boxtimes (q \circ \alpha): V' \to \mathbb{C}$. We denote by $(R', U', f \circ \iota, i')$ and $(S', V', g \circ j, j')$ the induced critical chart on $U'$ and $V'$. Using the fact that $PV_{\mathbb{C}^m, q} \simeq \mathbb{Q}_{\{0\}}$, one has a sequence of isomorphisms:

$$ \begin{aligned} \iota|_{R'}^*(PV_{U,f}) & \longrightarrow \iota|_{R'}^*(\mathrm{Id}_U \times 0)|_R^*(PV_{U,f} \boxtimes PV_{\mathbb{C}^m,q}) \\ & \stackrel{\iota|_{R'}^*(\mathrm{Id}_U \times 0)|_R^*(\mathcal{S}_{U,f,\mathbb{C}^m,q}^{-1})}{\longrightarrow} \Phi'|_{R'}^*(\alpha \times \beta)|_{S'}^*(PV_{U \times \mathbb{C}^m, f \boxtimes q}) \\ & \stackrel{\Phi'|_{R'}^*(\Xi_{\alpha \times \beta})}{\longrightarrow} \iota|_{R'}^*\Phi|_R^*(PV_{V,g}) \end{aligned} $$

However, these isomorphisms defined étale-locally do not glue into an isomorphism between $PV_{U,f}$ and $\Phi|_R^*(PV_{V,g})$, due to orientation issue. Namely, in the choice of the étale local description above, one can replace $\beta$ by $M \circ \beta$, with $M$ any matrix which is orthogonal for the quadratic form $q$. According to [8, Theo 3.1], the morphisms $PV_{\alpha \times \beta}$ and $PV_{\alpha \times M \circ \beta}$
differ then by det($M$) = ±1, i.e. the étale local isomorphisms constructed above differ by a sign if the corresponding trivialization corresponds to different choice of local orientations of the normal bundle of $\Phi(U)$ in $V$, i.e. to different choices of local isomorphisms between $K_U|_{R^{red}}$ and $\Phi^*(K_V)|_{R^{red}}$. One can deal with this sign ambiguity by tensoring by $P_\Phi$, the $\mathbb{Z}/2\mathbb{Z}$ principal bundle on $R$ parametrizing the local square roots of the natural isomorphism $J_\Phi$: [8, Theo 5.4] gives a natural isomorphism of perverse sheaves on $R$:

$$ \Theta(\Phi) : \mathcal{P}\mathcal{V}_{U,f} \to \Phi|_{R}^{*}(\mathcal{P}\mathcal{V}_{V,g}) \otimes_{\mathbb{Z}/2\mathbb{Z}} P_{\Phi} \quad (4.7) $$

The twist by $P_\Phi$ shows that the naive construction of $P_{X,s}$ does not work: we cannot glue the vanishing cycles on intersections of critical charts. One must then twist the vanishing cycles sheaf by a $\mathbb{Z}/2\mathbb{Z}$ principal bundle that will compensate the twist by $P_\Phi$. It is done using an orientation data, i.e. a choice of square root $K_{X,s}^{1/2}$ of $K_{X,s}$. On a critical chart $(R, U, f, i)$, one can construct a $\mathbb{Z}/2\mathbb{Z}$ principal bundle $Q_{R,U,f,i}$ on $R$ which parametrizes local square roots of $\iota_{R,U,f,i}$. Informally, $K_{X,s}^{1/2}$ defines a canonical orientations on the critical charts, and $Q_{R,U,f,i}$ presents the difference between the orientation of a chart and this canonical orientation. One has also from equation (4.4) (see [8, Theo 6.9, ii])) a natural isomorphism:

$$ \Lambda_{\Phi} : Q_{S,V,g,j}|_{R} \to P_{\Phi} \otimes_{\mathbb{Z}/2\mathbb{Z}} Q_{R,U,f,i} \quad (4.8) $$

One has then an isomorphism:

$$ \Theta_{\Phi} \otimes_{\mathbb{Z}/2\mathbb{Z}} \Lambda_{\Phi}^{-1} : \mathcal{P}\mathcal{V}_{U,f} \otimes_{\mathbb{Z}/2\mathbb{Z}} Q_{R,U,f,i} \to (\mathcal{P}\mathcal{V}_{V,g} \otimes_{\mathbb{Z}/2\mathbb{Z}} Q_{S,V,g,j})|_{R} \quad (4.9) $$

One can then use the fact that, up to shrinking them, two intersecting critical charts embeds in a single critical chart, to glue these sheaves. [8, Theo 6.9] shows that it defines a perverse sheaf (or strongly polarized monodromic mixed Hodge module) $P_{X,s}$ on $X$, called the Donaldson-Thomas sheaf, defined locally by the natural isomorphisms:

$$ \begin{gathered} \omega_{R,U,f,i} : P_{X,s}|_{R} \to \mathcal{P}\mathcal{V}_{U,f} \otimes_{\mathbb{Z}/2\mathbb{Z}} Q_{R,U,f,i} \\ (\Theta_{\Phi} \otimes_{\mathbb{Z}/2\mathbb{Z}} \Lambda_{\Phi}^{-1}) \circ \omega_{R,U,f,i} = \omega_{S,V,g,j}|_{R} \end{gathered} \quad (4.10) $$

We will from now omit the subscript $\mathbb{Z}/2\mathbb{Z}$ in the tensor product of $\mathbb{Z}/2\mathbb{Z}$ principal bundles for readability.

## 4.3 Białynicki-Birula decomposition

### 4.3.1 Torus-equivariant d-critical algebraic space

An action $\mu : T \times X \to X$ of a one dimensional torus $\mathbb{C}^*$ 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 $\mu(\gamma)^*(s) = s$ (resp. $\mu(\gamma)^*K_{X,s}^{1/2} = K_{X,s}^{1/2}$) for $\gamma \in T$. In particular, the d-critical structure of the classical truncation of a $(-1)$-shifted symplectic scheme with a $\mathbb{C}^*$-action leaving the $(-1)$-shifted symplectic structure invariant is $\mathbb{C}^*$-invariant. If the action is étale locally linearizable, then translating [6, Prop 2.43, 2.44] from the Zariski to the étale topology, we can then work with $\mathbb{C}^*$-equivariant critical charts. Namely, we can cover $X$ in the étale topology by charts $(R, U, f, i)$ such that $U$ has a $\mathbb{C}^*$-action for which $i$ is equivariant and $f$ is invariant. Moreover, considering $(R, U, f, i)$ and $(S, V, g, j)$ two $\mathbb{C}^*$-equivariant critical charts, and $x \in R \cap S$,
one has étale restrictions $(R', U', f', i') \to (R, U, f, i)$ and $(S', V', g', j') \to (S, V, g, j)$ with $x \in R' \cap S'$, and $\mathbb{C}^*$-equivariant critical chart $(T, W, h, k)$ and $\mathbb{C}^*$-equivariant embeddings $\Phi : (R', U', f', i') \hookrightarrow (T, W, h, k)$ and $\Psi(S', V', g', j') \hookrightarrow (T, W, h, k)$.

As explained in [6, Cor 2.45], considering the closed embedding $\xi_X : X^0 \hookrightarrow X$, $X^0$ admits a natural d-critical structure $(X^0, s^0 := \xi_X^*(s))$. Indeed, for $(R, U, f, i)$ a $\mathbb{C}^*$-equivariant d-critical chart, denoting by $R^0$, $U^0$ the $\mathbb{C}^*$-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 $s_{R^0, U^0}^0 = f^0 + I_{R^0, U^0}^2$, and then $X^0$ is covered by d-critical charts. Because $\xi_X : X^0 \hookrightarrow X$ is a closed embedding, $T_{X^0/X}^* = 0$, and then $K_{X^0, s^0} = \xi_X^*(K_{X,s})$ using [6, Prop 2.30]. One can then naturally define the orientation $K_{X^0, s^0}^{1/2} := \xi_X^*(K_{X,s}^{1/2})$ on $(X^0, s^0)$.

Consider the decomposition of $X^0$ into connected components $X^0 = \bigsqcup_{\pi \in \Pi} X_{\pi}^0$: one can further consider the oriented d-critical schemes $(X_{\pi}^0, s_{\pi}^0, K_{X_{\pi}^0, s_{\pi}^0}^{1/2})$. Denote by $\mathrm{Ind}_{\pi}$ the number of contracting weight in the tangent-obstruction complex of $X$ at $X_{\pi}^0$. In a critical chart $(R, U, f, i)$, the tangent-obstruction complex is given by $0 \to TU \to T^*U \to 0$, and then, denoting by $d_+$, $d_0$, $d_-$ respectively the number of contracting, invariant and repelling weights in $TU$ at $i(X_{\pi}^0 \cap R)$, one has $\mathrm{Ind}_{\pi} = d_+ - d_-$, i.e. this definition is consistent with the previous definition on a critical chart.

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

$$ \beta_{X,s}^{\pm} : (p_X^{\pm})!(\eta_X^{\pm})^* P_{X,s} \xrightarrow{\sim} \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} P_{X_{\pi}^0, s_{\pi}^0} \quad (4.11) $$

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

$$ \begin{tikzcd} (p_X^+)! (\eta_X^+)^* P_{X,s} \arrow[r, "\beta_{X,s}^+"'] & \bigoplus_{\pi \in \Pi} \mathbb{L}^{\mathrm{Ind}_{\pi}/2} P_{X_{\pi}^0, s_{\pi}^0} \arrow[d, "S_X^{-1} \circ (p_X^+)! (\eta_X^+)^* (\Sigma_{X,s})"] \\ \mathbb{D}(p_X^-)! (\eta_X^-)^* P_{X,s} \arrow[r, "\mathbb{D}\beta_{X,s}^-"'] & \mathbb{D} \bigoplus_{\pi \in \Pi} \mathbb{L}^{-\mathrm{Ind}_{\pi}/2} P_{X_{\pi}^0, s_{\pi}^0} \arrow[d, "\bigoplus_{\pi \in \Pi} \mathbb{L}^{\mathrm{Ind}_{\pi}/2} \Sigma_{X_{\pi}^0, s_{\pi}^0}"] \end{tikzcd} $$

**Proof:** Because we have a local description of the above sheaves in critical charts, we will first build isomorphisms:

$$ \beta_R^{\pm} : ((p_X^{\pm})!(\eta_X^{\pm})^* P_{X,s})|_{R^0} \to \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} P_{X^0, s^0}|_{R^0} \quad (4.12) $$

for each $\mathbb{C}^*$-equivariant critical chart $(R, U, f, i)$ compatible with the monodromy and the duality. Because morphisms in the abelian category $\mathrm{Perv}(X_{\pi}^0)[-\mathrm{Ind}_{\pi}]$ (resp. $\mathrm{MMHM}(X_{\pi}^0)[-\mathrm{Ind}_{\pi}]$) satisfy étale descent by [8, Theo 2.7] (resp. [8, Theo 2.21]), it suffices to show that two of these local isomorphisms agree on their restriction to show that they glue uniquely into isomorphisms $\beta_{X,s}^{\pm}$ compatible with monodromy and duality.
### 4.3.2 Isomorphisms on critical charts

Consider a $\mathbb{C}^*$-equivariant critical chart $(R, U, f, i)$. Because $R \to X$ is étale, in particular it is smooth of relative dimension 0, and $\mathbb{C}^*$-equivariant:

$$ ((p_X^{\pm})!(\eta_X^{\pm})^*P_{X,s})|_{R^0} \simeq (p_R^{\pm})!(\eta_R^{\pm})^*P_{X,s}|_R \\ \simeq (p_R^{\pm})!(\eta_R^{\pm})^*(\mathcal{PV}_{U,f} \otimes Q_{R,U,f,i}) \\ = (p_R^{\pm})!((\eta_R^{\pm})^*\mathcal{PV}_{U,f} \otimes (\eta_R^{\pm})^*Q_{R,U,f,i}) \quad (4.13) $$

where we have used the commutation of hyperbolic localization with smooth maps in the first line, the isomorphism $\omega_{R,U,f,i}$ in the second line and the fact that the pullback commutes with the tensor product in the last line.

A local section of $Q_{R,U,f,i}$ is a local isomorphism between $i^*(K_U)|_{R^{red}}$ and $K_X^{1/2}|_{R^{red}}$ which is a square root of $\iota_{R,U,f,i}$: a local section of $\xi_R^*Q_{R,U,f,i}$ gives then an isomorphism between $\xi_R^*i^*K_U = (i^0)^*K_{U^0}$ and $\xi_R^*K_X^{1/2}|_R = K_{X^0}^{1/2}|_{R^0}$ which is a square root of $\xi_R^*\iota_{R,U,f,i} = \iota_{R^0,U^0,f^0,i^0}$, i.e.:

$$ \xi_R^*Q_{R,U,f,i} \simeq Q_{R^0,U^0,f^0,i^0} \quad (4.14) $$

Hence, recalling $\xi_R = \eta_R^{\pm} \circ \zeta_R^{\pm} : R^0 \to R$, one has:

$$ (p_R^{\pm})^*Q_{R^0,U^0,f^0,i^0} \simeq (p_R^{\pm})^*(\zeta_R^{\pm})^*(\eta_R^{\pm})^*Q_{R,U,f,i} \\ \simeq (\eta_R^{\pm})^*Q_{R,U,f,i} \quad (4.15) $$

where the second line follows from the fact that $p_R^{\pm} : R^{\pm} \to R^0$ is an affine fibration. Hence one obtains:

$$ ((p_X^{\pm})!(\eta_X^{\pm})^*P_{X,s})|_{R^0} \simeq (p_R^{\pm})!((\eta_R^{\pm})^*\mathcal{PV}_{U,f} \otimes (p_R^{\pm})^*Q_{R^0,U^0,f^0,i^0}) \\ \simeq (p_R^{\pm})!(\eta_R^{\pm})^*\mathcal{PV}_{U,f} \otimes Q_{R^0,U^0,f^0,i^0} \\ \simeq \bigoplus_{\pi \in \pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} \mathcal{PV}_{U_{\pi}^0, f_{\pi}^0} \otimes Q_{R_{\pi}^0, U_{\pi}^0, f_{\pi}^0, i_{\pi}} \\ \simeq \bigoplus_{\pi \in \pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} P_{X_{\pi}^0, s_{\pi}^0}|_{R_{\pi}^0} \quad (4.16) $$

Here the second line follows from the projection formula, the second line is the isomorphism $\beta_{U,f} \otimes \mathrm{Id}$, and the last line follows from the isomorphisms $\omega_{R_{\pi}^0, U_{\pi}^0, f_{\pi}^0, i_{\pi}}^{-1}$. We have the built an isomorphism $\beta_R^{\pm}$.

According to the square 6.8 in [8, Theo 6.9], the isomorphisms $\omega_{R,U,f,i}$ and $\omega_{R_{\pi}^0, U_{\pi}^0, f_{\pi}^0, i_{\pi}}^{-1}$ commute with the monodromy, and $\beta_{U,f}^{\pm}$ commutes also with the monodromy, hence $\beta_R^{\pm}$ commute with the monodromy.

Consider now the following diagram:

$$ \begin{tikzcd} ((p_X^{\pm})!(\eta_X^{\pm})^*P_{X,s})|_{R^0} \arrow[r, "\simeq"] & (p_R^{\pm})!(\eta_R^{\pm})^*P_{X,s}|_R \arrow[d, "(p_R^{\pm})!(\eta_R^{\pm})^*\omega_{R,U,f,i}"] & (p_R^{\pm})!(\eta_R^{\pm})^*(\mathcal{PV}_{U,f} \otimes Q_{R,U,f,i}) \\ \arrow[d, "((p_X^{\pm})!(\eta_X^{\pm})^*\Sigma_{X,s})|_{R^0}"] & \arrow[d, "(p_R^{\pm})!(\eta_R^{\pm})^*\Sigma_R"] & \arrow[d, "(p_R^{\pm})!(\eta_R^{\pm})^*(\sigma_{U,f} \times \mathrm{Id})"] \\ ((p_X^{\pm})!(\eta_X^{\pm})^*\mathcal{D}P_{X,s})|_{R^0} \arrow[r, "\simeq"] & (p_R^{\pm})!(\eta_R^{\pm})^*\mathcal{D}P_{X,s}|_R \arrow[d, "(p_R^{\pm})!(\eta_R^{\pm})^*\mathcal{D}\omega_{R,U,f,i}"] & (p_R^{\pm})!(\eta_R^{\pm})^*(\mathcal{D}\mathcal{PV}_{U,f} \otimes Q_{R,U,f,i}) \\ \arrow[d, "S_X^{-1}|_{R^0}"] & \arrow[d, "S_R^{-1}"] & \arrow[d, "S_R^{-1}"] \\ (\mathcal{D}(p_X^{\pm})!(\eta_X^{\pm})^*P_{X,s})|_{R^0} \arrow[r, "\simeq"] & \mathcal{D}(p_R^{\pm})!(\eta_R^{\pm})^*P_{X,s}|_R \arrow[d, "(\mathcal{D}(p_R^{\pm})!(\eta_R^{\pm})^*\omega_{R,U,f,i}")", "\simeq"'] & \mathcal{D}(p_R^{\pm})!(\eta_R^{\pm})^*(\mathcal{PV}_{U,f} \otimes Q_{R,U,f,i}) \end{tikzcd} $$
the top left square is commutative because the commutation of the hyperbolic localization with pullback by smooth maps is functorial, the bottom left square commutes because the isomorphism $S$ is compatible with pullback by smooth maps, the top right square is $(p_R^+)! (\eta_R^+)^*$ applied to the commutative square 6.8 in [8, Theo 6.9], and the bottom right diagram commutes from the functoriality of $S_R$, hence the whole diagram is commutative. Gluing horizontally this commutative diagram, the commutative diagram expressing the compatibility of $\beta_{U,f}^\pm$ with the duality $\sigma$, and the commutative diagram 6.8 of [8, Theo 6.9] applied to the charts $(R_\pi^0, U_\pi^0, f_\pi^0, i_\pi^0)$ in $(X_\pi^0, s_\pi^0)$, one obtains that the following square is commutative:

$$ ((p_X^+)! (\eta_X^+)^* P_{X,s})|_{R^0} \xrightarrow{\beta_{R,U,f,i}^+} \bigoplus_{\pi \in \Pi} \mathbb{L}^{\mathrm{Ind}_\pi/2} P_{X_\pi^0, s_\pi^0}|_{R_\pi^0} \\ \xleftarrow{\leftarrow} \leftarrow_{(S_X^{-1} \circ (p_X^+)! (\eta_X^+)^* (\Sigma_{X,s}))|_{R^0}} \bigoplus_{\pi \in \Pi} \mathbb{L}^{\mathrm{Ind}_\pi/2} \Sigma_{X_\pi^0, s_\pi^0} \\ \xrightarrow{\xleftarrow{\leftarrow} \leftarrow_{\mathcal{D}\beta_{R,U,f,i}^-}} \mathcal{D} \bigoplus_{\pi \in \Pi} \mathbb{L}^{-\mathrm{Ind}_\pi/2} P_{X_\pi^0, s_\pi^0}|_{R_\pi^0} $$

hence $\beta_R^\pm$ are compatible with duality. This construction can be lifted to mixed Hodge modules, and the compatibility for monodromy and strong polarization can be checked at the level of perverse sheaves.

### 4.3.3 Compatibility with embeddings

Consider an equivariant embedding of $\mathbb{C}^*$-equivariant charts $\Phi : (R, U, f, i) \hookrightarrow (S, V, g, j)$ We denote by $\Phi^0 : (R^0, U^0, f^0, i^0) \hookrightarrow (S^0, V^0, g^0, j^0)$ the induced embeddings of $\mathbb{C}^*$-fixed charts. The principal $\mathbb{Z}/2\mathbb{Z}$ bundle $P_{\Phi^0}$ on $R^0$ gives local isomorphisms between $(i^0)^*(K_{U^0}) = (i^0)^*\xi_U^*(K_U) = \xi_R^*i^*(K_U)$, and $(i^0)^*(\Phi^0)^*(K_{V^0}) = (i^0)^*(\Phi^0)^*\xi_V^*(K_V) = \xi_R^*i^*\Phi^*(K_V)$ which are the square root of the natural isomorphism $J_{\Phi^0} = \beta_R^*(J_\Phi)$, i.e. :

$$ \xi_R^* P_\Phi \simeq P_{\Phi^0} \qquad (4.17) $$

Hence, recalling $\xi_R = \eta_R^\pm \circ \zeta_R^\pm : R^0 \to R$, one has:

$$ \begin{aligned} (p_R^\pm)^* P_{\Phi^0} &\simeq (p_R^\pm)^* (\zeta_R^\pm)^* (\eta_R^\pm)^* P_\Phi \\ &\simeq (\eta_R^\pm)^* P_\Phi \end{aligned} \qquad (4.18) $$

where the second line follows from the fact that $p_R^\pm : R^\pm \to R^0$ is an affine fibration. In particular, one has obviously from the definitions that the following square of isomorphisms is commutative:

$$ \begin{array}{ccc} (\eta_R^\pm)^* P_\Phi \otimes (\eta_R^\pm)^* Q_{S,V,g,j}|_R & \xrightarrow{\simeq} & (p_R^\pm)^* P_{\Phi^0} \otimes (p_R^\pm)^* Q_{S^0,V^0,g^0,j^0}|_R \\ (\eta_R^\pm)^* \Lambda_\Phi \uparrow & & (p_R^\pm)^* \Lambda_{\Phi^0} \uparrow \\ (\eta_R^\pm)^* Q_{R,U,f,i} & \xrightarrow{\simeq} & (p_R^\pm)^* P_{\Phi^0} \otimes (p_R^\pm)^* Q_{R^0,U^0,f^0,i^0} \end{array} $$

In the sequel, we will use implicitly the functorial isomorphisms giving the commutation of restriction with hyperbolic localisation, namely $(p_R^+)! (\eta_R^+)^* F|_R \simeq ((p_S^+)! (\eta_S^+)^* F)_R$ for $R \to S$
étale i.e. smooth of relative dimension 0, and $F \in D_c^b(S)$. Consider the following diagram:

<figure><img src="image_33.png" /></figure>

the left triangle is commutative because it is $(p_R^{\pm})!(\eta_R^{\pm})^*$ applied to the commutative triangle 6.10 in [8, Theo 6.9]. The upper right square is commutative from the projection formula behaves well with composition, i.e. the isomorphisms $p_!(A \otimes p^*B \otimes p^*C) \simeq p_!(A \otimes p^*B) \otimes C \simeq p_!A \otimes B \otimes C$ and $p_!(A \otimes p^*B \otimes p^*C) \simeq p_!(A \otimes p^*(B \otimes C)) \simeq p_!A \otimes B \otimes C$ agree in the six-functor formalism. The bottom right square is commutative because the diagram 4.3.3 is commutative and the projection formula is functorial. Hence the whole diagram is commutative.

Consider now the following diagram of isomorphisms:

<figure><img src="image_34.png" /></figure>

the top left square is $- \otimes Q_{R^0, U^0, f^0, i^0}$ applied to the diagram 4.3.5, which we will prove to be commutative, hence is commutative, the bottom left square is commutative because of the properties of the tensor product, and the right triangle is commutative because it is a sum of the commutative diagram 6.10 in [8, Theo 6.9] for each $R_{\pi}^0$, hence the whole diagram is commutative.

Gluing the two commutative diagram above, one obtains that for $\Phi : (R, U, f, i) \hookrightarrow (S, V, g, j)$ a $\mathbb{C}^*$-equivariant embedding of critical charts:

$$ \beta_S^{\pm}|_{R^0} = \beta_R^{\pm} \qquad (4.19) $$

### 4.3.4 Gluing of local isomorphisms

Having built the isomorphisms $\beta_R^{\pm}$ for each equivariant critical charts $(R, U, f, i)$, we only need to show that these isomorphisms agree on intersections of such charts (we will denote by abuse of notation $R \times_X S$ by $R \cap S$ when $R \to X, S \to X$ are étale maps). Consider two such critical charts $(R, U, f, i)$ and $(S, V, g, j)$, and $x \in R^0 \cap S^0$: we must show that $\beta_R^{\pm}$ and $\beta_S^{\pm}$ agree locally near $x$. Let's take $\mathbb{C}^*$-equivariant subcharts $(R', U', f', i') \subseteq (R, U, f, i)$ and
$(S', V', g', j') \subseteq (S, V, g, j)$ with $x \in R' \cap S'$, a $\mathbb{C}^*$-equivariant critical chart $(T, W, h, k)$ and $\mathbb{C}^*$-equivariant embeddings $\Phi : (R', U', f', i') \hookrightarrow (T, W, h, k)$ and $\Psi : (S', V', g', j') \hookrightarrow (T, W, h, k)$.

The proof that the isomorphisms $\beta$ agrees on embedding gives that:

$$ \begin{align} \beta_R^{\pm}|_{R'0} &= \beta_{R'}^{\pm} \nonumber \\ \beta_S^{\pm}|_{S'0} &= \beta_{S'}^{\pm} \nonumber \\ \beta_T^{\pm}|_{R'0} &= \beta_{R'}^{\pm} \nonumber \\ \beta_T^{\pm}|_{S'0} &= \beta_{S'}^{\pm} \tag{4.20} \end{align} $$

where we have used in the first two lines the fact the restriction of a critical chart is a particular type of embedding, and then:

$$ \begin{align} \beta_R^{\pm}|_{R' \cap S'} &= \beta_{R'}^{\pm}|_{R' \cap S'} \nonumber \\ &= \beta_T^{\pm}|_{R' \cap S'} \nonumber \\ &= \beta_{S'}^{\pm}|_{R' \cap S'} \nonumber \\ &= \beta_S^{\pm}|_{R' \cap S'} \tag{4.21} \end{align} $$

hence $\beta_R^{\pm}$ and $\beta_S^{\pm}$ agree locally near $x$ for each $x \in U \cap V$, hence agree in $U \cap V$, hence they glue into morphisms $\beta_{X,s}^{\pm}$ compatible with monodromy and duality. Because they agree at the level of perverse sheaves, the corresponding isomorphisms of mixed Hodge modules agree.

### 4.3.5 Proof of the claim

We prove here the commutativity of the following diagram:

<figure><img src="image_35.png" /></figure>

Because the abelian category $\bigoplus_{\pi \in \Pi} \mathrm{Perv}(R_{\pi}^0)[-\mathrm{Ind}_{\pi}]$ satisfies étale descent from [8, Theo 2.7], it suffices to show that the diagram commutes étale locally to show that it commutes, hence one can use étale charts where the embedding is trivialized. For this, use the $\mathbb{C}^*$-equivariant version of [8, Theo 5.1] proven in 6.1, giving that for $x \in i(R^0)$, there is smooth $\mathbb{C}^*$-equivariant algebraic spaces $U', V'$, $x' \in U'$, $\mathbb{C}^*$-equivariant morphisms $\iota : U' \to U$ with $\iota(x') = x$, $\jmath : V' \to V$, $\Phi' : U' \to V'$, $\alpha : V' \to U$ and $\beta : V' \to \mathbb{C}^m$ with $\mathbb{C}^m$ provided with a $\mathbb{C}^*$-action such that coordinates are equivariant, with $\iota, \jmath$ and $\alpha \otimes \beta$ being étale, such that the following square commutes:
<figure><img src="image_36.png" /></figure>

with $g \circ j = f \circ \iota + q \circ \alpha : V' \to \mathbb{C}$, with $q$ a $\mathbb{C}^*$-invariant quadratic form on $\mathbb{C}^m$. Consider also the $\mathbb{C}^*$-fixed version of this diagram, which is still commutative, with $\iota^0, j^0, \alpha^0 \times \beta^0$ étale:

<figure><img src="image_37.png" /></figure>

with $g^0 \circ j^0 = f^0 \circ \iota^0 + q^0$. We denote then $R' := \iota^{-1}(R)$, $S' := j^{-1}(S)$. Consider the following diagram:

<figure><img src="image_38.png" /></figure>

The first square commutes because the diagram 3.5 commutes. The second square commutes because it is the restriction to $R'^0$ of the commutative square expressing the naturality of $\beta$ with respect to the Thom-Sebastiani isomorphism in proposition 3.5. The third square and the last square commutes because they are the restriction to $R'^0$ of the commutative squares expressing the commutation of $\beta$ with the pullback by the étale morphism $\alpha \times \beta$ and $j$ in proposition 3.4, then the whole diagram commutes.
Because for $a : Y \to Z$ étale, $a^*K_Z = K_Y$, the construction of the $\mathbb{Z}/2\mathbb{Z}$ bundle $P_\Phi$ is functorial with étale maps of critical locus, i.e. one has isomorphisms:

$$ P_{\Phi}|_{R'} = P_{\Phi'} = P_{\mathrm{Id}_{U} \times 0}|_{R'} \qquad (4.22) $$

the canonical orientation of $\mathbb{C}^m$ gives an isomorphism $K_U|_R \to (K_U|_R \times K_{\mathbb{C}^m}|_0)$, hence a section of $(P_{\mathrm{Id}_{U} \times 0})$. We denote by $\delta$ the induced trivialization of $P_\Phi|_{R'}$. Recall that $\xi_R^*P_\Phi = P_{\Phi^0}$, hence $\xi_{R'}^*P_\Phi|_{R'} = P_{\Phi^0}|_{R'}$, and $\delta^0 := \xi_{R'}^*(\delta)$ is the trivialization of $P_{\Phi^0}|_{R'} = P_{\Phi^0} = P_{\mathrm{Id}_{U^0} \times 0}|_{R'}$ obtained by considering the canonical orientation of $\mathbb{C}^r = (\mathbb{C}^m)^0$. One obtains also that the following diagram commutes:

<figure><img src="image_39.png" /></figure>

hence tensoring $\beta_{U,f}|_{R'}$ and applying the projection formula, one obtains that the following diagram is commutative:

<figure><img src="image_40.png" /></figure>

According to the étale local definition of $\Theta(\Phi)$ given in the diagram 5.17 of [8], the restriction of the diagram 4.3.5 on $R'^0$ is obtained by gluing vertically the diagrams 4.3.5 and 4.3.5, hence the diagram 4.3.5 commutes étale locally, and then commutes on $\mathbb{R}^0$. $\square$

## 4.4 Compatibility with smooth pullbacks

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 $\phi^*(t) = s$). [9, Cor 3.8] shows that:

$$ K_{X,s} = K_{Y,t} \otimes (\Lambda^{\mathrm{top}} T_{X/Y}^*)^{\otimes 2} |_{X^{\mathrm{red}}} \qquad (4.23) $$

A smooth map of oriented d-critical scheme $\phi : (X, s, K_{X,s}^{1/2}) \to (Y, t, K_{Y,t}^{1/2})$ 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:

$$ K_{X,s}^{1/2} = \phi^*(K_{Y,t}^{1/2}) \otimes (\Lambda^{\mathrm{top}} T_{X/Y}^*)|_{X^{\mathrm{red}}}. \qquad (4.24) $$
which is a square root of the above isomorphism. Giving such a map, [9, Prop 4.5] built an isomorphism:

$$ \Delta_{\phi}: \phi^{*}[d]P_{Y,t} \simeq P_{X,s} \qquad (4.25) $$

Consider now a $\mathbb{C}^*$-equivariant smooth map of oriented d-critical schemes $\phi: (X, s, K_{X,s}^{1/2}) \to (Y, t, K_{Y,t}^{1/2})$ of relative dimension $d$. The induced morphism $\phi^0: X^0 \to Y^0$ is smooth of dimension $d$. One has:

$$ \begin{aligned} (\phi^0)^*(t^0) &= (\phi^0)^*\xi_Y^*(t) \\ &= \xi_X^*\phi^*(t) \\ &= \xi_X^*(s) \\ &= s^0 \end{aligned} \qquad (4.26) $$

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. Moreover, consider the following commutative diagram of isomorphisms:

<figure><img src="image_41.png" /></figure>

where the left vertical isomorphism is defined such that the diagram is commutative. The isomorphism of the left vertical arrow together with the map $\phi^0$ defines a smooth morphism of oriented d-critical scheme $\phi^0: (X^0, s^0, K_{X^0,s^0}^{1/2}) \to (Y^0, t^0, K_{Y^0,t^0}^{1/2})$ of relative dimension $d$, hence one can consider the isomorphism:

$$ \Delta_{\phi^0}: (\phi^0)^*[d]P_{Y^0,t^0} \to P_{X^0,s^0} \qquad (4.27) $$

**Proposition 4.2.** 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 $\mathbb{C}^*$-equivariant smooth map of oriented critical algebraic space of relative dimension $d$, the following diagram of isomorphisms commutes:

<figure><img src="image_42.png" /></figure>

Proof: It suffices to show that this square commutes étale locally on $X^0$. Consider $x \in X^0$, and denote $y := \phi(x) \in Y^0$. According to lemma 6.2, we can then choose two $\mathbb{C}^*$-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 $\Phi \circ i = j \circ \phi$. Notice that:

$$ 
\begin{aligned}
i^*(K_U)|_{R^{\text{red}}} &= i^*(\Phi^*[d](K_V) \otimes (\Lambda^{\text{top}}T_{U/V}^*))|_{R^{\text{red}}} \\
&= (\Phi|_R)^*(j^*(K_V)|_{S^{\text{red}}} \otimes (\Lambda^{\text{top}}T_{R/S}^*))|_{R^{\text{red}}} \\
K_{X,s}^{1/2}|_{R^{\text{red}}} &= (\Phi|_R)^*(K_{Y,t}^{1/2}|_{S^{\text{red}}} \otimes (\Lambda^{\text{top}}T_{R/S}^*)|_{R^{\text{red}}})
\end{aligned} 
\quad (4.28) $$

hence there is a natural isomorphism $\alpha_{\Phi}: (\Phi|_R)^*Q_{S,V,g,j} \simeq Q_{R,U,f,i}$.

Consider now the following diagram:

<figure><img src="image_43.png" /></figure>

the top square is commutative because the isomorphism giving the commutation of hyperbolic localization and smooth pullbacks is functorial, and the bottom square is commutative because it is $(p_R^{\pm})!(\eta_R^{\pm})^*$ applied to the commutative square 4.6 in [9, Prop 4.5]. From the proposition 2.4, we can deduce that the following square of isomorphism is commutative:

<figure><img src="image_44.png" /></figure>

And then using the natural commutations of the six-functor formalism, one obtains that the following diagram is commutative:

<figure><img src="image_45.png" /></figure>

Gluing horizontally the two above commutative diagrams with the product of the commutative diagram of proposition 3.4 with $\alpha_{\Phi^0}$, and finally with the sum of the commutative square 4.6 in [9, Prop 4.5] applied to the morphisms $\phi_{\pi}^0$, one obtains that the square of the proposition is commutative when it is restricted on $R$, hence it commutes étale locally on $X$, and then it commutes. The commutation of this diagram for the corresponding morphism of monodromic mixed Hodge modules can be checked at the level of perverse sheaves. $\square$

# 5 Hyperbolic localization on a d-critical algebraic stack

## 5.1 Hyperbolic localization on an algebraic stack

**Definition 5.1.** An algebraic stack $\mathcal{X}$ with an action of a 1-dimensional torus $\mathbb{C}^*$ is said to be lisse-étale locally linearizable if for every point $x$ there is a $\mathbb{C}^*$-equivariant smooth affine
neighborhood $(X, u)$ of $(\mathcal{X}, x)$ inducing an isomorphism of stabilizers of $u$ and $x$.

According to [16, Theo 20.1], a quasi-separated algebraic stack with affine stabilizers locally of finite presentation over $\mathbb{C}$ is lisse-étale locally linearizable over each point $x \in \mathcal{X}$ such that, denoting by $G_x$ the stabilizer subgroup of $x$, $G_y$ the stabilizer subgroup of the image $y$ of $x$ in $\mathcal{Y} := \mathcal{X}/T$, and $T_x \subset T$ the subgroup stabilizing $x$, the following exact sequence is split:

$$1 \to G_x \to G_y \to T_x \to 1 \qquad (5.1)$$

Consider now the functors of groupoids:

$$ \begin{align} \mathcal{X}^0 : Y &\mapsto \operatorname{Hom}_S^T(Y, \mathcal{X}) \nonumber \\ \mathcal{X}^+ : Y &\mapsto \operatorname{Hom}_S^T((\mathcal{A}_Y^1)^+, \mathcal{X}) \nonumber \\ \mathcal{X}^- : Y &\mapsto \operatorname{Hom}_S^T((\mathcal{A}_Y^1)^-, \mathcal{X}) \tag{5.2} \end{align} $$

where the superscript $\mathbb{C}^*$ denotes the $\mathbb{C}^*$-equivariant morphism, and $T, (\mathcal{A}_Y^1)^+, (\mathcal{A}_Y^1)^-$ has the trivial, resp. usual, resp. opposite $\mathbb{C}^*$-action. We can define the whole hyperbolic localization diagram by the same way than for algebraic spaces, and we have:

**Proposition 5.2.** For $\mathcal{X}$ an $S$ algebraic stack with a lisse-étale locally linearizable $\mathbb{C}^*$-action, $\mathcal{X}^0$ and $\mathcal{X}^\pm$ are representable as algebraic stacks, and for any smooth $\mathbb{C}^*$-equivariant 1-morphism $Y \to \mathcal{X}$, $Y^0 \to \mathcal{X}^0$ and $Y^\pm \to \mathcal{X}^\pm$ are also smooth and the natural morphism of algebraic spaces $Y^\pm \to Y^0 \times_{\mathcal{X}_0} \mathcal{X}^\pm$ is an affine fiber bundle.

**Proof:** $\mathcal{X}^0$ and $\mathcal{X}^\pm$ are stacks because the above functors satisfy descent. Two $\mathbb{C}^*$-equivariant 1-morphisms $Y \to \mathcal{X}$ are 2-isomorphic if and only if they are 2-isomorphic to a single 1-morphism, hence as functors of groupoids the diagonal $\mathcal{X}^0 \to \mathcal{X}^0 \times \mathcal{X}^0$ is the pullback of the diagonal $\mathcal{X} \to \mathcal{X} \times \mathcal{X}$, hence it is representable because $\mathcal{X}$ is an algebraic stack. Similarly, two $\mathbb{C}^*$-equivariant morphisms $y, y' : \mathbb{A}_Y^1$ are 2-isomorphic if and only if their restriction to 0 and to 1 are 2-isomorphic to single 1-morphisms, hence as functors of groupoids the diagonal $\mathcal{X}^\pm \to \mathcal{X}^\pm \times \mathcal{X}^\pm$ is the pullback of the diagonal of $\mathcal{X}^0 \times \mathcal{X}$, hence it is representable because $\mathcal{X}$ and $\mathcal{X}^0$ has representable diagonal.

Consider a smooth $\mathbb{C}^*$-equivariant 1-morphism $Y \to \mathcal{X}$ from an algebraic space $Y$. Then $Y^0 \to Y \to X$ is $\mathbb{C}^*$-equivariant, and then factorize as a morphism $Y^0 \to X^0$. Consider a 1-morphism $U \to \mathcal{X}^0$ from an algebraic space, and the following commutative cube:

<figure><img src="image_46.png" /></figure>
the above, below, left and right squares are Cartesian and in the front and back squares the right vertical arrow is the restriction of the left vertical arrow on the $\mathbb{C}^*$-fixed domains and codomains. Because $Y \to \mathcal{X}$ is presentable and the left square is Cartesian, then $U \times_{\mathcal{X}} Y$ is an algebraic space, and then because the above square is Cartesian $U \times_{\mathcal{X}^0} Y^0$ is an algebraic space. Because $Y \to \mathcal{X}$ is smooth and the left square is Cartesian, $U \times_{\mathcal{X}} Y \to U$ is smooth, and then proposition 2.4 applied to the back square, which is a square of algebraic spaces, gives that $U \times_{\mathcal{X}^0} Y^0 \to U$ is smooth. Hence $Y^0 \to \mathcal{X}^0$ is smooth. As morphism of functors of groupoids $Y^\pm \to \mathcal{X}$ factorize as a morphism $Y^\pm \to \mathcal{X}^\pm$. Exactly the same proof as above, replacing the superscripts 0 by $\pm$ gives that $Y^\pm \to \mathcal{X}^\pm$ is smooth.

Consider the smooth and surjective $\mathbb{C}^*$-equivariant presentation $Y$ of $\mathcal{X}$ obtained by taking the disjoint union of $\mathbb{C}^*$-equivariant smooth affine neighborhood $(Y_x, y_x)$ inducing an isomorphism of stabilizers for each point $x$ of $X$. A point $x \in \mathcal{X}^0$ has stabilizer $\mathbb{C}^*$, and then the point $y_x$ has stabilizer $\mathbb{C}^*$, hence $y_x \in Y^0$, i.e. $Y^0 \to \mathcal{X}^0$ is surjective, and smooth from above. Then $\mathcal{X}^0$ is an algebraic stack. Consider a point $x \in \mathcal{X}^\pm$, i.e. a $\mathbb{C}^*$-equivariant morphism $x : (\mathbb{A}^1)^\pm \to \mathcal{X}$. Consider the smooth $\mathbb{C}^*$-equivariant morphism of algebraic spaces $Y \times_{\mathcal{X}} (\mathbb{A}^1)^\pm \to (\mathbb{A}^1)^\pm$ obtained by base change from $Y \to \mathcal{X}$. As any smooth $\mathbb{C}^*$-equivariant morphism of algebraic spaces can be étale-locally written as a $\mathbb{C}^*$-equivariant fibration with $\mathbb{C}^*$-linear affine fiber, then it admits étale locally a smooth $\mathbb{C}^*$-equivariant section. Any $\mathbb{C}^*$-equivariant étale neighborhood of 0 in $(\mathbb{A}^1)^\pm$ is $(\mathbb{A}^1)^\pm$ itself, there is a $\mathbb{C}^*$-invariant section $s$ of $Y \times_{\mathcal{X}} (\mathbb{A}^1)^\pm \to (\mathbb{A}^1)^\pm$, giving a $\mathbb{C}^*$-equivariant morphism $y : (\mathbb{A}^1)^\pm \to Y$ lifting $x : (\mathbb{A}^1)^\pm \to \mathcal{X}$, i.e. a point of $y \in Y^\pm$ above $x \in \mathcal{X}^\pm$. Hence $Y^\pm \to \mathcal{X}^\pm$ is surjective, and smooth from above, i.e. $\mathcal{X}^\pm$ is an algebraic stack.

The proof of the fact that $Y^\pm \to Y^0 \times_{\mathcal{X}_0} \mathcal{X}^\pm$ is an affine fiber bundle is similar to the proof of the corresponding fact in proposition 2.4. □

For a smooth $\mathbb{C}^*$-equivariant 1-morphism $\phi : Y \to \mathcal{X}$ from an algebraic space, denote by $\phi_\pi^0$ the component of $\phi^0$ where the cocharacter induced by the action of $\mathbb{C}^*$ on the relative tangent complex is $\pi$. Exactly the same manipulations than in the proof of proposition 2.5 gives the natural functor in the six-functor formalism:

$$ (p_Y^{\pm})!(\eta_Y^{\pm})^* \phi^*[d] \simeq \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_\pi/2} (\phi_\pi^0)^*[d_\pi^0] (p_X^{\pm})!(\eta_X^{\pm})^* \quad (5.3) $$

The smooth 1-morphisms $\phi^0 : Y^0 \to \mathcal{X}^0$ coming from a smooth $\mathbb{C}^*$-equivariant 1-morphism $\phi : Y \to \mathcal{X}$ from an algebraic space covers $X$, hence a sheaf on $X^0$ is determined by its pullback by such $\phi^0$. In particular, the morphisms $S_Y$ of proposition 2.3 glue into a morphism:

$$ S_X : \mathbb{D}(p_X^-)!(\eta_X^-)^* \mathbb{D} \to (p_X^+)! (\eta_X^+)^* \quad (5.4) $$

which is an isomorphism on $\mathbb{C}^*$-equivariant sheaves.

## 5.2 D-critical stack

As defined in section 2.8 of [6], a d-critical structure on an algebraic stack $X$ is equivalent to the data of d-critical structure $y^*(s)$ on the scheme $Y$ for each smooth 1-morphism $y : Y \to X$ such that for each 2-morphism $(\phi, \eta)$ between the 1-morphism $y : Y \to X$ and $z : Z \to X$, $\phi^*z^*(s) = y^*(s)$. In particular, [9, Theo 3.18] shows that a (-1)-shifted symplectic algebraic
stack has a canonical d-critical structure. On a d-critical stack $X$, Joyce built its canonical sheaf $K_X$, which is the determinant of the tangent complex if $X$ is the classical truncation of a (-1)-shifted symplectic stack. An orientation $K_X^{1/2}$ is then a square root of $K_X$, and [6, Lem 2.57] shows that an orientation $K_X^{1/2}$ determines an orientation of $(Y, y^*(s))$ for each scheme $Y$ and 1-morphism $y: Y \to X$. In particular, a smooth 2-morphism $(\phi, \eta)$ of relative dimension $d$ gives then a smooth morphism of oriented critical scheme $\phi: (X, y^*(s), K_Y^{1/2}) \to (Y, z^*(s); K_Z^{1/2})$ of relative dimension $d$.

A perverse sheaf $\mathcal{P}$ (and an underlying strongly polarized mixed Hodge module) on an algebraic stack $X$ is given by the data:

* For each smooth 1-morphism $y: Y \to X$ from a scheme $Y$, a perverse sheaf $y^*\mathcal{P}$ on $Y$.

* For each commutative diagram in the 2-category of $\mathbb{C}$-algebraic stacks:

<figure><img src="image_47.png" /></figure>

with $Y, Z$ being schemes, and $y, z$ smooth and $\phi$ smooth of relative dimension $d$, an isomorphism $\mathcal{P}(\phi, \eta) : \phi^*[d]z^*\mathcal{P} \to y^*\mathcal{P}$.

with a compatibility condition on composition of 2-morphisms, see definition 4.6 of [9]. A morphism of perverse sheaf $\beta: \mathcal{P} \to \mathcal{Q}$ is then given by the data of a morphism of perverse sheaves $y^*\beta: y^*\mathcal{P} \to y^*\mathcal{Q}$ for each smooth 1-morphism $y: Y \to X$ from a scheme $Y$. The compatibility condition is that for of the above diagram, the following diagram is commutative:

$$ \begin{tikzcd} \phi^*[d]z^*\mathcal{P} \arrow[r, "\mathcal{P}(\phi)"'] & y^*\mathcal{P} \arrow[d, "\phi^*[d]z^*\beta"] \\ \phi^*[d]z^*\mathcal{Q} \arrow[r, "\mathcal{Q}(\phi, \eta)"'] & y^*\mathcal{Q} \arrow[d, "y^*\beta"'] \end{tikzcd} $$

In [9, Theo 4.8], the authors show that for an oriented d-critical stack $(X, s, K_{X,s}^{1/2})$ the data $P_{Y,y^*}$ and $\Delta_\phi: \phi^*[d](P_{Z,z^*(s)}) \to P_{Y,y^*(s)}$ defines a perverse sheaf $P_{X,s}$ on $X$.

When $(X, s, K_{X,s}^{1/2})$ is a $\mathbb{C}^*$-equivariant d-critical algebraic stack with a lisse-étale linearizable $\mathbb{C}^*$-action, the smooth 1-morphisms $y^0: Y^0 \to \mathcal{X}^0$ and 2-morphisms $(\phi^0, \eta^0): y^0, Y^0 \to z^0, Z^0$ arising from a smooth $\mathbb{C}^*$-equivariant 1-morphism $y: Y \to \mathcal{X}$ or 2-morphism $(\phi, \eta): y, Y \to z, Z$ cover $\mathcal{X}^0$, hence a sheaf or a morphism of sheaf on $\mathcal{X}^0$ can be defined using those 1-morphisms and 2-morphisms. For each $y^0: Y^0 \to \mathcal{X}^0$ arising from a smooth $\mathbb{C}^*$-equivariant 1-morphism $y: Y \to \mathcal{X}$, $(y^0)^*(\xi_\mathcal{X})^*(s) = (\xi_Y)^*y^*(s)$ defines a d-critical structure on $Y^0$, hence $s^0 := (\xi_\mathcal{X})^*(s)$ defines a d-critical structure on $\mathcal{X}^0$. Similarly, $K_{\mathcal{X}^0, s^0} := (\xi_\mathcal{X})^*K_{\mathcal{X}, s}^{1/2}$ defines an orientation on $(\mathcal{X}^0, (\xi_\mathcal{X})^*(s))$, hence $\mathcal{X}^0$ carries a natural structure of oriented d-critical stack.

According to proposition 4.2 and equation (5.3), the isomorphisms $\beta_{Y,y^*(s)}$ glue into an isomorphism $\beta_{\mathcal{X},s}^\pm$ commuting with monodromy and duality:
**Theorem 5.3.** For $\mathcal{X}$ an oriented d-critical algebraic stack with a lisse-étale linearizable $\mathbb{C}^*$-action leaving the d-critical structure and the orientation invariant, one has natural isomorphisms of perverse sheaves and monodromic mixed Hodge modules on $\mathcal{X}^0$:

$$ \beta_{\mathcal{X},s}^{\pm} : (p_{\mathcal{X}}^{\pm})!(\eta_{\mathcal{X}}^{\pm})^* P_{\mathcal{X},s} \to \bigoplus_{\pi \in \Pi} \mathbb{L}^{\pm \mathrm{Ind}_{\pi}/2} P_{\mathcal{X}_{\pi}^0, s_{\pi}^0} \quad (5.5) $$

Which are compatible with strong polarization in the sense that the following diagram is commutative:

$$ \begin{tikzcd} (p_{\mathcal{X}}^{+})!(\eta_{\mathcal{X}}^{+})^{*} P_{\mathcal{X},s} \arrow[r, "\beta_{\mathcal{X},s}^{+}"] & \bigoplus_{\pi \in \Pi} \mathbb{L}^{\mathrm{Ind}_{\pi}/2} P_{\mathcal{X}_{\pi}^{0}, s_{\pi}^{0}} \arrow[d, "S_{\mathcal{X}}^{-1} \circ (p_{\mathcal{X}}^{+})!(\eta_{\mathcal{X}}^{+})^{*}(\Sigma_{\mathcal{X},s})"] & \bigoplus_{\pi \in \Pi} \mathbb{L}^{\mathrm{Ind}_{\pi}/2} \Sigma_{\mathcal{X}_{\pi}^{0}, s_{\pi}^{0}} \arrow[d, "\bigoplus_{\pi \in \Pi} \mathbb{L}^{\mathrm{Ind}_{\pi}/2} P_{\mathcal{X}_{\pi}^{0}, s_{\pi}^{0}}"] \\ \mathbb{D}(p_{\mathcal{X}}^{-})!(\eta_{\mathcal{X}}^{-})^{*} P_{\mathcal{X},s} \arrow[r, "\mathbb{D}\beta_{\mathcal{X},s}^{-}"] & \mathbb{D} \bigoplus_{\pi \in \Pi} \mathbb{L}^{-\mathrm{Ind}_{\pi}/2} P_{\mathcal{X}_{\pi}^{0}, s_{\pi}^{0}} \end{tikzcd} $$

# 6 Technical lemmas

## 6.1 Trivialization of torus-equivariant embeddings

**Lemma 6.1.** Let $\Phi : (R, U, f, i) \to (S, V, g, j)$ be a $\mathbb{C}^*$-equivariant embedding of $\mathbb{C}^*$-equivariant critical charts. For $x \in i(R^0)$, there are smooth $\mathbb{C}^*$-equivariant algebraic space $U'$, $V'$, $x' \in U'$, $\mathbb{C}^*$-equivariant morphisms $\iota : U' \to U$ with $\iota(x') = x$, $j : V' \to V$, $\Phi' : U' \to V'$, $\alpha : V' \to U$ and $\beta : V' \to \mathbb{C}^n$ with $\mathbb{C}^n$ provided with a $\mathbb{C}^*$ action leaving the coordinates equivariant, with $\iota, j$ and $\alpha \otimes \beta$ being étale, such that the following square commutes:

$$ \begin{tikzcd} U \arrow[r, "\iota"] \arrow[d, "\Phi"] & U' \arrow[r, "\iota"] \arrow[d, "\Phi'"] & U \arrow[d, "\mathrm{Id}_U \times \mathbb{C}^n"] \\ V \arrow[r, "j"] & V' \arrow[r, "\alpha \times \beta"] & U \times \mathbb{C}^n \end{tikzcd} $$

and $g \circ j = f \circ \iota + (2 \sum_{b=1}^r z_{2b-1} z_{2b} + \sum_{b=2r+1}^n z_b^2) \circ \alpha : V' \to \mathbb{C}$, with $z_{2b-1}, z_{2b} \mathbb{C}^*$-equivariant of opposite nonzero character for $b \le r$, and $z_b \mathbb{C}^*$-invariant for $b \ge 2r+1$.

**Proof:** We follow here closely the proof of [6, Prop 2.22, 2.23], adapting it to the equivariant case. The map $\Phi : U \to V$ is a $\mathbb{C}^*$-equivariant embedding of smooth schemes of respective dimension $\dim(U) = m$, $\dim(V) = m+n$. We can then find $\mathbb{C}^*$-equivariant étale coordinates $(\dot{y}_1, \dots, \dot{y}_m, \dot{z}_1, \dots, \dot{z}_n)$ on a $\mathbb{C}^*$-equivariant Zariski open neighborhood $\dot{V}$ of $j(x)$ in $V$ such that $j(x) = (0, \dots, 0)$ and $\Phi(U) \cap \dot{V}$ is the locus $\dot{z}_1 = \dots = \dot{z}_n = 0$ in $\dot{V}$. Set $\dot{U} = \Phi^{-1}(\dot{V})$ and $\dot{x}_a = \dot{y}_a \circ \Phi|_{\dot{U}}$ for $a = 1, \dots, m$. Then $\dot{U}$ is an open neighborhood of $i(x)$ in $U$, and $(\dot{x}_1, \dots, \dot{x}_m)$ are étale coordinates on $\dot{U}$ with $i(x) = (0, \dots, 0)$. Then the ideal $I_{R,U} = I_{(df)}$ is on $\dot{U}$ the ideal generated by the $\mathbb{C}^*$-equivariant functions $\frac{\partial f}{\partial \dot{x}_a}$ for $a = 1, \dots, m$, and the ideal $I_{S,V} = I_{(dg)}$ is on $\dot{V}$ the ideal generated by the $\mathbb{C}^*$-equivariant functions $\frac{\partial g}{\partial \dot{y}_a}$ for $a = 1, \dots, m$ and $\frac{\partial g}{\partial \dot{z}_b}$ for $b = 1, \dots, n$. Since $\Phi$ maps $U$ to $\dot{z}_1 = \dots = \dot{z}_n = 0$ and $i(R)$ to $j(R) \subset j(S)$, we have
$I(df) \simeq I(dg)|_{\dot{z}_1=\dots=\dot{z}_n=0}$, that is:

$$ (\frac{\partial f}{\partial \dot{x}_a}(\dot{y}_1, \dots, \dot{y}_m) : a = 1, \dots, m) = (\frac{\partial g}{\partial \dot{y}_a}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0) : a = 1, \dots, m, \frac{\partial g}{\partial \dot{z}_b}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0) : b = 1, \dots, n) \quad (6.1) $$

this holds provided each $\frac{\partial g}{\partial \dot{z}_b}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0)$ lies in $(\frac{\partial g}{\partial \dot{y}_a}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0) : a = 1, \dots, m$. Thus, making $\dot{U}, \dot{V}$ smaller if necessary, we can suppose there exist étale locally $\mathbb{C}^*$-equivariant functions $A_{ab}(\dot{y}_1, \dots, \dot{y}_m)$ on $\dot{U}$ for $a = 1, \dots, m, b = 1, \dots, n$ such that for each b:

$$ \frac{\partial g}{\partial \dot{z}_b}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0) = \sum_{a=1}^{m} A_{ab}(\dot{y}_1, \dots, \dot{y}_m) \frac{\partial g}{\partial \dot{y}_a}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0) \quad (6.2) $$

Defining the $\mathbb{C}^*$-equivariant functions $\tilde{y}_a = \dot{y}_a - \sum a = 1^m A_{ab}(\dot{y}_1, \dots, \dot{y}_m) \dot{z}_b$, and $\tilde{z}_b = \dot{z}_b$, they give $\mathbb{C}^*$-equivariant étale coordinates on a $\mathbb{C}^*$-invariant neighborhood $\tilde{V}$ of $j(x)$. We also define the $\mathbb{C}^*$-invariant subspace $\tilde{U} = \Phi^{-1}(\tilde{V})$, and define on it the $\mathbb{C}^*$-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:

$$ \begin{aligned} \frac{\partial g}{\partial \tilde{z}_b}(\tilde{y}_1, \dots, \tilde{y}_m, 0, \dots, 0) &= \sum_{a=1}^{m} \frac{\partial g}{\partial \dot{y}_a}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0) \cdot \frac{\partial \dot{y}_a}{\partial \tilde{z}_b} + \sum c = 1^n \frac{\partial g}{\partial \dot{z}_c}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0) \cdot \frac{\partial \dot{z}_c}{\partial \tilde{z}_b} \\ &= \sum_{a=1}^{m} \frac{\partial g}{\partial \dot{y}_a}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0) (-A_{ab}(\dot{y}_1, \dots, \dot{y}_m)) \\ &\quad + \sum c = 1^n \left( \sum_{a=1}^{n} A_{ac}(\dot{y}_1, \dots, \dot{y}_m) \right) \cdot \frac{\partial g}{\partial \dot{y}_a}(\dot{y}_1, \dots, \dot{y}_m, 0, \dots, 0) \cdot \delta_{bc} \\ &= 0 \end{aligned} \quad (6.3) $$

We define the $\mathbb{C}^*$-equivariant space $\tilde{V}$ and the $\mathbb{C}^*$-equivariant étale morphisms $\tilde{j}, \tilde{\alpha}, \tilde{\beta}$ by the $\mathbb{C}^*$-equivariant Cartesian square:

<figure><img src="image_48.png" /></figure>

where $\mathbb{C}^{m+n}$ is provided with the $\mathbb{C}^*$-action induced from the action of $\mathbb{C}^*$ on the étale coordinates. There is a unique $\tilde{v} \in \tilde{V}$ with $\tilde{\alpha}(\tilde{v}) = i(x), \tilde{\beta}(\tilde{v}) = (0, \dots, 0)$ and $\tilde{j}(\tilde{v}) = j(x)$. We can regard $\tilde{j}: \tilde{V} \to \tilde{V} \subset V$ as an étale $\mathbb{C}^*$-invariant open set in $V$. Define $\mathbb{C}^*$-equivariant étale coordinates $(\tilde{y}_1, \dots, \tilde{y}_m, \tilde{z}_1, \dots, \tilde{z}_n): \tilde{V} \to \mathbb{C}^{m+n}$ by $\tilde{y}_a = \tilde{y}_a \circ \tilde{j}, \tilde{z}_b = \tilde{z}_b \circ \tilde{j}$, and define the $\mathbb{C}^*$-invariant functions $\tilde{f}, \tilde{g}: \tilde{V} \to \mathbb{C}$ by $\tilde{f} = f \circ \tilde{\alpha}, \tilde{g} = g \circ \tilde{j}$, and $\tilde{h} = \tilde{g} - \tilde{f}: \tilde{V} \to \mathbb{C}$. The previous argument now shows that on the smooth subscheme $\tilde{U} \subset \tilde{V}$ defined by $\tilde{z}_1 = \dots = \tilde{z}_n = 0$ we have $\tilde{h}|_{\tilde{U}}$ and $\frac{\partial \tilde{h}}{\partial \tilde{z}_b}|_{\tilde{U}} = 0$ for $b = 1, \dots, n$. Therefore the $\mathbb{C}^*$ invariant function lies in the ideal $(\tilde{z}_1, \dots, \tilde{z}_n)^2$ generated by $\mathbb{C}^*$-equivariant functions on $\tilde{V}$. So making $\tilde{U}, \tilde{V}, \tilde{U}, \tilde{V}$ smaller but still $\mathbb{C}^*$-invariant, we may write $\tilde{h} = \sum_{b,c=1}^n \tilde{z}_b \tilde{z}_c Q_{bc}$ for some $\mathbb{C}^*$-equivariant $Q_{bc}: \tilde{V} \to \mathbb{C}$ with $Q_{bc} = Q_{cb}$.
$$ \sum_{b,c=1}^{n} \tilde{z}_b \tilde{z}_c Q_{bc}(0, \dots, 0) = 2 \sum_{b=1}^{r} \tilde{z}_{2b-1} \tilde{z}_{2b} + \sum_{b=2r+1}^{n} \tilde{z}_b^2 \quad (6.4) $$

with $n_{2i-1} = -n_{2i} > 0$ for $i \le r$ and $n_i = 0$ for $i \ge 2r+1$. Suppose that $n \ge 2r+1$: the $\mathbb{C}^*$-invariant function $Q_{nn}(\tilde{y}_1, \dots, \tilde{y}_m, \tilde{z}_1, \dots, \tilde{z}_n)$ is invertible near $\tilde{v}$, hence there is a small étale open neighborhood of $\tilde{v}$ such that $Q_{nn}$ has a square root $Q_{nn}^{1/2}$ with strictly positive imaginary part. For $v \in \tilde{V}$ such that $Q_{nn}^{1/2}(v)$ and $Q_{nn}^{1/2}(t.v)$ are defined:

$$ \begin{gathered} (Q_{nn}^{1/2}(t.v))^2 = Q_{nn}(t.v) = Q_{nn}(v) = (Q_{nn}^{1/2}(v))^2 \\ \Rightarrow Q_{nn}^{1/2}(t.v) = \pm Q_{nn}^{1/2}(t.v) \Rightarrow Q_{nn}^{1/2}(t.v) = Q_{nn}^{1/2}(t.v) \end{gathered} \quad (6.5) $$

where the last equality holds because $Q_{nn}^{1/2}$ is assumed to have strictly positive imaginary part. Hence defining $Q_{nn}^{1/2}(t.v) := Q_{nn}^{1/2}(v)$ when $Q_{nn}^{1/2}(v)$ is defined, there is a $\mathbb{C}^*$-invariant open neighborhood of $\tilde{v}$ such that $Q_{nn}$ has a $\mathbb{C}^*$-invariant square root $Q_{nn}^{1/2}$. We replace then $\tilde{V}$ by this étale open neighborhood. We can then write:

$$ \begin{aligned} h &= \sum_{b,c=1}^{n} \tilde{z}_b \tilde{z}_c Q_{bc} \\ &= \sum_{b,c=1}^{n-1} \tilde{z}_b \tilde{z}_c (Q_{bc} - Q_{nn}^{-1} Q_{bn} Q_{cn}) + (Q_{nn}^{1/2} \tilde{z}_n + \sum_{b=1}^{n-1} Q_{nn}^{-1/2} Q_{bn} \tilde{z}_b)^2 \\ &= \sum_{b,c=1}^{n-1} \tilde{z}_b \tilde{z}_c \hat{Q}_{bc} + z_n^2 \end{aligned} \quad (6.6) $$

where $\hat{Q}_{bc} = Q_{bc} - Q_{nn}^{-1} Q_{bn} Q_{cn}$ is $\mathbb{C}^*$-equivariant of character $-n_b - n_c$ and $z_n = Q_{nn}^{1/2} \tilde{z}_n + \sum_{b=1}^{n-1} Q_{nn}^{-1/2} Q_{bn} \tilde{z}_b$ is $\mathbb{C}^*$-invariant, with $\frac{\partial z_n}{\partial \tilde{z}_b}(\tilde{v}) = \delta_{nb}$ and $\hat{Q}_{bc}(0, \dots, 0) = \delta_{bc}$. By recursion we can find an étale open neighborhood of $\tilde{v}$ in $V$, $\mathbb{C}^*$-invariant functions on it $z_b$ for $b \ge 2r+1$ with $\frac{\partial z_b}{\partial \tilde{z}_c}(\tilde{v}) = \delta_{bc}$, and $\mathbb{C}^*$-equivariant functions on it $\tilde{Q}_{bc}$ for $b, c \ge 2r$ with $\tilde{Q}_{bc}(0, \dots, 0) = Q_{bc}(0, \dots, 0)$, such that:

$$ h = \sum_{b,c=1}^{2r} \tilde{z}_b \tilde{z}_c \tilde{Q}_{bc} + \sum_{b=2r+1}^{n} z_b^2 \quad (6.7) $$

On a $\mathbb{C}^*$-invariant open neighborhood of $\tilde{v}$ such that $\tilde{Q}_{2r-1,2r}$ is invertible:

$$ \begin{aligned} h &= \sum_{b,c=1}^{2r-2} \tilde{z}_b \tilde{z}_c (\tilde{Q}_{bc} - \sum_{b,c=1}^{2r-2} \tilde{Q}_{2r-1,2r}^{-1} \tilde{Q}_{b,2r} \tilde{Q}_{2r-1,c}) + \sum_{b=2r+1}^{n} z_b^2 \\ &\quad + 2(\tilde{z}_{2r-1} + \sum_{b=1}^{2r-2} \tilde{Q}_{2r-1,2r}^{-1} \tilde{Q}_{b,2r} \tilde{z}_b)(\tilde{Q}_{2r-1,2r} \tilde{z}_{2r} + \sum_{c=1}^{2r-2} \tilde{Q}_{2r-1,c} \tilde{z}_c) \\ &= \sum_{b,c=1}^{2r-2} \tilde{z}_b \tilde{z}_c \tilde{Q}_{bc} + 2z_{2r-1}z_{2r} + \sum_{b=2r+1}^{n} z_b^2 \end{aligned} \quad (6.8) $$

where $\tilde{Q}_{bc} = \tilde{Q}_{bc} - \sum_{b,c=1}^{2r-2} \tilde{Q}_{2r-1,2r}^{-1} \tilde{Q}_{b,2r} \tilde{Q}_{2r-1,c}$ is $\mathbb{C}^*$-equivariant of character $-n_b - n_c$ with $\tilde{Q}_{bc}(0, \dots, 0) = Q_{bc}(0, \dots, 0)$, $z_{2r-1} = \tilde{z}_{2r-1} + \sum_{b=1}^{2r-2} \tilde{Q}_{2r-1,2r}^{-1} \tilde{Q}_{b,2r} \tilde{z}_b$ is $\mathbb{C}^*$-equivariant of character
$n_{2r-1}$ with $\frac{\partial z_{2r-1}}{\partial \check{z}_b}(\check{v}) = \delta_{2r-1,b}$, and $z_{2r} = \check{Q}_{2r-1,2r} \check{z}_{2r} + \sum_{c=1}^{2r-2} \check{Q}_{2r-1,c} \check{z}_c$ is $\mathbb{C}^*$-equivariant of weight $n_{2r}$ with $\frac{\partial z_{2r}}{\partial \check{z}_b}(\check{v}) = \delta_{2r,b}$. By recursion, there is then on an $\mathbb{C}^*$-invariant étale open neighborhood $j' : V' \to \check{V}$ of $\check{v}$ and $\mathbb{C}^*$-equivariant functions $z_b$ with $\frac{\partial z_b}{\partial \check{z}_c}(\check{v}) = \delta_{bc}$ (i.e. by restricting $V'$ to a $\mathbb{C}^*$-invariant open neighborhood of $\check{v}$, $(y_1 := y_1 \circ j', ..., y_m := y_m \circ j', z_1, ..., z_n$ forms a system of étale coordinates of $V'$), such that:

$$h = 2 \sum_{b=1}^{r} z_{2b-1} z_{2b} + \sum_{b=2r+1}^{n} z_b^2 \qquad (6.9)$$

Define the $\mathbb{C}^*$-invariant subset $U' = \{v' \in V' : z_1(v') = ... = z_n(v') = 0\}$. Define the $\mathbb{C}^*$-equivariant maps $\iota : U' \to U, j' : V' \to V, \Phi' : U' \to V', \alpha : V' \to U, \beta : V' \to \mathbb{C}^n$ by $\iota := \check{\alpha} \circ j'|_U, j := \check{j} \circ j', \Phi' = \mathrm{Id}_{U'}, \alpha = \check{\alpha} \circ j'$ and $\beta = (z_1, ..., n)$. As $j' : V' \to \check{V}$ is an étale open neighborhood of $\check{v}$ in $\check{V}$, there exists $u' \in V'$ with $j'(u') = \check{v}$, and $z_b(u' = 0)$ for $b = 1, ..., n$ as $\check{z}_b(\check{v}) = 0$, so $u' \in U$ with $\iota(u') = \check{\alpha} \circ j'(u') = \check{\alpha}(\check{v}) = i(x)$. Also $\iota, j, \alpha \times \beta$ are étale as $\alpha\beta\times\beta, j, j'$ are. The same computations as in the proof of [6, Prop 2.23] show that in a neighborhood of $u'$, which can be taken to be $\mathbb{C}^*$-invariant, we have $\Phi \circ \iota = j \circ \Phi$, so this making $u'$ smaller and still $\mathbb{C}^*$-invariant it holds on $U'$. The equations $\alpha \circ \Phi' = \iota, \beta \circ \Phi' = 0$ are immediate, and $g \circ j = f \circ \alpha + (2 \sum_{b=1}^{r} z_{2b-1} z_{2b} + \sum_{b=2r+1}^{n} z_b^2) \circ \beta$ follows from (6.9). $\square$

## 6.2 Smooth torus-equivariant morphism of d-critical scheme

**Lemma 6.2.** Consider a smooth morphism of d-critical scheme $\phi : (X, s) \to (Y, t)$ of relative dimension $d$. For $x \in X$, we can then choose two $\mathbb{C}^*$-equivariant critical charts $(R, U, f, i)$ and $(S, V, g, j)$ of $X$ and $Y$ containing $x$ and $y := f(x)$ with a smooth map $\Phi : U \to V$ of relative dimension $d$ such that $f = g \circ \Phi$ and $\Phi \circ i = j \circ \phi$.

**Proof:** This is a $\mathbb{C}^*$-equivariant version of the result used in the proof of [9, Prop 4.5]. As in the proof of [6, Prop 2.43], because the $\mathbb{C}^*$ action on $Y$ is étale-locally linearizable, we can find an étale neighborhood $S'$ of $y$ with a closed $\mathbb{C}^*$-equivariant embedding into a smooth algebraic space with $\mathbb{C}^*$-action $V'$, with $\dim(V') = \dim(T_y Y)$. According to the proof of the proposition 2.4, the $\mathbb{C}^*$-equivariant smooth map $\phi^{-1}(S') \to S'$ is étale locally a $\mathbb{C}^*$-equivariant affine fibration, hence we can find a $\mathbb{C}^*$-equivariant étale neighborhood $S$ of $y$, $\mathbb{C}^*$-equivariant embeddings $i : R := \phi^{-1}(S) \to U, j : S \to V$ into smooth algebraic spaces with $\mathbb{C}^*$-action $U, V$ with dimension $\dim(U) = \dim(T_x X)$, $\dim(V) = \dim(T_y Y)$, with a $\mathbb{C}^*$-equivariant smooth map of relative dimension $d$ $\Phi : U \to V$ such that $\Phi \circ i = j \circ \phi|_R$. Hence as in the proof of [6, Prop 2.43], up to shrinking $S, V$ (and then $R, U$) to a $\mathbb{C}^*$-invariant open subspace we can find a $\mathbb{C}^*$ invariant regular function $g : V \to \mathbb{C}$ such that $\iota_{S,V}(t|_S) = j^{-1}(g) + I_{S,V}^2$, and $S = \mathrm{Crit}(g)$, i.e. $(S, V, g, j)$ is a $\mathbb{C}^*$-equivariant critical chart of $Y$ near $y$. Defining the $\mathbb{C}^*$-equivariant regular function $f := g \circ \Phi : U \to \mathbb{C}$, because $s = \phi^*(t)$, one has $\iota_{R,U}(s|_R) = i^{-1}(f) + I_{R,U}^2$. Also $\Phi \circ i = j \circ \phi|_R$ and $\Phi, \phi$ are smooth of relative dimension $d$, then $i(R) = \Phi^{-1}(j(S))$ in a neighborhood of $i(x)$: taking the union of all these neighborhood, one can shrink $U$ (and then $R$) to $\mathbb{C}^*$-invariant open neighborhood satisfying $i(R) = \Phi^{-1}(j(S))$. Because $\Phi$ is smooth, one has:

$$\mathrm{Crit}(f) = \Phi^{-1}(\mathrm{Crit}(g)) = \Phi^{-1}(j(S)) = i(R) \qquad (6.10)$$

i.e. $(R, U, f, i)$ is a $\mathbb{C}^*$-equivariant critical chart near $x$. We have then built $\mathbb{C}^*$-equivariant critical charts $(R, U, f, i)$ and $(S, V, g, j)$ of $X$ and $Y$ near $x$ and $y$ with a smooth map $\Phi : U \to V$ of relative dimension $d$ such that $f = g \circ \Phi$ and $\Phi \circ i = j \circ \phi$. $\square$
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