2.1 Shifted symplectic geometry

Here we briefly recall some notions in derived algebraic geometry and shifted symplectic geometry.

Let $U$ be a smooth scheme, and $s \in \Gamma (U, E)$ be a section of a vector bundle $E$ on $U$. Write $\boldsymbol {Z}(s)$ for the derived zero locus of $s$. We have the following Cartesian diagram in $\mathop {\mathbf {dSch}}\nolimits$.

Since $\mathbb {L}_g[-1]$ and $g^{*} \mathbb {L}_U$ are locally free sheaves concentrated in degree zero, we conclude that $\boldsymbol {Z}(s)$ is quasi-smooth.

Proof. (i) is a direct consequence of [Reference Brav, Bussi and JoyceBBJ19, Theorem 4.1].

(ii) follows from [Reference Brav, Bussi and JoyceBBJ19, Theorem 4.2] except for $\eta _i$ being étale and $\tau _i$ being isomorphism. Since $(Z_i, U_i, E_i, s_i, \boldsymbol {f}_i)$ is minimal at $q_i$, we have

\[ \mathrm{H}^{0}(\mathbb{L}_{\boldsymbol{Z}(s_i)} |_{q_i}) \cong \Omega_{U_i} |_{q_i},\quad \mathrm{H}^{{-}1}(\mathbb{L}_{\boldsymbol{Z}(s_i)} |_{q_i}) \cong E_i^{{\vee}} |_{q_i}. \]

Similarly, we have

\[ \mathrm{H}^{0}(\mathbb{L}_{\boldsymbol{Z}(s')} |_{q'}) \cong \Omega_{U'} |_{q'},\quad \mathrm{H}^{{-}1}(\mathbb{L}_{\boldsymbol{Z}(s')} |_{q'}) \cong E'^{{\vee}} |_{q'}. \]

Since the open immersion $\boldsymbol {Z}(s') \to \boldsymbol {Z}(s_i)$ induces a quasi-isomorphism $\mathbb {L}_{\boldsymbol {Z}(s_i)} |_{q_i} \simeq \mathbb {L}_{\boldsymbol {Z}(s')} |_{q'}$, we see that $\eta _i$ is étale at $q'$ and $\tau _i$ is invertible at $q'$. Thus by shrinking $U'$ around $q'$ if necessary, we obtain the required properties.

Let $A$ be a cdga, and take a semi-free resolution $A' \to A$. We define the space of $n$-shifted $p$-forms and the space of $n$-shifted closed $p$-forms by

\begin{align*} \mathcal{A}^{p}(A, n) & := \lvert ({\wedge} ^{p} \Omega_{A'} [n], d) \rvert,\\ \mathcal{A}^{p,cl}(A, n) & := \bigg\lvert \bigg(\prod_{i \geq 0} \wedge^{p + i} \Omega_{A'} [{-}i + n], d + {d_{\mathrm{dR}}}\bigg) \bigg\rvert ,\end{align*}

respectively, where $d$ is the internal differential induced by the differential of $\Omega _{A'}$, and ${d_{\mathrm {dR}}}$ is the de Rham differential. Here for a differential graded (dg) module $E$, $\lvert E \rvert$ denotes the simplicial set corresponding to the truncation $\tau ^{\leq 0}(E)$ by the Dold–Kan correspondence. These constructions can be made $\infty$-functorial, and they satisfy the sheaf condition with respect to the étale topology (see [Reference Pantev, Toën, Vaquié and VezzosiPTVV13, Proposition 1.11]). Therefore we obtain $\infty$-functors

\[ {\mathcal{A}^{p}(-, n), \mathcal{A}^{p,cl}(-, n)} \colon {\mathop{\mathbf{dSt}}\nolimits^{\mathrm{op}}} \to {\mathbb{S}}. \]

We write $\boldsymbol {f}^{\star }$ for $\mathcal {A}^{p}(\boldsymbol {f}, n)$ and also for $\mathcal {A}^{p,cl}(\boldsymbol {f}, n)$. For a derived Artin stack $\boldsymbol {\mathfrak {X}}$, it is shown in [Reference Pantev, Toën, Vaquié and VezzosiPTVV13, Proposition 1.14] that we have an equivalence

(2.1)\begin{equation} \mathcal{A}^{p}(\boldsymbol{\mathfrak{X}}, n) \simeq \mathrm{Map}(\mathcal{O}_{\boldsymbol{\mathfrak{X}}}, \wedge^{p} \mathbb{L}_{\boldsymbol{\mathfrak{X}}[n]}). \end{equation}

We have natural morphisms

\begin{align*} \pi &\colon \mathcal{A}^{p,cl}(-, n) \to \mathcal{A}^{p}(-, n), \\ {{\mathrm d}}_{\mathrm{dR}} &\colon \mathcal{A}^{p}(-, n) \to \mathcal{A}^{p+1,cl}(-, n), \end{align*}

where $\pi$ is induced by the projection $\prod _{i \geq 0} \wedge ^{p + i} \Omega _{A'} [-i + n] \to \wedge ^{p} \Omega _{A'} [p]$, and ${{\mathrm d}}_{\mathrm {dR}}$ is induced by the map

\[ \mathcal{A}^{p}(A, n) \ni \omega^{0} \mapsto ({d_{\mathrm{dR}}} \omega^{0},0,0,\ldots) \in \mathcal{A}^{p+1,cl}(A, n). \]

We say that two shifted symplectic derived Artin stacks $(\boldsymbol {\mathfrak {X}}_1, \omega _{\boldsymbol {X}_1})$ and $(\boldsymbol {\mathfrak {X}}_2, \omega _{\boldsymbol {\mathfrak {X}}_2})$ are equivalent if there exist an equivalence $\boldsymbol {\Phi } \colon \boldsymbol {\mathfrak {X}}_1 \xrightarrow {\sim } \boldsymbol {\mathfrak {X}}_2$ as derived stacks and an equivalence $\boldsymbol {\Phi } ^{\star } \omega _{\boldsymbol {\mathfrak {X}}_2} \sim \omega _{\boldsymbol {\mathfrak {X}}_1}$.

Example 2.5 Let $U = \mathop {\mathrm {Spec}}\nolimits A$ be a smooth affine scheme and ${f} \colon {U} \to {\mathbb {C}}$ be a regular function on it. Denote by $\boldsymbol {X}$ the derived critical locus $\mathop {\mathbf {Crit}}\nolimits (f)$. Since $\boldsymbol {X}$ is the derived zero locus of the section ${{d_{\mathrm {dR}}}f} \colon {U} \to {\Omega _U}$, the derived scheme $\boldsymbol {X}$ is equivalent to $\mathop {\mathbf {Spec}}\nolimits B$ where $B$ is a cdga defined by the Koszul complex

\[ B := ( \cdots \to \wedge^{2} \Omega_A ^{{\vee}} \xrightarrow{\cdot {d_{\mathrm{dR}}}f} \Omega_A ^{{\vee}} \xrightarrow{\cdot {d_{\mathrm{dR}}}f} A). \]

Assume there exists a global étale coordinate $(x_1, \ldots, x_n)$ on $U$. We write the dual basis of ${d_{\mathrm {dR}}}x_1, \ldots, {d_{\mathrm {dR}}}x_n$ as ${\partial }/{\partial x_1}, \ldots, {\partial }/{\partial x_n}$, and $y_i \in B^{-1}$ denotes the element of degree $-1$ corresponding to ${\partial }/{\partial x_i}$ for each $i=1,\ldots, n$. Although $B$ is not semi-free in general, one can see that $\Omega _B$ gives a model for $\mathbb {L}_B$. Define an element $\omega _{\boldsymbol {X}}' \in (\wedge ^{2} \Omega _B)^{-1}$ of degree $-1$ by

\[ \omega_{\boldsymbol{X}}' := {d_{\mathrm{dR}}}x_1 \wedge {d_{\mathrm{dR}}}y_1 + \cdots + {d_{\mathrm{dR}}}x_n \wedge {d_{\mathrm{dR}}}y_n. \]

This defines a $-1$-shifted $2$-form, which is clearly non-degenerate. Since we also have ${d_{\mathrm {dR}}} \omega _{\boldsymbol {X}}' =0$, the closed form $\omega _{\boldsymbol {X}} := (\omega _{\boldsymbol {X}}', 0, \ldots )$ defines a $-1$-shifted symplectic structure on $\boldsymbol {X}$.

It is proved in [Reference Brav, Bussi and JoyceBBJ19, Theorem 5.18] that any $-1$-shifted symplectic derived scheme is Zariski locally modeled on a derived critical locus. For a $-1$-shifted symplectic derived scheme of the form $\mathbf {T}^{*}[-1]\boldsymbol {Y}$ for some quasi-smooth derived scheme $\boldsymbol {Y}$, its local model as a derived critical locus can be described by combining Proposition 2.3 and the following lemma.

Lemma 2.7 Let $U = \mathop {\mathrm {Spec}}\nolimits A$ be a smooth affine scheme admitting a global étale coordinate, $E$ be a trivial vector bundle on $U$, and $s \in \Gamma (U, E)$ be a section. Denote by $\bar {s}$ the regular function on $\mathop {\mathrm {Tot}}\nolimits _U(E ^{\vee })$ corresponding to $s$. Then we have an equivalence of $-1$-shifted symplectic derived schemes

(2.2)\begin{equation} (\mathop{\mathbf{Crit}}\nolimits(\bar{s}), \omega_{\mathop{\mathbf{Crit}}\nolimits(\bar{s})}) \simeq (\mathbf{T}^{*}[{-}1]{\boldsymbol{Z}(s)}, \omega_{\mathbf{T}^{*}[{-}1]{\boldsymbol{Z}(s)}}) \end{equation}

equipped with the $-1$-shifted symplectic structures constructed in Examples 2.5 and 2.6, respectively.

Proof. Fix a global étale coordinate $x_1, \ldots, x_n$ on $U$ and a basis $e_1, \ldots, e_m$ of $M := \Gamma (U, E)$. Write $s = a_1 e_1 + \cdots + a_m e_m$. If we write $\alpha _i := {d_{\mathrm {dR}}}x_i$, then $\alpha _1, \ldots, \alpha _n$ defines a basis of $\Omega _U$. Denote by $e_1^{\vee }, \ldots, e_m^{\vee }$ and $\alpha _1^{\vee }, \ldots, \alpha _n^{\vee }$ the dual bases of $e_1, \ldots, e_m$ and $\alpha _1, \ldots, \alpha _n$, respectively. Define a cdga $C$ by the Koszul complex

\[ C := (\cdots \to (\Omega_A ^{{\vee}} \oplus M^{{\vee}}) \otimes _A A[z_1, \ldots, z_m] \xrightarrow{\alpha_i^{{\vee}} \otimes 1 \mapsto \sum (\partial a_j / \partial x_i) z_j, e_j^{{\vee}} \otimes 1 \mapsto a_j} A[z_1, \ldots, z_m]). \]

Now it is clear that $\mathop {\mathbf {Spec}}\nolimits C$ gives models for both $\mathop {\mathbf {Crit}}\nolimits (\bar {s})$ and $\mathbf {T}^{*}[-1]{\boldsymbol {Z}(s)}$. The tautological $1$-form $\lambda _{\mathbf {T}^{*}[-1]{\boldsymbol {Z}(s)}}$ is represented by

\[ \sum _{i=1} ^{n} \alpha_i^{{\vee}} ({d_{\mathrm{dR}}}x_i ) + \sum _{j=1} ^{m} z_j({d_{\mathrm{dR}}}e_j ^{{\vee}}) \in \mathcal{A}^{1}(\mathop{\mathbf{Spec}}\nolimits C, -1). \]

A direct computation shows that ${{\mathrm d}}_{\mathrm {dR}} \lambda _{\mathbf {T}^{*}[-1]{\boldsymbol {Z}(s)}} \sim \omega _{\mathop {\mathbf {Crit}}\nolimits (\bar {s})}$, which implies the lemma.

2.2 D-critical schemes

In this section we briefly recall the notion of d-critical structures introduced in [Reference JoyceJoy15], which constitute a classical model for $-1$-shifted symplectic structures. D-critical structures are easier to treat than $-1$-shifted symplectic structures and are enough to apply in cohomological Donaldson–Thomas theory.

For any complex analytic space $X$, Joyce in [Reference JoyceJoy15, Theorem 2.1] introduced a sheaf

(2.3)\begin{equation} \mathcal{S}_X \in \mathrm{Mod}(\mathbb{C}_{X}) \end{equation}

of $\mathbb {C}$-vector space on $X$ with the following property: for any open subset $R \subset X$ and any closed embedding $i \colon R \hookrightarrow U$ where $U$ is a complex manifold, we have an exact sequence of sheaves on $R$,

\[ 0 \to \mathcal{S}_X \vert _R \to i^{{-}1} \mathcal{O}_U/I_{R, U} ^{2} \xrightarrow{{d_{\mathrm{dR}}}} i^{{-}1} \Omega_U / (I_{R, U} \cdot i^{{-}1} \Omega_U). \]

Here $I_{R, U}$ is the ideal sheaf of $i^{-1} \mathcal {O}_U$ corresponding to $R$. The composition

\[ \mathcal{S}_X \vert _R \to i^{{-}1} \mathcal{O_U}/I_{R, U} ^{2} \to i^{{-}1}\mathcal{O_U}/I_{R, U} \cong \mathcal{O}_R \]

glues to define a morphism ${\beta _X} \colon {\mathcal {S}_X} \to {\mathcal {O}_X}$, and we define a subsheaf $\mathcal {S}_X ^{0}$ of $\mathcal {S}_X$ by the kernel of the composition

\[ \mathcal{S}_X \xrightarrow{\beta_X} \mathcal{O}_X \to \mathcal{O}_{X^{\mathrm{red}}}. \]

It can be shown that we have a decomposition $\mathcal {S}_X = \mathbb {C}_X \oplus \mathcal {S}_X ^{0}$ where $\mathbb {C}_X$ is the constant sheaf and $\mathbb {C}_X \hookrightarrow \mathcal {S}_X$ is induced by the inclusion $\mathbb {C}_U \hookrightarrow \mathcal {O}_U$ identifying $\mathbb {C}_U$ with the sheaf of locally constant functions on $U$. If $X$ is the critical locus of some function $f$ on a complex manifold $U$ such that $f \vert _{X ^{\mathrm {red}}} = 0$, then $f + I_{X, U}^{2}$ defines an element of $\Gamma (X, \mathcal {S}_X^{0})$ since ${d_{\mathrm {dR}}}f \vert _X = 0$.

For a d-critical chart $(R, U, f, i)$ of a d-critical scheme $(X, s)$ and an open subset $U' \subset U$, define $R' = i^{-1}(U')$, $f' = f|_{U'}$, and $i' = i |_{R'} \colon R' \hookrightarrow U'$. Then $(R', U', f', i')$ defines a d-critical chart on $(X, s)$. We call $(R', U', f', i')$ an open subchart of $(R, U, f, i)$.

In order to compare two d-critical charts, Joyce introduced the notion of embedding: for a d-critical scheme $(X, s)$ and its d-critical charts $(R_1, U_1, f_1, i_1)$ and $(R_2, U_2, f_2, i_2)$ such that $R_1 \subset R_2$, an embedding ${(R_1, U_1, f_1, i_1)}\hookrightarrow {(R_2, U_2, f_2, i_2)}$ is defined by a locally closed embedding $\Phi \colon {U_1}\hookrightarrow {U_2}$ such that $f_1 = f_2 \circ {\Phi }$ and $\Phi \circ i_1 = i_2 \vert _{R_2}$. The following theorem is useful when comparing two d-critical charts.

For an embedding of d-critical charts $\Phi \colon (R_1, U_1, f_1, i_1) \hookrightarrow (R_2, U_2, f_2, i_2)$ of a d-critical scheme $(X, s)$, Joyce defined in [Reference JoyceJoy15, Definition 2.26] a natural isomorphism

\[ J_{\Phi} \colon i_1^{*} (K_{U_1}^{{\otimes}^{2}})|_{R_1 ^{\mathrm{red}}} \cong i_2^{*} (K_{U_2}^{{\otimes}^{2}}) |_{R_1 ^{\mathrm{red}}}. \]

If there exist $\alpha, \beta$ as in Theorem 2.10(ii), $J_{\Phi }$ is defined as follows. Firstly, we have isomorphisms

\[ K_{U_2} \cong (\alpha, \beta)^{*}(K_{U_1 \times \mathbb{C}^{n}}) \cong \alpha^{*}K_{U_1} \otimes \beta^{*}K_{\mathbb{C}^{n}} \cong \alpha^{*}K_{U_1} \]

where the final isomorphism is defined by the trivialization

\[ K_{\mathbb{C}^{n}} \cong \mathcal{O}_{\mathbb{C}^{n}} \cdot ({{\mathrm d}}z_1 \wedge \cdots \wedge {{\mathrm d}}z_n). \]

Then, by taking the square of this composition and pulling back to $R_1$, we obtain the desired isomorphism.

Using this preparation, we can construct a natural line bundle $K_{X, s}$ on $X^{\mathrm {red}}$, which is a d-critical version of the canonical line bundle as follows.

Definition 2.12 [Reference JoyceJoy15, Definition 2.31]

An orientation $o$ of a d-critical scheme $(X, s)$ is a choice of a line bundle $L$ on $X^{\mathrm {red}}$ and an isomorphism

\[ o \colon L^{{\otimes}^{2}} \xrightarrow{\cong} K_{X, s}. \]

As we have seen in the previous section, $-1$-shifted symplectic derived schemes are locally modeled on derived critical loci. Therefore we can regard the notion of d-critical schemes as an underived version of the $-1$-shifted symplectic derived scheme. The following theorem gives the rigorous statement of this fact.

Theorem 2.13 [Reference Brav, Bussi and JoyceBBJ19, Theorem 6.6]

Let $(\boldsymbol {X}, \omega _{\boldsymbol {X}})$ be a $-1$-shifted symplectic derived scheme. Then its underlying scheme $X = t_0 (\boldsymbol {X})$ carries a canonical d-critical structure $s_X$ with the following property: for any $-1$ shifted symplectic derived scheme $(\boldsymbol {R}, \omega _{\boldsymbol {R}})$ of the form $\boldsymbol {R} = \mathop {\mathbf {Crit}}\nolimits (f)$ where $f$ is a regular function on a smooth scheme $U$, $\omega _{\boldsymbol {R}}$ the $-1$-shifted symplectic form on $\boldsymbol {R}$ constructed in Example 2.5, and an open inclusion $\boldsymbol {\iota } \colon \boldsymbol {R} \hookrightarrow \boldsymbol {X}$ such that $\boldsymbol {\iota } ^{*} \omega _{\boldsymbol {X}} \sim \omega _{\boldsymbol {R}}$, the tuple $(R, U, f, i)$ gives a d-critical chart for $(X, s_X)$ where we write $R = t_0(\boldsymbol {R})$ and $i \colon R \hookrightarrow U$ for the natural closed embedding. Furthermore, there exists a canonical isomorphism of line bundles

\[ \Lambda_{\boldsymbol{X}} \colon \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{X}}) \cong K_{X, s_X}, \]

where $\mathop {\widehat {\mathrm {det}}}\nolimits (\mathbb {L}_{\boldsymbol {X}}) = \mathop {\mathrm {det}}\nolimits (\mathbb {L}_{\boldsymbol {X}} |_{X^{\mathrm{red}}})$ by definition.

We define the notion of orientations for $-1$-shifted symplectic derived schemes to be that of the underlying d-critical schemes.

For later use, we explain the construction of $\Lambda _{\boldsymbol {X}}$ in the above theorem for $\boldsymbol {X} = \mathop {\mathbf {Crit}}\nolimits (f)$ where $f$ is a regular function on a smooth scheme $U$. In this case, $\mathbb {L}_{\boldsymbol {X}}|_{X}$ is represented by the two-term complex

\[ (T_U |_X \xrightarrow{\mathrm{Hess}(f)} \Omega_U |_X), \]

where $\mathrm {Hess}(f)$ denotes the Hessian of $f$. We define $\Lambda _{\boldsymbol {X}}$ by the following composition:

(2.4)\begin{align} \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{X}}) &\cong \mathop{\mathrm{det}}\nolimits(\Omega_U \vert _{X^{\mathrm{red}}}) \otimes \mathop{\mathrm{det}}\nolimits(T_U \vert _{X^{\mathrm{red}}})^{{-}1} \nonumber\\ &\cong (i^{*} K_U) \vert _{X^{\mathrm{red}}} ^{{\otimes}^{2}} \xrightarrow{\cdot \big(\frac{1}{2}\big)^{\dim U}} (i^{*} K_U) \vert _{X^{\mathrm{red}}} ^{{\otimes}^{2}} \cong K_{X, s}. \end{align}

where $\mathop {\mathrm {det}}\nolimits (T_U)^{-1} \cong K_U$ is locally defined by

(2.5)\begin{equation} (\partial/ \partial z_1 \wedge \cdots \wedge \partial / \partial z_n)^{{\vee}} \mapsto {{\mathrm d}}z_1 \wedge \cdots \wedge {{\mathrm d}}z_n. \end{equation}

The constant $(\frac {1}{2})^{\dim U}$ is just a convention, and it corresponds to the fact that $\mathrm {Hess} (z^{2}) = 2({{\mathrm d}}z)^{\otimes ^{2}}$.

We now we discuss the canonical orientation for $-1$-shifted cotangent schemes. To do this we recall basic facts on the determinant of perfect complexes, which are proved in Appendix A.

We write $\hat {\chi }_E = \hat {\chi }_E^{(1)}$. The following example defines the canonical orientation for $-1$-shifted cotangent schemes.

Example 2.15 [Reference TodaTod19, Lemma 4.3]

Let $\boldsymbol {Y}$ be a quasi-smooth derived scheme, ${\boldsymbol {\pi }} \colon {\mathbf {T}^{*}[-1]\boldsymbol {Y}} \to {\boldsymbol {Y}}$ be the projection, and write $s_{\mathbf {T}^{*}[-1]\boldsymbol {Y}}$ for the d-critical structure associated with the canonical $-1$-shifted symplectic form constructed in Example 2.6. We have a natural distinguished triangle

(2.6)\begin{equation} \Delta_{\boldsymbol{\pi} } \colon {\boldsymbol{\pi} ^{*}} \mathbb{L}_{\boldsymbol{Y}} \to \mathbb{L}_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}} \to {\boldsymbol{\pi} ^{*}} \mathbb{T}_{\boldsymbol{Y}}[1] \to \mathbb{L}_{\boldsymbol{Y}}[1] . \end{equation}

Define an isomorphism

\[ o_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}}' \colon (\pi^{\mathrm{red}} )^{*} \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{Y}})^{{\otimes}^{2}} \cong \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}}) \]

by the composition of the isomorphisms

\begin{align*} (\pi^{\mathrm{red}} )^{*} \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{Y}})^{{\otimes}^{2}} &\xrightarrow{\mathop{\mathrm{id}}\nolimits \otimes \hat{\eta}_{\boldsymbol{\pi}^{*} \mathbb{T}_{\boldsymbol{Y}}}} (\pi^{\mathrm{red}} )^{*} \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{Y}}) \otimes (\pi^{\mathrm{red}} )^{*} \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{T}_{\boldsymbol{Y}})^{{-}1} \\ &\xrightarrow{\mathop{\mathrm{id}}\nolimits \otimes \hat{\chi}_{\boldsymbol{\pi}^{*} \mathbb{T}_{\boldsymbol{Y}}}^{{-}1}} (\pi^{\mathrm{red}} )^{*} \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{Y}}) \otimes (\pi^{\mathrm{red}} )^{*} \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{T}_{\boldsymbol{Y}}[1]) \\ &\xrightarrow{\hat{\imath}(\Delta_{\boldsymbol{\pi}})} \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}}) \end{align*}

where we write $\pi = t_0(\boldsymbol {\pi } )$. Define

(2.7)\begin{equation} o^{}_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}}\colon (\pi^{\mathrm{red}} )^{*} \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{Y}})^{{\otimes}^{2}} \cong K_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}, s_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}}} \end{equation}

by the composition $\Lambda _{\mathbf {T}^{*}[-1]\boldsymbol {Y}} \circ o_{\mathbf {T}^{*}[-1]\boldsymbol {Y}}'$, and we call this the canonical orientation for $\mathbf {T}^{*}[-1]\boldsymbol {Y}$.

2.3 Vanishing cycle complexes

Let $f$ be a holomorphic function on a complex manifold $U$, and set $U_0 = f^{-1}(0)$ and $U_{\leq 0} = f^{-1}(\{z\in \mathbb {C} \mid \mathrm {Re}(z) \leq 0\})$. The (shifted) vanishing cycle functor ${\varphi _f ^{p}} \colon {D^{b} _c(U, \mathbb {Q})} \to {D^{b} (U_0, \mathbb {Q})}$ is defined by the composition of the functors

\[ \varphi_f ^{p} := (U_0 \hookrightarrow U_{{\leq} 0})^{*} (U_{{\leq} 0} \hookrightarrow U)^{!}. \]

The functor $\varphi _f ^{p}$ preserves the constructibility (see, for example, [Reference DimcaDim04, Definition 4.2.4, Proposition 4.2.9]). The canonical morphism $(U_{\leq 0} \hookrightarrow U)^{!} \to (U_{\leq 0} \hookrightarrow U)^{*}$ induces a natural transform

(2.8)\begin{equation} \gamma_f \colon \varphi_f ^{p} \to (U_0 \hookrightarrow U)^{*}. \end{equation}

Here we list basic properties of the functor $\varphi _f ^{p}$ we use later.

By an abuse of notation, we write $\varphi _f ^{p} = \varphi _f ^{p} (\mathbb {Q}_U[\dim U])$ if there is no confusion. We identify the $i$th cohomology of the stalk of $\varphi _f ^{p}$ at some point with the $(i + \dim U)$th relative cohomology of a ball modulo the Milnor fiber at a small positive value. We regard $\varphi _f ^{p}$ as a perverse sheaf on $U$, $U_0$, or $\mathop {\mathrm {Crit}}\nolimits (f)$ depending on each situation.

If we write $z \colon \mathbb {C} \to \mathbb {C}$ for the identity map, we have natural isomorphisms

(2.11)\begin{align} (\varphi _{z^{2}} ^{p})_0 &\cong \mathrm{H}^{1} (\mathbb{C}, \{z \in \mathbb{C} \mid \mathrm{Re}(z^{2}) >0 \}; \mathbb{Q}) \nonumber\\ &\cong \mathrm{H}^{1} (\mathbb{R}, \mathbb{R} \setminus 0; \mathbb{Q}). \end{align}

The latter isomorphism is induced by the inclusion

\[ (\mathbb{R}, \mathbb{R} \setminus 0) \hookrightarrow (\mathbb{C}, \{z \in \mathbb{C} \mid \mathrm{Re}(z^{2}) >0 \}). \]

The orientation of $\mathbb {R}$ given by the positive direction defines a cohomology class $a_+ \in \mathrm {H}^{1} (\mathbb {R}, \mathbb {R} \setminus 0; \mathbb {Q})$, hence a trivialization

(2.12)\begin{equation} h_{1,z} \colon \varphi _{z^{2}} ^{p} \cong \mathbb{Q}_{0}. \end{equation}

Let $(z_1, \ldots, z_n)$ be the standard coordinate of $\mathbb {C}^{n}$. Then the Thom–Sebastiani theorem and (2.12) give an isomorphism

(2.13)\begin{equation} h_{n,(z_1, \ldots, z_n)} \colon \varphi _{z_1^{2}+\cdots+z_n ^{2}} ^{p} |_{(0, \ldots, 0)} \cong \mathbb{Q}_{(0,\ldots,0)}. \end{equation}

We now recall the construction of the globalization of the vanishing cycle complexes associated with oriented d-critical schemes introduced in [Reference Brav, Bussi, Dupont, Joyce and SzeondröiBBDJS15]. To do this we introduce the following notation. For a scheme $X$, a principal $\mathbb {Z}/2\mathbb {Z}$-bundle $P$ on $X$, and $F \in D^{b} _c (X, \mathbb {Q})$, one defines

\[ F \otimes _{\mathbb{Z}/2\mathbb{Z}} P := F \otimes _{\mathbb{Q}_X} (\mathbb{Q}_X \otimes _{\mathbb{Z}_X/2\mathbb{Z}_X} P) \]

where the $\mathbb {Z}_X/2\mathbb {Z}_X$-module structure on $\mathbb {Q}_X$ is defined by multiplication by $-1$. For a d-critical scheme $(X, s)$ with a fixed orientation $o \colon {L^{\otimes ^{2}}} \xrightarrow {\sim }{K_{X,s}}$ and its d-critical chart $\mathscr {R} = (R, U, f, i)$, we define a principal $\mathbb {Z}/2\mathbb {Z}$-bundle

(2.14)\begin{equation} Q_{\mathscr{R}}^{o} \end{equation}

over $R$ whose sections are local isomorphisms

\[ {a} \colon {L} \to {(i^{*} K_U)\vert _{R^{\mathrm{red}}}} \]

such that $a^{\otimes ^{2}} = \iota _{R, U, f, i} \circ o$. For an embedding of d-critical charts

\[ \Phi \colon \mathscr{R}_1 = (R, U_1, f_1, i_1) \hookrightarrow \mathscr{R}_2 = (R, U_2, f_2, i_2) \]

such that $\alpha \colon U_2 \to U_1$ and $\beta \colon U_1 \to \mathbb {C}^{n}$ as in Theorem 2.10(ii) exist,

\[ (Q_{\mathscr{R}_1}^{o})^{{-}1} \otimes Q_{\mathscr{R}_2}^{o} \]

parameterizes square roots of

\[ i_1^{*} \beta^{*} ({{\mathrm d}}z_1\wedge \cdots \wedge {{\mathrm d}}z_n) |_{R^{\mathrm{red}}}^{{\otimes}^{2}}. \]

Thus the choice $i_1^{*} \beta ^{*} ({{\mathrm d}}z_1\wedge \cdots \wedge {{\mathrm d}}z_n) |_{R^{\mathrm{red}}}$ gives an isomorphism

(2.15)\begin{equation} Q_{\mathscr{R}_1}^{o} \cong Q^{o}_{\mathscr{R}_2}. \end{equation}

On the other hand, we have isomorphisms

(2.16)\begin{equation} i_1 ^{*} \varphi_{f_1} ^{p}\cong i_2^{*} \alpha^{*} \varphi_{f_1} ^{p} \otimes \beta^{*} \varphi_{z_1^{2} + \cdots + z_n^{2}} ^{p} \cong i_2^{*} \varphi_{f_2}^{p} \end{equation}

where the first one is defined using (2.12) and the second one is a Thom–Sebastiani isomorphism.

With this notation, the globalized vanishing cycle complex is defined by the following theorem.

Theorem 2.17 [Reference Brav, Bussi, Dupont, Joyce and SzeondröiBBDJS15, Theorem 6.9]

Let $(X, s, o)$ be an oriented d-critical scheme, and let $\mathscr {R}_1 = (R_1, U_1, f_1, i_1)$ and $\mathscr {R}_2 = (R_2, U_2, f_2, i_2)$ be any d-critical charts with $R_1 \subset R_2$. Then there exists a natural isomorphism

\[ {\Upsilon_{\mathscr{R}_2, \mathscr{R}_1}} {i_1^{*}\varphi_{f_1}^{p}\otimes _{\mathbb{Z}/2\mathbb{Z}}Q^{o}_{\mathscr{R}_2}} {i_2^{*}\varphi_{f_2}^{p}\otimes _{\mathbb{Z}/2\mathbb{Z}}Q^{o}_{\mathscr{R}_1} \vert _{R_1}} \]

with the following properties.

1. (i) Given another d-critical chart $\mathscr {R}_3 = (R_3, U_3, f_3, i_3)$ with $R_2 \subset R_3$, we have

   \[ \Upsilon_{\mathscr{R}_3, \mathscr{R}_1} =\Upsilon_{\mathscr{R}_3, \mathscr{R}_2} \vert _{R_1} \circ \Upsilon_{\mathscr{R}_2, \mathscr{R}_1}. \]

2. (ii) If $\mathscr {R}_1$ is an open subchart of $\mathscr {R}_2$, then $\Upsilon _{\mathscr {R}_2, \mathscr {R}_1}$ is defined by the canonical isomorphisms

   \[ \varphi_{f_1}^{p} \cong \varphi_{f_2}^{p} |_{U_1},\quad Q^{o}_{\mathscr{R}_1} \cong Q^{o}_{\mathscr{R}_1} |_{U_1}. \]

3. (iii) For an embedding of d-critical charts

   \[ \mathscr{R}_1 = (R, U_1, f_1, i_1) \hookrightarrow \mathscr{R}_2 = (R, U_2, f_2, i_2) \]
   such that $\alpha \colon U_2 \to U_1$ and $\beta \colon U_1 \to \mathbb {C}^{n}$ as in Theorem 2.10(ii) exist, $\Upsilon _{\mathscr {R}_2, \mathscr {R}_1}$ is defined by isomorphisms (2.15) and (2.16).

Using (i), we can define a perverse sheaf $\varphi _{X, s, o}^{p}$ on $X$ such that for a given d-critical chart $\mathscr {R} = (R, U, f, i)$ there exists a natural isomorphism

\[ \omega_{\mathscr{R}} \colon \varphi _{X, s, o}^{p} |_R \cong i^{*} \varphi_f^{p} \otimes _{\mathbb{Z}/2\mathbb{Z}} Q^{o}_{\mathscr{R}}. \]

Moreover, there exists an isomorphism $\sigma _{X, s, o} \colon \mathbb {D}_X(\varphi _{X, s, o}^{p}) \cong \varphi _{X, s, o}^{p}$.

For later use, we recall the construction of $\sigma _{X, s, o}$. For a d-critical chart $\mathscr {R} = (R, U, f, i)$, the Verdier self-duality of $\varphi _f^{p}$ induces an isomorphism

(2.17)\begin{equation} \sigma'_{\mathscr{R}} \colon \mathbb{D}_X(\varphi_{X, s, o}^{p})|_R \cong \varphi_{X, s, o}^{p}|_R. \end{equation}

If we define

\[ \sigma_{\mathscr{R}} = ({-}1)^{\dim U \cdot (\dim U -1)/2} \sigma'_{\mathscr{R}}, \]

we can show that it glues to define an isomorphism $\sigma _{X, s, o}$ (the necessity of the sign intervention is due to the fact that the first diagram in [Reference Brav, Bussi, Dupont, Joyce and SzeondröiBBDJS15, Theorem 2.13] commutes up to the sign $(-1)^{\dim U \cdot \dim V}$). If there is no confusion, we write $\sigma _{X} = \sigma _{X, s, o}$.

For an oriented $-1$-shifted symplectic derived scheme $(\boldsymbol {X}, \omega _{\boldsymbol {X}}, o)$, define $\varphi _{\boldsymbol {X}, \omega _{\boldsymbol {X}}, o}$ to be the perverse sheaf $\varphi ^{p}_{X, s_X, o}$ on $X = t_0(\boldsymbol {X})$ where $s_X$ is the d-critical structure associated with $\omega _{\boldsymbol {X}}$. If there is no confusion, we simply write $\varphi _{\boldsymbol {X}, o}$ or $\varphi _{\boldsymbol {X}}$ instead of $\varphi _{\boldsymbol {X}, \omega _{\boldsymbol {X}}, o}$.

2.4 Dimensional reduction

Let $U$ be a smooth variety of dimension $n$, and $s$ be a section of a trivial vector bundle $E$ of rank $r$ on $U$. Denote by $\bar {s} \colon \mathop {\mathrm {Tot}}\nolimits _U(E^{\vee }) \to \mathbb {A}^{1}$ the regular function corresponding to $s$. We have a canonical morphism

(2.18)\begin{equation} \gamma_{\bar{s}} (\mathbb{Q}_{\mathop{\mathrm{Tot}}\nolimits_U(E^{{\vee}})}[n + r]) \colon \varphi_{\bar{s}}^{p} \to \mathbb{Q}_{{\bar{s}}^{{-}1}(0)}[n + r]. \end{equation}

Define $Z := Z(s)$ to be the zero locus of $s$, and $\tilde {Z} := (\pi _{E^{\vee }})^{-1}(Z)$ where $\pi _{E^{\vee }} \colon \mathop {\mathrm {Tot}}\nolimits _U(E^{\vee }) \to U$ is the projection. By restricting (2.18) to $\tilde {Z}$, we obtain

(2.19)\begin{equation} \gamma_{\bar{s}} := \gamma_{\bar{s}} (\mathbb{Q}_{\mathop{\mathrm{Tot}}\nolimits_U(E^{{\vee}})}[n + r]) |_{\tilde{Z}} \colon \varphi_{\bar{s}}^{p} \to \mathbb{Q}_{\tilde{Z}}[n + r]. \end{equation}

Here we identify $\varphi _{\bar {s}}^{p}$ and $\varphi _{\bar {s}}^{p} | _{\tilde {Z}}$ since the support of $\varphi _{\bar {s}}^{p}$ is contained in $\tilde {Z}$.

Theorem 2.18 [Reference DavisonDav17, Theorem A.1]

The natural map

(2.20)\begin{equation} \bar{\gamma}_{\bar{s}} := (\pi_{E^{{\vee}}})_! \gamma_{\bar{s}} \colon (\pi_{E^{{\vee}}})_! \varphi_{\bar{s}}^{p} \to (\pi_{E^{{\vee}}})_! \mathbb{Q}_{\tilde{Z}}[n + r] \cong \mathbb{Q}_{Z}[n - r] \end{equation}

is an isomorphism.

We want to globalize this statement for the $-1$-shifted cotangent space using Lemma 2.7. To do this, we first need to prove the triviality of the $\mathbb {Z} / 2\mathbb {Z}$-bundle introduced in (2.14) associated with the canonical orientations for shifted cotangent schemes and certain d-critical charts.

Lemma 2.19 Let $U, s$ be as above, and $\boldsymbol {Z} := \boldsymbol {Z}(s)$ be the derived zero locus of $s$. Assume $U$ is affine and carries a global étale coordinate. Denote by $o^{} _{\mathbf {T}^{*}[-1]\boldsymbol {Z}}$ the canonical orientation constructed in Example 2.15 and by

\[ \tilde{\mathscr{Z}} = (\mathrm{Crit}(\bar{s}), \mathop{\mathrm{Tot}}\nolimits_U(E^{{\vee}}), \bar{s}, i) \]

the d-critical chart induced by the equivalence (2.2). Then $Q^{o^{} _{\mathbf {T}^{*}[-1]\boldsymbol {Z}}}_{\tilde {\mathscr {Z}}}$ is a trivial $\mathbb {Z} / 2\mathbb {Z}$-bundle.

Proof. The distinguished triangle (2.6) for $\boldsymbol {Z}$ restricted to $\mathop {\mathrm {Crit}}\nolimits (\bar {s})$ is represented by the following short exact sequence of two-term complexes:

(2.21)

Thus the canonical orientation (2.7) for $\mathop {\mathbf {Crit}}\nolimits (\bar {s})$ is identified with

(2.22)\begin{equation} \mu \colon (\mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} \Omega_U) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} E^{{\vee}})^{{-}1})^{{\otimes}^{2}} \cong \mathop{\widehat{\mathrm{det}}}\nolimits(\Omega_{\mathop{\mathrm{Tot}}\nolimits_U(E^{{\vee}})})^{{\otimes}^{2}}, \end{equation}

where

\[ \mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} E) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} T_U)^{{-}1} \cong \mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} \Omega_U) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} E^{{\vee}})^{{-}1} \]

and

\[ \mathop{\mathrm{det}}\nolimits(T_U)^{{-}1} \cong \mathop{\mathrm{det}}\nolimits(\Omega_U) \]

are defined in the same manner as (2.5). On the other hand, we have an isomorphism

(2.23)\begin{align} \nu \colon \mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} \Omega_U) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} E^{{\vee}})^{{-}1} &\cong \mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} \Omega_U) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\pi_{E^{{\vee}}}^{*} E) \nonumber\\ &\cong \mathop{\widehat{\mathrm{det}}}\nolimits(\Omega_{\mathop{\mathrm{Tot}}\nolimits_U(E^{{\vee}})}), \end{align}

where the first isomorphism is defined as (2.5) and the second isomorphism is induced by the lower short exact sequence in (2.21). By the definition of $o^{} _{\mathbf {T}^{*}[-1]\boldsymbol {Z}}$ and (2.4) we see that

(2.24)\begin{equation} \mu = ({-}1)^{ (n+r)(n+r-1)/2 } (1/2)^{n + r} \cdot \nu ^{{\otimes}^{2}}, \end{equation}

where we write $n = \dim U$ and $r = \mathop {\mathrm {rank}}\nolimits E$. The appearance of the sign $(-1)^{ (n+r)(n+r-1)/2 }$ is caused by the difference of the maps (2.5) and (A.2), and the difference of the symmetric monoidal structure for the category of graded line bundles (A.1) and the standard symmetric monoidal structure for the category of ungraded line bundles. Equation (2.24) implies the triviality of $Q^{o^{} _{\mathbf {T}^{*}[-1]\boldsymbol {Z}}}_{\tilde {\mathscr {Z}}}$.

For later use, we explicitly choose a trivialization of $Q^{o^{} _{\mathbf {T}^{*}[-1]\boldsymbol {Z}}}_{\tilde {\mathscr {Z}}}$. For each $(a, b) \in \mathbb {Z}_{\geq 0}^{2}$, take $\epsilon _{a, b} \in \{1, -1, \sqrt {-1}, -\sqrt {-1}\}$ so that

• • $\epsilon _{0, 0} = 1$,

• • $\epsilon _{a, b} = (-1)^{b} \sqrt {-1} \epsilon _{a-1, b-1}$,

• • $\epsilon _{a+1, b} = (-\sqrt {-1})^{a - b}\epsilon _{a, b}$.

Then

(2.25)\begin{equation} \epsilon_{n, r} (1/\sqrt{2})^{n+r} \cdot \nu \end{equation}

gives a square root of $\mu$, hence a trivialization of $Q^{o^{} _{\mathbf {T}^{*}[-1]\boldsymbol {Z}}}_{\tilde {\mathscr {Z}}}$.

4.1 Lisse-analytic topology

We briefly recall the theory of lisse-analytic topos introduced in [Reference SunSun17], which is a complex analytic analogue of the lisse-étale topos. All statements in this section can be deduced in the same manner as in [Reference OlssonOls07] or [Reference Laszlo and OlssonLO08], so we do not give detailed proofs.

Let $\mathbf {AnSp}$ denote the site of complex analytic spaces equipped with the analytic topology. A stack in groupoid $\mathcal {X}$ over $\mathbf {AnSp}$ is called a complex analytic stack if the following conditions hold.

1. (i) The diagonal morphism $\mathcal {X} \to \mathcal {X} \times \mathcal {X}$ is representable by complex analytic spaces.

2. (ii) There exists a smooth surjection $U \to \mathcal {X}$ from a complex analytic space $U$.

It can easily be seen that a sheaf $F \in \mathcal {X}_{\text {lis-an}}$ is given by the following data:

• • a sheaf $F_{(U, u)}$ on $U$ for each $(u \colon U \to \mathcal {X}) \in \mathop {\text {Lis-An}} \nolimits (\mathcal {X})$ and

• • a morphism $c_f \colon f^{-1} F_{(V, v)} \to F_{(U, u)}$ for each $f \colon (u \colon U \to \mathcal {X}) \to (v \colon V \to \mathcal {X})$ in $\mathop {\text {Lis-An}} \nolimits (\mathcal {X})$

such that the following conditions hold.

• – $c_f$ is an isomorphism if $f$ is an open immersion.

• – Given a composition

  \[ (u \colon U \to \mathcal{X}) \xrightarrow{f} (v \colon V \to \mathcal{X}) \xrightarrow{g} (w \colon W \to \mathcal{X}), \]
  we have $c_{g \circ f} = c_f \circ f^{-1}c_g$.

Denote by $\mathrm {Mod}(\mathcal {X}_{\text {lis-an}}, \mathbb {Q})$ the category of sheaves of $\mathbb {Q}$-vector spaces over $\mathcal {X}$ and by $D(\mathcal {X}_{\text {lis-an}}, \mathbb {Q})$ the derived category of $\mathrm {Mod}(\mathcal {X}_{\text {lis-an}}, \mathbb {Q})$.

Denote by $D_{\mathrm {cart}}(\mathcal {X}_{\text {lis-an}}, \mathbb {Q})$ (respectively, $D_c(\mathcal {X}_{\text {lis-an}}, \mathbb {Q})$) the full subcategory of $D(\mathcal {X}_{\text {lis-an}}, \mathbb {Q})$ spanned by complexes whose cohomologies are Cartesian sheaves (respectively, constructible sheaves).

For an Artin stack $\mathfrak {X}$, one can define its associated complex analytic stack $\mathfrak {X}^{\mathrm {an}}$ as in [Reference SunSun17, 3.2.2]. By an abuse of notation, we write $\mathop {\text {Lis-An}} \nolimits (\mathfrak {X})$ (respectively, $\mathfrak {X}_{\text {lis-an}}$) instead of $\mathop {\text {Lis-An}} \nolimits (\mathfrak {X}^{\mathrm {an}})$ (respectively, ${\mathfrak {X}}^{\mathrm {an}}_{\text {lis-an}}$). For $* \in \{b, +, -\}$, $D^{(*)}(\mathfrak {X}_{\text {lis-an}}, \mathbb {Q})$ denotes the full subcategory of $D(\mathfrak {X}_{\text {lis-an}}, \mathbb {Q})$ consisting of complexes $K$ such that $K|_\mathfrak {U} \in D^{*}(\mathfrak {U}_{\text {lis-an}}, \mathbb {Q})$ for any quasi-compact Zariski open subset $\mathfrak {U} \subset \mathfrak {X}$. Define $D^{(*)}_{\mathrm {cart}}(\mathfrak {X}_{\text {lis-an}}, \mathbb {Q})$ and $D^{(*)}_c(\mathfrak {X}_{\text {lis-an}}, \mathbb {Q})$ in a similar manner.

Arguing as in [Reference Laszlo and OlssonLO08], if we are given a morphism $f \colon \mathfrak {X} \to \mathfrak {Y}$ of finite type between Artin stacks, we can construct six functors:

\begin{gather*} Rf_* \colon D^{(+)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}) \to D^{(+)}_{c}(\mathfrak{Y}_{\text{lis-an}}, \mathbb{Q}),\quad f^{*} \colon D_{c}(\mathfrak{Y}_{\text{lis-an}}, \mathbb{Q}) \to D_{c} (\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}),\\ Rf_! \colon D^{(-)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}) \to D^{(-)}_{c}(\mathfrak{Y}_{\text{lis-an}}, \mathbb{Q}),\quad f^{!} \colon D_{c}(\mathfrak{Y}_{\text{lis-an}}, \mathbb{Q}) \to D_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}),\\ (-) \otimes (-) \colon D^{(-)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}) \times D^{(-)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}) \to D^{(-)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}), \\ \mathop{\mathit{R}\mathcal{H}\! \mathit{om}}\nolimits \colon D^{(-)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q})^{\mathrm{op}} \times D^{(+)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}) \to D^{(+)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}). \end{gather*}

We briefly recall the construction of $Rf_*$, $f^{*}$, $Rf_!$ and $f^{!}$. Firstly define

\[ f_* \colon \mathrm{Mod}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}) \to \mathrm{Mod}(\mathfrak{Y}_{\text{lis-an}}, \mathbb{Q}) \]

by the rule that $f_* F(U) = F(U \times _{\mathfrak {Y}} \mathfrak {X})$. By taking the derived functor of $f_*$ we obtain $Rf_*$. When $f$ is a smooth morphism, $f^{*}$ is nothing but the restriction functor. In general, $f^{*}$ is constructed by taking simplicial covers, but we use pullback functors only for smooth morphisms in this paper, so we do not need this general construction. To define $Rf_!$ and $f^{!}$ we use the Verdier duality functor. Arguing as in [Reference Laszlo and OlssonLO08, § 3], we can construct the dualizing complex

\[ \omega_{\mathfrak{X}} \in D^{(b)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}) \]

and define the Verdier duality functor

\[ \mathbb{D}_{\mathfrak{X}} := \mathop{\mathit{R}\mathcal{H}\! \mathit{om}}\nolimits(-, \omega_{\mathfrak{X}}) \colon D^{(-)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q})^{\mathrm{op}} \to D^{(+)}_{c}(\mathfrak{X}_{\text{lis-an}}, \mathbb{Q}). \]

Now define $Rf_! := \mathbb {D}_{\mathfrak {Y}} \circ f_* \circ \mathbb {D}_{\mathfrak {X}}$ and $f^{!} := \mathbb {D}_{\mathfrak {X}} \circ f^{*} \circ \mathbb {D}_{\mathfrak {Y}}$. If $f$ is a smooth morphism of relative dimension $d$, we have natural isomorphisms

(4.1)\begin{align} f^{*}\mathbb{D}_{\mathfrak{X}}(F) &= f^{*}\mathop{\mathit{R}\mathcal{H}\! \mathit{om}}\nolimits(F, \omega_{\mathfrak{Y}}) \xrightarrow{\sim} \mathop{\mathit{R}\mathcal{H}\! \mathit{om}}\nolimits(f^{*}F, f^{*}\omega_{\mathfrak{Y}}) \nonumber\\ &\xrightarrow{\sim} \mathop{\mathit{R}\mathcal{H}\! \mathit{om}}\nolimits(f^{*}F, \omega_{\mathfrak{X}}[{-}2d]) \cong \mathbb{D}_{\mathfrak{Y}}(f^{*}F)[{-}2d] \end{align}

for $F \in D_c(\mathfrak {Y}, \mathbb {Q})$. Therefore we have $f^{!} \cong f^{*}[2d]$.

If we are given a $2$-morphism $\xi \colon f \Rightarrow g$ between morphisms of finite type of Artin stacks, we have a natural isomorphism $\xi _* \colon Rf_* \Rightarrow Rg_*$ compatible with the vertical and horizontal compositions. The same statement also holds for $Rf_!$, $f^{*}$ and $f^{!}$.

We now discuss the base-change isomorphisms. Consider the following $2$-Cartesian diagram of Artin stacks.

By adjunction, we have the base-change map

(4.2)\begin{equation} \mathfrak{bc}_{\eta} \colon g^{*} Rf_* \to Rf'_*g'^{*} \end{equation}

which is an isomorphism if $g$ is smooth. Now assume that $f$ is of finite type and $g$ is smooth with relative dimension $d$, and take $F \in D^{(-)}_c(\mathfrak {X}, \mathbb {Q})$. Then we can construct the proper base-change map

\[ \mathfrak{pbc}_{\eta} \colon g^{*} Rf_! F \xrightarrow{\sim} Rf'_!g'^{*} F \]

by the composition

\begin{align*} g^{*} Rf_! F &= g^{*} \mathbb{D}_{\mathfrak{Y}} Rf_* \mathbb{D}_{\mathfrak{X}} F \\ &\cong \mathbb{D}_{\mathfrak{Y}'} g^{*} Rf_* \mathbb{D}_{\mathfrak{X}} F [2d] \\ &\cong \mathbb{D}_{\mathfrak{Y}'} Rf'_* g'^{*} \mathbb{D}_{\mathfrak{X}} F [2d] \\ &\cong \mathbb{D}_{\mathfrak{Y}'} Rf'_* \mathbb{D}_{\mathfrak{X}'} g'^{*} F = Rf'_! g'^{*} F, \end{align*}

where the first and third isomorphisms are defined by using (4.1) and the second isomorphism is the base-change map (4.2). Now consider the following composition of $2$-Cartesian diagrams:

where $f$ is of finite type, and $g$ and $h$ are smooth. We define $\eta \colon f \circ k' \Rightarrow k \circ f''$ by composing $2$-morphisms in the diagram. Arguing as [Reference Laszlo and OlssonLO08, Lemma 5.1.2], we can show the commutativity of the following diagram.

(4.3)

For an Artin stack $\mathfrak {X}$, we define a full subcategory $\mathrm {Perv}(\mathfrak {X}) \hookrightarrow D_c(\mathfrak {X}, \mathbb {Q})$ consisting of objects $K$ such that, for any smooth morphism $f \colon U \to \mathfrak {X}$ from a scheme, $f^{*} K [\dim f]$ is a perverse sheaf on $U$. An object in $\mathrm {Perv}(\mathfrak {X})$ is called a perverse sheaf on $\mathfrak {X}$. Arguing as [Reference Laszlo and OlssonLO09, Proposition 7.1], we see that $U \mapsto \mathrm {Perv}(U)$ defines a stack on $\mathop {\text {Lis-An}} \nolimits (\mathfrak {X})$ whose global section category is $\mathrm {Perv}(\mathfrak {X})$.

4.2 D-critical stacks

In this section we first recall the notion of d-critical stacks introduced in [Reference JoyceJoy15, § 2.8], which is a stacky generalization of d-critical schemes. And then we prove that the canonical d-critical structures and the canonical orientations for $-1$-shifted cotangent stacks are preserved under smooth base changes. This is the key ingredient in the proof of the dimensional reduction theorem for quasi-smooth derived Artin stacks.

We first explain the functorial behavior of the d-critical structure.

Proposition 4.3 [Reference JoyceJoy15, Propositions 2.3, 2.8]

Let $f \colon X \to Y$ be a morphism of complex analytic spaces, and $\mathcal {S}_X$ (respectively, $\mathcal {S}_Y$) be the sheaf on $X$ (respectively, $Y$) defined in (2.3). Then there exists a natural map $\theta _f \colon f^{-1}\mathcal {S}_Y \to \mathcal {S}_X$ with the following property. If $R \subset X$ and $S \subset Y$ are open subsets with $f(R) \subset S$, $i \colon R \hookrightarrow U$ and $j \colon S \hookrightarrow V$ are closed embeddings into complex manifolds, and $\tilde {f} \colon U \to V$ is a holomorphic map with $j \circ f |_R = \tilde {f} \circ i$, then the following diagram commutes.

Here horizontal maps are induced by the natural inclusions, and the right vertical map is induced by $\tilde {f}^{\sharp } \colon \tilde {f}^{-1}\mathcal {O}_V \to \mathcal {O}_U$. The map $\theta _f$ induces natural map $f^{-1}\mathcal {S}_Y ^{0} \to \mathcal {S}_X ^{0}$ (also written as $\theta _f$). If $f$ is smooth and $s \in \Gamma (Y, \mathcal {S}_Y^ 0)$ is a d-critical structure, $f^{\star } s := \theta _f(f^{-1}s)$ is also a d-critical structure.

We now explain that the definition of the sheaf $\mathcal {S}_X ^{0}$ can be extended to complex analytic stacks.

We have a stacky version of Theorem 2.13.

Proof. The uniqueness part is proved in [Reference Ben-Bassat, Brav, Bussi and JoyceBBBJ15, Theorem 3.18(a)]. We now verify that the d-critical structure constructed there satisfies the property as above. Using [Reference Ben-Bassat, Brav, Bussi and JoyceBBBJ15, Theorem 2.10], we have derived schemes $\boldsymbol {U}$ and $\hat {\boldsymbol {U}}$, a smooth surjection $\boldsymbol {u} \colon \boldsymbol {U} \to \boldsymbol {\mathfrak {X}}$, a morphism $\boldsymbol {\tau }_{\boldsymbol {U}} \colon \boldsymbol {U} \to \hat {\boldsymbol {U}}$, and a $-1$-shifted symplectic form $\omega _{\hat {\boldsymbol {U}}}$ on $\hat {\boldsymbol {U}}$ such that $\boldsymbol {u}^{\star } \omega _{\boldsymbol {X}} \sim \boldsymbol {\tau }_{\boldsymbol {U}}^{\star } \omega _{\hat {\boldsymbol {U}}}$. Further, if we write $s_{\hat {U}}$ for the d-critical structure associated with $\omega _{\hat {\boldsymbol {U}}}$, we may assume $t_0(\boldsymbol {u})^{\star } s_{\mathfrak {X}} = t_0(\boldsymbol {\tau }_{\boldsymbol {U}})^{\star } s_{\hat {U}}$. We have the following diagram of derived stacks.

Now take any point $x \in t_0(\boldsymbol {X})$ and an étale morphism from a derived scheme $\boldsymbol {\eta } \colon \boldsymbol {W} \to \boldsymbol {U} \times _{\boldsymbol {\mathfrak {X}}} \boldsymbol {X}$ such that the image of $t_0(\boldsymbol {u}' \circ \boldsymbol {\eta })$ contains $x$. Since $t_0(\boldsymbol {u}' \circ \boldsymbol {\eta })$ is a smooth morphism, it suffices to show that

\[ t_0(\boldsymbol{\tau}_{\boldsymbol{U}} \circ \boldsymbol{g}' \circ \boldsymbol{\eta})^{{\star}} s_{\hat{U}} = t_0(\boldsymbol{\tau} \circ \boldsymbol{u}' \circ \boldsymbol{\eta})^{{\star}} s_{\hat{X}}. \]

This follows by arguing as in [Reference Brav, Bussi and JoyceBBJ19, Example 5.22] since we have $(\boldsymbol {\tau }_{\boldsymbol {U}} \circ \boldsymbol {g}' \circ \boldsymbol {\eta })^{\star } \omega _{\hat {\boldsymbol {U}}} \sim (\boldsymbol {\tau } \circ \boldsymbol {u}' \circ \boldsymbol {\eta }) ^{\star } \omega _{\hat {\boldsymbol {X}}}$.

We now discuss the behavior of the d-critical structure associated with the canonical $-1$-shifted symplectic structures on $-1$-shifted cotangent stacks constructed in Example 2.6 under smooth pullbacks. Let $\boldsymbol {f}\colon \boldsymbol {Y} \to \boldsymbol {\mathfrak {Y}}$ be a smooth morphism from a derived scheme $\boldsymbol {Y}$ to a quasi-smooth derived Artin stack $\boldsymbol {\mathfrak {Y}}$. Consider the following diagram.

(4.4)

Here $\boldsymbol {f}^{*}\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}$ is the total space $\mathop {\mathrm {Tot}}\nolimits _{\boldsymbol {Y}}(\mathbb {L}_{\boldsymbol {\boldsymbol {f}^{*} \mathfrak {Y}}}[-1])$, $\pi _{\boldsymbol {Y}}$, $\pi _{\boldsymbol {\mathfrak {Y}}}$, and $\pi _{\boldsymbol {\mathfrak {Y}}, \boldsymbol {f}}$ are the projections, $\boldsymbol {\tau }$ is induced by the canonical map $\boldsymbol {f}^{*} \mathbb {L}_{\boldsymbol {\mathfrak {Y}}}[-1] \to \mathbb {L}_{\boldsymbol {Y}}[-1]$, and $\tilde {\boldsymbol {f}} \colon \boldsymbol {f}^{*}\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}} \to \mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}$ is the base change of $\boldsymbol {f}$. The smoothness of $\boldsymbol {f}$ implies that $\boldsymbol {\tau }$ induces an isomorphism on underlying schemes, so we use the identification

(4.5)\begin{equation} t_0(\boldsymbol{f}^{*}\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}) = t_0(\mathbf{T}^{*}[{-}1]\boldsymbol{Y}) \end{equation}

throughout the paper.

Proof. By Theorem 4.6, we only need to show that $\boldsymbol {\tau }^{\star } \omega _{\mathbf {T}^{*}[-1]\boldsymbol {Y}} \sim \tilde {\boldsymbol {f}}^{\star } \omega _{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}}$. If we write $\lambda _{\mathbf {T}^{*}[-1]\boldsymbol {Y}}$ and $\lambda _{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}}$ for the tautological $1$-forms on $\mathbf {T}^{*}[-1]\boldsymbol {Y}$ and $\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}$ respectively, we have ${{\mathrm d}}_{\mathrm {dR}} \lambda _{\mathbf {T}^{*}[-1]\boldsymbol {Y}} = \omega _{\mathbf {T}^{*}[-1]\boldsymbol {Y}}$ and ${d_{\mathrm {dR}}} \lambda _{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}} = \omega _{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}}$ by definition. Therefore we only need to prove

\[ \boldsymbol{\tau}^{{\star}} \lambda_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}} \sim \tilde{\boldsymbol{f}}^{{\star}} \lambda_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}}. \]

By the functoriality of the cotangent complex, we have the following homotopy commutative diagram.

Here $a$ and $b$ are defined by using $\boldsymbol {f} \circ \boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}, \boldsymbol {f}} \simeq \boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}} \circ \tilde {\boldsymbol {f}}$ and $\boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}, \boldsymbol {f}} \simeq \boldsymbol {\pi }_{\boldsymbol {Y}} \circ \boldsymbol {\tau }$ respectively, and other morphisms are induced by the functoriality of the cotangent complex. Now write $\gamma _{\boldsymbol {f}^{*}\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}}$, $\gamma _{\mathbf {T}^{*}[-1]\boldsymbol {Y}}$, and $\gamma _{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}}$ for the tautological sections of $(\boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}, \boldsymbol {f}})^{*} \boldsymbol {f}^{*} \mathbb {L}_{\boldsymbol {\mathfrak {Y}}}[-1]$, $\boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}}^{*} \mathbb {L}_{\boldsymbol {\mathfrak {Y}}}[-1]$, and $\boldsymbol {\pi }_{\boldsymbol {Y}}^{*} \mathbb {L}_{\boldsymbol {Y}}[-1]$, respectively. By definition, we have

\begin{align*} \tilde{\boldsymbol{f}}^{{\star}} \lambda_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}} \sim \theta_{\tilde{\boldsymbol{f}}}(\tilde{\boldsymbol{f}}^{*} \lambda_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}}) &\sim \theta_{\tilde{\boldsymbol{f}}} \circ \tilde{\boldsymbol{f}}^{*} \theta_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}}} (\tilde{\boldsymbol{f}}^{*} \gamma_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}}), \\ \boldsymbol{\tau}^{{\star}} \lambda_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}} \sim \theta_{\boldsymbol{\tau}}(\boldsymbol{\tau}^{*} \lambda_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}}) &\sim \theta_{\boldsymbol{\tau}} \circ \boldsymbol{\tau}^{*} \theta_{\boldsymbol{\pi}_{\boldsymbol{Y}}} (\boldsymbol{\tau}^{*} \gamma_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}}). \end{align*}

Since we have the tautological relations

\[ \tilde{\boldsymbol{f}}^{*} \gamma_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}} \sim a( \gamma_{\boldsymbol{f}^{*}\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}}),\quad \boldsymbol{\tau}^{*} \gamma_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}} \sim b \circ (\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}})^{*} \theta_{\boldsymbol{f}} (\gamma_{\boldsymbol{f}^{*}\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}}), \]

the proposition follows.

We now discuss the notion of the virtual canonical bundles and the orientations for d-critical stacks. Before doing this, we recall a property of the virtual canonical bundle of d-critical schemes. For a d-critical chart $(R, U, f, i)$ of a d-critical scheme $(X, s)$ and a point $x \in R$, consider the following complex concentrated in degree $-1$ and $0$:

\[ L_x := (\mathrm{T}_U |_x \xrightarrow{\mathrm{Hess}(f) |_x} \Omega_U |_x). \]

Since $\mathrm {H}^{0}(L_x) \cong \Omega _X |_x$ and $\mathrm {H}^{-1}(L_x) \cong (\Omega _X |_x)^{\vee }$, we can define an isomorphism

(4.6)\begin{equation} \kappa_x \colon K_{X, s} |_x \cong \mathop{\mathrm{det}}\nolimits(L_x) \cong \mathop{\mathrm{det}}\nolimits(\Omega_X|_x) \otimes \mathop{\mathrm{det}}\nolimits((\Omega_X|_x)^{{\vee}})^{{-}1} \cong \mathop{\mathrm{det}}\nolimits(\Omega_X|_x)^{{\otimes}^{2}}. \end{equation}

Here the final isomorphism is defined in the same manner as (2.5). It is proved in [Reference JoyceJoy15, Theorem 2.28] that $\kappa _x$ does not depend on the choice of a d-critical chart. Now the virtual canonical bundle for a d-critical stack is defined by the following proposition.

An orientation $o$ of a d-critical stack $(\mathfrak {X}, s)$ is the choice of a line bundle $L$ on $\mathfrak {X}^{\mathrm{red}}$ and an isomorphism $o \colon L^{\otimes ^{2}} \cong K_{\mathfrak {X}, s}$. An isomorphism between orientations $o_1\colon L_1^{\otimes ^{2}} \cong K_{\mathfrak {X}, s}$ and $o_2 \colon L_2^{\otimes ^{2}} \cong K_{\mathfrak {X}, s}$ is defined by an isomorphism $\phi \colon L_1 \cong L_2$ such that $o_1 = o_2 \circ {\phi }^{\otimes ^{2}}$. If there exists a smooth morphism $u \colon U \to \mathfrak {X}$, we define an orientation $u^{\star } o$ for $(U, u^{\star } s)$ by the following composition:

\[ u^{{\star}} o \colon ((u^{\mathrm{red}})^{*}L \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\Omega_{U/\mathfrak{X}}))^{{\otimes} ^{2}} \cong (u^{\mathrm{red}})^{*} K_{\mathfrak{X}, s} \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\Omega_{U{/}\mathfrak{X}})^{{\otimes} ^{2}} \xrightarrow{\Gamma_{U, u}} K_{U ,u^{{\star}} s}. \]

If we are given a smooth morphism $q \colon (u \colon U \to \mathfrak {X}) \to (v \colon V \to \mathfrak {X})$ in $\mathop {\text {Lis-An}} \nolimits (\mathfrak {X})$, define an isomorphism

(4.9)\begin{equation} u^{{\star}} o \cong q^{{\star}} v^{{\star}} o \end{equation}

by using the natural isomorphism

\[ \mathop{\widehat{\mathrm{det}}}\nolimits(\Omega_{U/\mathfrak{X}}) \cong (f^{\mathrm{red}})^{*}\mathop{\widehat{\mathrm{det}}}\nolimits(\Omega_{V/\mathfrak{X}}) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\Omega_{U/V}). \]

We now discuss the relation of the cotangent complex of a $-1$-shifted symplectic derived Artin stack and the virtual canonical bundle of the associated d-critical stack.

Theorem 4.9 [Reference Ben-Bassat, Brav, Bussi and JoyceBBBJ15, Theorem 3.18(b)]

Let $(\boldsymbol {\mathfrak {X}}, \omega _{\boldsymbol {\mathfrak {X}}})$ be a $-1$-shifted symplectic derived Artin stack, and $(\mathfrak {X}, s_{\mathfrak {X}})$ the associated d-critical stack. Then there exists a natural isomorphism

(4.10)\begin{equation} \Lambda_{\boldsymbol{\mathfrak{X}}} \colon \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{\mathfrak{X}}}) \cong K_{\mathfrak{X}, s_{\mathfrak{X}}} \end{equation}

characterized by the following property.

Assume we are given derived schemes $\boldsymbol {X}$ and $\hat {\boldsymbol {X}}$, and morphisms $\boldsymbol {g} \colon \boldsymbol {X} \to \boldsymbol {\mathfrak {X}}$ and $\boldsymbol {\tau } \colon \boldsymbol {X} \to \hat {\boldsymbol {X}}$ such that $\boldsymbol {g}$ is smooth and $\mathbb {L}_{\boldsymbol {\tau }} |_{x}$ is concentrated in degree $-2$ for each $x \in \boldsymbol {X}$. Note that it automatically follows that $\tau = t_0(\boldsymbol {\tau })$ is étale. Further assume that there exist a $-1$-shifted symplectic structure $\omega _{\hat {\boldsymbol {X}}}$ on $\hat {\boldsymbol {X}}$ with associated d-critical locus $(\hat {X}, s_{\hat {X}})$ and an equivalence $\boldsymbol {g}^{\star } \omega _{\boldsymbol {\mathfrak {X}}} \sim \boldsymbol {\tau }^{\star } \omega _{\hat {\boldsymbol {X}}}$. This equivalence induces a homotopy between the composition

\[ \mathbb{T}_{\boldsymbol{g}} \to \mathbb{T}_{\boldsymbol{X}} \to \boldsymbol{\tau}^{*}\mathbb{T}_{\hat{\boldsymbol{X}}} \xrightarrow{\boldsymbol{\tau}^{{\star}} \omega_{\hat{\boldsymbol{X}}}} \boldsymbol{\tau}^{*}\mathbb{L}_{\hat{\boldsymbol{X}}}[{-}1] \to \mathbb{L}_{\boldsymbol{X}}[{-}1] \]

and $0$, hence an isomorphism

(4.11)\begin{equation} \ell \colon \mathbb{T}_{\boldsymbol{g}} \xrightarrow{\sim} \mathbb{L}_{\boldsymbol{\tau}}[{-}2]. \end{equation}

Then the composition

\begin{align*} \mathop{\widehat{\mathrm{det}}}\nolimits(\boldsymbol{g}^{*} \mathbb{L}_{\boldsymbol{\mathfrak{X}}}) &\cong \mathop{\widehat{\mathrm{det}}} (\mathbb{L}_{\boldsymbol{X}}) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{g}})^{{-}1} \\ &\cong \mathop{\widehat{\mathrm{det}}}\nolimits(\boldsymbol{\tau}^{*} \mathbb{L}_{\hat{\boldsymbol{X}}}) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{\tau}}) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{g}})^{{-}1} \\ &\cong \mathop{\widehat{\mathrm{det}}}\nolimits(\boldsymbol{\tau}^{*} \mathbb{L}_{\hat{\boldsymbol{X}}}) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{g}})^{{\otimes} ^{{-}2}}\\ &\cong (\tau^{\mathrm{red}})^{*} K_{\hat{X}, s_{\hat{X}}} \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{g}})^{{\otimes} ^{{-}2}} \\ &\cong K_{X,\tau^{{\star}} s_{\hat{X}}} \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\boldsymbol{g}})^{{\otimes} ^{{-}2}}\\ &\cong (g^{\mathrm{red}})^{*}K_{\mathfrak{X}, s_{\mathfrak{X}}} \end{align*}

is equal to $(-1)^{\mathop {\mathrm {rank}}\nolimits (\Omega _{g})} {(g^{\mathrm{red}})}^{*}\Lambda _{\boldsymbol {\mathfrak {X}}}$, where we write $g = t_0(\boldsymbol {g})$. Here the first and second isomorphisms are defined by using $\hat {\imath }(\Delta _{\boldsymbol {g}})$ and $\hat {\imath }(\Delta _{\boldsymbol {\tau }})$ respectively, where $\Delta _{\boldsymbol {g}}$ and $\Delta _{\boldsymbol {\tau }}$ are distinguished triangles

\[ \Delta_{\boldsymbol{g}} \colon \boldsymbol{g}^{*} \mathbb{L}_{\boldsymbol{\mathfrak{X}}} \to \mathbb{L}_{\boldsymbol{X}} \to \mathbb{L}_{\boldsymbol{g}} \to \boldsymbol{g}^{*} \mathbb{L}_{\boldsymbol{\mathfrak{X}}}[1],\quad \Delta_{\boldsymbol{\tau}} \colon \boldsymbol{\tau}^{*} \mathbb{L}_{\hat{\boldsymbol{X}}} \to \mathbb{L}_{\boldsymbol{X}} \to \mathbb{L}_{\boldsymbol{\tau}} \to \boldsymbol{\tau}^{*} \mathbb{L}_{\hat{\boldsymbol{X}}}[1]. \]

The third isomorphism is defined in the same manner as (2.5) using $\ell$ (without any sign intervention used in Appendix A), the fourth isomorphism is $\Lambda _{\hat {\boldsymbol {X}}}$ in Theorem 2.13, and the fifth isomorphism is $\Gamma _{X, \tau }$ defined in Proposition 4.8. The final isomorphism is $\Gamma _{X, g}$, where we use the fact that $\tau ^{\star } s_{\hat {X}} = g^{\star } s_{\mathfrak {X}}$ proved in Theorem 4.6.

Proof. The proof is essentially same as one in [Reference Ben-Bassat, Brav, Bussi and JoyceBBBJ15], but we include it for the reader's convenience and to fix the sign. Suppose we are given $\boldsymbol {X}$, $\hat {\boldsymbol {X}}$, $\boldsymbol {g}$, and $\boldsymbol {\tau }$ as above. Define

\[ \Lambda_{\boldsymbol{X}, \hat{\boldsymbol{X}}, \boldsymbol{g}, \boldsymbol{\tau}} \colon \mathop{\widehat{\mathrm{det}}}\nolimits(\boldsymbol{g}^{*} \mathbb{L}_{\boldsymbol{\mathfrak{X}}}) \to (g^{\mathrm{red}})^{*}K_{\mathfrak{X}, s_{\mathfrak{X}}} \]

by the composition as above multiplied by $(-1)^{\mathop {\mathrm {rank}}\nolimits (\Omega _{g})}$. Write $\mathrm {pr}_1, \mathrm {pr}_2 \colon R = X \times _{\mathfrak {X}} X \rightrightarrows X$ for the first and second projections. We have a natural $2$-morphism $\xi \colon g \circ \mathrm {pr}_1 \Rightarrow g \circ \mathrm {pr}_2$. We now prove the commutativity of the following diagram.

(4.12)

By the reducedness of $R^{\mathrm{red}}$, we only need to prove the commutativity at each point $r \in R$. Write $\mathrm {pr}_1(r) = x_1$ and $\mathrm {pr}_2(r) = x_2$. Now consider the following diagram.

Here (A)$_i$ is defined by the quasi-isomorphism $\boldsymbol {g}^{*} \mathbb {L}_{\boldsymbol {\mathfrak {X}}} |_{x_i} \simeq \mathrm {H}^{*}(\boldsymbol {g}^{*}\mathbb {L}_{\boldsymbol {\mathfrak {X}}} |_{x_i} )$ and (B)$_i$ is defined by $\kappa _{g(x_i)}$ in Proposition 4.8 and the quasi-isomorphism $\tau ^{\geq 0}(g^{*}\mathbb {L}_{\mathfrak {X}})|_{x_i} \simeq \mathrm {H}^{*}(\tau ^{\geq 0}(g^{*}\mathbb {L}_{\mathfrak {X}})|_{x_i})$. The map (C)$_i$ is defined in the same manner as (2.5) using the isomorphisms

\[ \mathrm{H}^{n}(\boldsymbol{g}^{*}\mathbb{L}_{\boldsymbol{\mathfrak{X}}} |_{x_i}) \cong \begin{cases} \mathrm{H}^{n}(\tau^{{\geq} 0}(g^{*}\mathbb{L}_{\mathfrak{X}})|_{x_i}) & n = 0,1 \\ \mathrm{H}^{{-}n-1}(\tau^{{\geq} 0}(g^{*}\mathbb{L}_{\mathfrak{X}})|_{x_i})^{{\vee}} & n ={-}2, -1. \end{cases} \]

The commutativity of the left trapezoid and middle square is obvious, and the commutativity of the right trapezoid follows from the proof of [Reference JoyceJoy15, Theorem 2.56]. It is easy to see that the upper and lower trapezoids commute up to the sign $(-1)^{\mathop {\mathrm {rank}}\nolimits (\mathrm {H}^{1}(\mathbb {L}_{\mathfrak {X}} |_{g(x_i)}))}$ by using the equality (A.3). These commutativity properties imply the commutativity of the outer square, and hence the commutativity of diagram (

4.12

).

By Darboux's theorem [Reference Ben-Bassat, Brav, Bussi and JoyceBBBJ15, Theorem 2.10], we can take $\boldsymbol {X}$, $\hat {\boldsymbol {X}}$, $\boldsymbol {g}$, and $\boldsymbol {\tau }$ in the proposition so that $g$ is surjective. By the commutativity of diagram (4.12), $\Lambda _{\boldsymbol {X}, \hat {\boldsymbol {X}}, \boldsymbol {g}, \boldsymbol {\tau }}$ descends to $\Lambda _{\boldsymbol {\mathfrak {X}}} \colon \mathop {\widehat {\mathrm {det}}}\nolimits (\mathbb {L}_{\boldsymbol {\mathfrak {X}}}) \cong K_{\mathfrak {X}, s_{\mathfrak {X}}}$ satisfying the property in the proposition. The uniqueness of $\Lambda _{\boldsymbol {\mathfrak {X}}}$ as in the theorem is clear from the construction.

The notion of orientation for $-1$-shifted symplectic derived Artin stacks is defined by that of the associated d-critical stacks. Let $\boldsymbol {\mathfrak {Y}}$ be a quasi-smooth derived Artin stack. The argument in Example 2.15 works also for the stacky case and defines a natural isomorphism

(4.13)\begin{equation} o_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}}' \colon \mathop{\widehat{\mathrm{det}}}\nolimits(\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}}^{*}\mathbb{L}_{\boldsymbol{\mathfrak{Y}}})^{{\otimes} ^{2}} \cong \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}}). \end{equation}

We define the canonical orientation $o^{}_{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}}$ for $\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}$ by the composition $o^{}_{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}} := \Lambda _{\boldsymbol {\mathfrak {Y}}} \circ o'_{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}}$.

Proposition 4.10 In the notation as in Proposition 4.7, we have an isomorphism

(4.14)\begin{equation} \Xi_{\boldsymbol{f}} \colon \tilde{f}^{{\star}} o^{}_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}} \cong o^{}_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}}. \end{equation}

Proof. Throughout the proof we use the following notation: for a morphism $\boldsymbol {h} \colon {\boldsymbol {\mathfrak {Z}}} \to \boldsymbol {\mathfrak {W}}$ of derived Artin stacks, we write

\[ \Delta_{\boldsymbol{h}} \colon \boldsymbol{h}^{*} \mathbb{L}_{\boldsymbol{\mathfrak{W}}} \xrightarrow{\theta_{\boldsymbol{h}}} \mathbb{L}_{\boldsymbol{\mathfrak{Z}}} \xrightarrow{\zeta_{\boldsymbol{h}}} \mathbb{L}_{\boldsymbol{h}} \xrightarrow{\delta_{\boldsymbol{h}}} \boldsymbol{h}^{*} \mathbb{L}_{\boldsymbol{\mathfrak{W}}}[1] \]

for the natural distinguished triangle of cotangent complexes.

Define

\[ \Xi_{\boldsymbol{f}}' \colon (\tilde{f}^{\mathrm{red}})^{*} \mathop{\widehat{\mathrm{det}}}\nolimits(\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}}^{*}\mathbb{L}_{\boldsymbol{\mathfrak{Y}}}) \otimes \mathop{\widehat{\mathrm{det}}}\nolimits(\mathbb{L}_{\tilde{\boldsymbol{f}}})\cong \mathop{\widehat{\mathrm{det}}}\nolimits(\boldsymbol{\pi}_{\boldsymbol{Y}}^{*}\mathbb{L}_{\boldsymbol{Y}}) \]

by using $\hat {\imath }(\Delta _{\boldsymbol {f}})$ and the identification $(\boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}, \boldsymbol {f}})^{*} \mathbb {L}_{\boldsymbol {f}} \cong \mathbb {L}_{\tilde {\boldsymbol {f}}}$. Write

(4.15)\begin{equation} \Xi_{\boldsymbol{f}} := \sqrt{-1}^{{\operatorname{vdim} \boldsymbol{f} \cdot (\operatorname{vdim} \boldsymbol{f} - 1) / 2} + \operatorname{vdim}{\boldsymbol{\mathfrak{Y}}} \cdot \operatorname{vdim}{\boldsymbol{f}}} \cdot \Xi_{\boldsymbol{f}}'. \end{equation}

Now it is enough to prove the commutativity of the following diagram of line bundles on $\mathbf {T}^{*}[-1]\boldsymbol {Y}^{\mathrm {red}}$.

(4.16)

Here $\Lambda _{\mathbf {T}^{*}[-1]\boldsymbol {Y}}$ and $\Lambda _{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}}$ are defined in Theorems 2.13 and 4.9 respectively, and the bottom arrow is defined by using $\Gamma _{t_0(\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}) ,\tilde {f}}$ in (4.8) and the identification

\[ \mathbb{L}_{\tilde{\boldsymbol{f}}}\vert_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}^{\mathrm{red}}} \cong \Omega_{\tilde{f}} \vert_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}^{\mathrm{red}}}. \]

Consider the following diagram in $\mathrm {Perf}(\boldsymbol {f}^{*} \mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}})$:

(4.17)

where the top vertical arrows are identified with a part of the natural morphism between distinguished triangles

\[ \boldsymbol{\tau}^{*} \Delta_{\boldsymbol{\pi}_{\boldsymbol{Y}}} \to \Delta_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}}}, \]

by the natural isomorphisms

\[ \boldsymbol{\tau}^{*} \boldsymbol{\pi}_{\boldsymbol{Y}}^{*} \mathbb{L}_{\boldsymbol{Y}} \cong (\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}})^{*} \mathbb{L}_{\boldsymbol{Y}},\quad \boldsymbol{\tau}^{*} \mathbb{L}_{\boldsymbol{\pi}_{\boldsymbol{Y}}} \cong (\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}})^{*} \mathbb{T}_{\boldsymbol{Y}}[1],\quad \mathbb{L}_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}}} \cong (\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}})^{*} \boldsymbol{f}^{*} \mathbb{T}_{\boldsymbol{\mathfrak{Y}}}[1], \]

and $k$ is taken so that the right horizontal arrows define a morphism between distinguished triangles

\[ \Delta_{\boldsymbol{\tau}} \to \Delta_{\boldsymbol{f}, \mathrm{rot}}^{{\vee}}. \]

Here $\Delta _{\boldsymbol {f}, \mathrm {rot}}^{\vee }$ denotes the right vertical distinguished triangle in the diagram above. Now we claim that

(4.18)\begin{equation} \widehat{\mathrm{det}}(k) \circ \widehat{\mathrm{det}}(\ell[2]) = \mathrm{det}(\phi[2]), \end{equation}

where $\ell \colon \mathbb {T}_{\tilde {\boldsymbol {f}}} \xrightarrow {\sim } \mathbb {L}_{\boldsymbol {\tau }}[-2]$ is defined in (4.11), and $\phi \colon (\boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}, \boldsymbol {f}})^{*} \mathbb {T}_{\tilde {\boldsymbol {f}}} \xrightarrow {\sim } \mathbb {T}_{\boldsymbol {f}}$ is the natural isomorphism. To see this, consider the following commutative diagram.

By using [Reference Knudsen and MumfordKM76, Proposition 6], it is enough to prove the equalities

\begin{gather*} \boldsymbol{\tau}^{*} \zeta_{\boldsymbol{\pi}_{\boldsymbol{Y}}} \circ ({\cdot} \boldsymbol{\tau}^{*} \omega_{\mathbf{T}^{*}[{-}1]\boldsymbol{Y}})[1] \circ \theta_{\boldsymbol{\tau}}^{{\vee}}[1]= \theta_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}}}^{{\vee}} [1], \\ \zeta_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}}} \circ \theta_{\tilde{\boldsymbol{f}}} \circ ({\cdot} \tilde{\boldsymbol{f}}^{*} \omega_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}})[1] = \tilde{\boldsymbol{f}}^{*} \theta_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}}}^{{\vee}} [1], \end{gather*}

but these are consequences of [Reference CalaqueCal19, Remark 2.5].

Now consider the following diagram of line bundles on $\boldsymbol {f}^{*} \mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}^{\mathrm {red}}$, in which we omit the pullback functors $\boldsymbol {\tau }^{*}$, $\boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}, \boldsymbol {f}}^{*}$, and $(\boldsymbol {f} \circ \boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}, \boldsymbol {f}} )^{*}$ to simplify the notation.

(4.19)

Here $\hat {\eta }, \hat {\chi }$ are defined in Lemma 2.14. The commutativity of diagram (4.17), equality (4.18), and [Reference Knudsen and MumfordKM76, Theorem 1] implies the commutativity of the upper square. By applying Propositions A.1 and A.4 we see that the lower square also commutes. Next consider the following commutative diagram in $\mathrm {Perf}(\boldsymbol {f}^{*} \mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}})$.

(4.20)

The upper vertical arrows are identified with a part of the natural morphism of distinguished triangles

\[ \tilde{\boldsymbol{f}}^{*} \Delta_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}}} \to \Delta_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}}} \]

by the natural isomorphisms

\[ \tilde{\boldsymbol{f}}^{*} \mathbb{L}_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}}} \cong \tilde{\boldsymbol{f}}^{*} \boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}}^{*} \mathbb{T}_{\boldsymbol{\mathfrak{Y}}}[1],\quad \mathbb{L}_{\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}}} \cong \tilde{\boldsymbol{f}}^{*} \boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}}^{*} \mathbb{T}_{\boldsymbol{\mathfrak{Y}}}[1], \]

and the left horizontal arrows are identified with a part of the natural morphism

\[ (\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}})^{*} \Delta_{\boldsymbol{f}} \to \Delta_{\tilde{\boldsymbol{f}}} \]

by the natural isomorphism

\[ (\boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{f}})^{*} \mathbb{L}_{\boldsymbol{\mathfrak{Y}}} \cong \tilde{\boldsymbol{f}}^{*} \boldsymbol{\pi}_{\boldsymbol{\mathfrak{Y}}}^{*} \mathbb{L}_{\boldsymbol{\mathfrak{Y}}}. \]

Now consider the following diagram of line bundles on $\boldsymbol {f}^{*} \mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}^{\mathrm {red}}$, in which we omit pullback functors as previous.

(4.21)

The commutativity of diagram (

4.20

) and [Reference Knudsen and MumfordKM76, Theorem 1] implies the commutativity of the upper square, and the commutativity of the lower square is obvious. By combining the commutativity of the diagrams (

4.19

) and (

4.21

), we obtain the commutativity of diagram (4.16) (the sign $(-1)^{{\operatorname {vdim} \boldsymbol {f} \cdot (\operatorname {vdim} \boldsymbol {f} - 1) / 2}}$ appears due to the difference of the maps (2.5) and (A.2)).

Remark 4.11 In the situation of the proposition above, assume further that there exists a smooth morphism $\boldsymbol {q} \colon \boldsymbol {Y}' \to \boldsymbol {Y}$, and write $\tilde {q} \colon t_0(\mathbf {T}^{*}[-1]\boldsymbol {Y}') \to t_0(\mathbf {T}^{*}[-1]\boldsymbol {Y})$ for the base change of $q = t_0(\boldsymbol {q})$. Then it is clear that the composition

is equal to $\Xi _{\boldsymbol {f} \circ \boldsymbol {q}}$.

4.3 Dimensional reduction for Artin stacks

We first recall the definition of the vanishing cycle complexes associated with d-critical stacks. To do this, we discuss the functorial behavior of the vanishing cycle complexes associated with d-critical schemes with respect to smooth morphisms.

Proposition 4.12 [Reference Ben-Bassat, Brav, Bussi and JoyceBBBJ15, Proposition 4.5]

Let $(Y, s, o)$ be an oriented d-critical scheme, and $q \colon X \to Y$ be a smooth morphism. Then there exists a natural isomorphism

\[ \Theta_q = \Theta_{q, s, o} \colon \varphi^{p} _{X, q^{{\star}} s, q^{{\star}} o} \cong q^{*} \varphi^{p} _{Y, s, o}[\dim q] \]

characterized by the following property: for a d-critical chart $\mathscr {R} = (R, U, f, i)$ of $(X, s)$, a d-critical chart $\mathscr {S} = (S, V, g, j)$ of $(Y, s)$ such that $q(R) \subset S$, and a smooth morphism $\tilde {q} \colon U \to V$ such that $f = g \circ \tilde {q}$ and $j \circ q = \tilde {q} \circ i$, the following diagram commutes:

where $\Theta _{\tilde {q}, f}$ is defined in Proposition 2.16(iii), and $\rho _q$ is defined by using the natural isomorphism $K_U \cong \tilde {q}^{*} K_V \otimes \mathop {\mathrm {det}}\nolimits (\Omega _{U/V})$.

Theorem 4.13 [Reference Ben-Bassat, Brav, Bussi and JoyceBBBJ15, Theorem 4.8]

Let $(\mathfrak {X}, s, o)$ be an oriented d-critical stack. Then there exists a natural perverse sheaf $\varphi _{\mathfrak {X}, s, o}$ with the following property: for each $(u \colon U \to \mathfrak {X}) \in \mathop {\text {Lis-An}} \nolimits (\mathfrak {X})$ there exists an isomorphism

\[ \Theta_u = \Theta_{u, s, o} \colon \varphi^{p}_{U, u^{{\star}} s, u^{{\star}} o} \cong u^{*} \varphi^{p}_{\mathfrak{X}, s, o}[\dim u] \]

satisfying $\Theta _{u, s, o} = q^{*}\Theta _{v, s, o}[\dim q] \circ \Theta _{q, v^{\star } s, v^{\star } o}$ for any smooth morphism $q \colon (u\colon U \to \mathfrak {X}) \to (v\colon V \to \mathfrak {X})$ in $\mathop {\text {Lis-An}} \nolimits (\mathfrak {X})$. Here we identify $q^{\star } v^{\star } o$ and $u^{\star } o$ by using (4.9).

Let $(\mathfrak {X}, s)$ be a d-critical stack, and $\Xi \colon o_1 \cong o_2$ be an isomorphism between orientations on $(\mathfrak {X},s)$. We write

\[ a_{\Xi} \colon \varphi_{\mathfrak{X}, s, o_1} \cong \varphi_{\mathfrak{X}, s, o_2} \]

for the isomorphism induced by $\Xi$.

We now state our main theorem.

Theorem 4.14 Let $\boldsymbol {\mathfrak {Y}}$ be a quasi-smooth derived Artin stack, and equip $\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}$ with the canonical $-1$-shifted symplectic structure and the canonical orientation. Then we have a natural isomorphism

\[ \bar{\gamma}_{\boldsymbol{\mathfrak{Y}}} \colon ({\pi_{\boldsymbol{\mathfrak{Y}}}})_! \varphi^{p}_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}} \cong \mathbb{Q}_{\mathfrak{Y}}[\operatorname{vdim} \boldsymbol{\mathfrak{Y}}], \]

where we write $\mathfrak {Y} = t_0(\boldsymbol {\mathfrak {Y}})$ and ${\pi _{\boldsymbol {\mathfrak {Y}}}} = t_0(\boldsymbol {\pi }_{\boldsymbol {\mathfrak {Y}}})$.

Proof. Take a smooth surjective morphism $\boldsymbol {v} \colon \boldsymbol {V} \to \boldsymbol {\mathfrak {Y}}$ and an étale morphism $\boldsymbol {\eta } \colon \boldsymbol {U} \to \boldsymbol {V} \times _{\boldsymbol {\mathfrak {Y}}} \boldsymbol {V}$ where $\boldsymbol {V}$ and $\boldsymbol {U}$ are derived schemes. $\boldsymbol {q}_1, \boldsymbol {q}_2 \colon \boldsymbol {U} \to \boldsymbol {V}$ denote the composition of $\boldsymbol {\eta }$ and the first and second projections, respectively. Write $U = t_0(\boldsymbol {U})$, $\tilde {U} = t_0(\mathbf {T}^{*}[-1]\boldsymbol {U})$, $V = t_0(\boldsymbol {V})$, $\tilde {V} = t_0(\mathbf {T}^{*}[-1]\boldsymbol {V})$, $\widetilde {\mathfrak {Y}} = t_0(\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}})$, $v = t_0(\boldsymbol {v})$, and $q_i = t_0(\boldsymbol {q}_i)$ for $i =1, 2$. Denote by $\pi _{\mathfrak {Y}} \colon \widetilde {\mathfrak {Y}} \to \mathfrak {Y}$, $\pi _{U} \colon \tilde {U} \to U$, and $\pi _{V} \colon \tilde {V} \to V$ the projections, and by $\tilde {v} \colon \tilde {V} \to \widetilde {\mathfrak {Y}}$ (respectively, $\tilde {q}_i \colon \tilde {U} \to \tilde {V}$) the base change of $v$ (respectively, $q_i$). Denote by $s$ the d-critical structure on $\widetilde {\mathfrak {Y}}$ associated with the canonical $-1$-shifted symplectic structure $\omega _{\mathbf {T}^{*}[-1]\boldsymbol {\mathfrak {Y}}}$.

Define

\[ \bar{\gamma}_{\boldsymbol{\mathfrak{Y}}, \boldsymbol{v}} \colon v^{*}(\pi_{\mathfrak{Y}})_! \varphi^{p}_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}} \cong v^{*} \mathbb{Q}_{\mathfrak{Y}}[\operatorname{vdim} \boldsymbol{\mathfrak{Y}}] \]

by the following composition:

\begin{align*} v^{*}({\pi_{\mathfrak{Y}}})_! \varphi^{p}_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}} &\cong (\pi_{{V}})_! \tilde{v}^{*} \varphi^{p}_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}} \\ &\cong (\pi_{{V}})_! \varphi^{p}_{\mathbf{T}^{*}[{-}1]\boldsymbol{V}}[-\dim v] \\ &\cong v^{*} \mathbb{Q}_{\mathfrak{Y}}[\operatorname{vdim} \boldsymbol{\mathfrak{Y}}], \end{align*}

where the first isomorphism is the proper base-change map, the second isomorphism is $(\pi _{V})_! ( a_{\Xi _{\boldsymbol {v}}} \circ \Theta _{\tilde {v}}^{-1}) [-\dim v]$, and the third isomorphism is $\bar {\gamma }_{\boldsymbol {V}} [-\dim v]$. By the sheaf property, it is enough to prove the commutativity of the diagram

(4.22)

where $\xi \colon v \circ q_1 \Rightarrow v \circ q_2$ is the natural $2$-morphism. We define

\[ \bar{\gamma}_{\boldsymbol{\mathfrak{Y}}, {\boldsymbol{v}} \circ \boldsymbol{q}_i} \colon (v \circ q_i)^{*}(\pi_{{\mathfrak{Y}}})_! \varphi^{p}_{\mathbf{T}^{*}[{-}1]\boldsymbol{\mathfrak{Y}}} \cong (v \circ q_i)^{*} \mathbb{Q}_{\mathfrak{Y}}[\operatorname{vdim} \boldsymbol{\mathfrak{Y}}] \]

for $i =1, 2$ in the same manner as $\bar {\gamma }_{\boldsymbol {\mathfrak {Y}}, \boldsymbol {v}}$. The commutativity of diagram (4.3) implies the commutativity of the following diagram.

Therefore the commutativity of diagram (

4.22

) follows once we prove the commutativity of the diagram

(4.23)

for each $i = 1, 2$. We drop $i$ from the notation, and write $q = q_i$ and $\tilde {q} = \tilde {q}_i$. By Remark 4.11, the following diagram commutes.

Using this and the commutativity of diagram (4.3) again, the commutativity of diagram (

4.23

) is implied by the commutativity of the diagram

(4.24)

where $\mathfrak {pbc}_q$ denotes the base-change map. Arguing as in the proof of [Reference Ben-Bassat, Brav, Bussi and JoyceBBBJ15, Theorem 2.9] and by shrinking if necessary, we may assume that there exist a smooth morphism $\hbox{$F \colon M \to N$}$ with a constant relative dimension between smooth schemes, a vector bundle $E$ of rank $r$ on $N$, and its section $e \in \Gamma (N, E)$ such that $\boldsymbol {V} = \boldsymbol {Z}(e)$, $\boldsymbol {U} = \boldsymbol {Z}(F^{*} e)$, and $\boldsymbol {q} \colon \boldsymbol {Z}(F^{*} e) \to \boldsymbol {Z}(e)$ is the base change of $F$. Write $\tilde {F} \colon \mathop {\mathrm {Tot}}\nolimits _M(F^{*}E^{\vee }) \to \mathop {\mathrm {Tot}}\nolimits _N(E^{\vee })$ for the base change of $F$, and denote by $\bar {e} \colon \mathop {\mathrm {Tot}}\nolimits _N(E^{\vee }) \to \mathbb {A}^{1}$ the regular function corresponding to $e$. Then

\[ \mathscr{U} = (\tilde{U}, \mathop{\mathrm{Tot}}\nolimits_M(F^{*} E^{{\vee}}), \bar{e} \circ \tilde{F}, i),\quad \mathscr{V} = (\tilde{V}, \mathop{\mathrm{Tot}}\nolimits_N(E^{{\vee}}), \bar{e}, j) \]

define d-critical charts on $\tilde {U}$ and $\tilde {V}$ respectively, where $i$ and $j$ denote the natural embeddings. Consider the composition

\[ \rho_{\tilde{q}}' \colon Q_{\mathscr{U}}^{o^{}_{\mathbf{T}^{*}[{-}1]\boldsymbol{U}}} \cong Q_{\mathscr{U}}^{q^{{\star}} o^{}_{ \mathbf{T}^{*}[{-}1]\boldsymbol{V}}} \xrightarrow{\rho_{\tilde{q}}} \tilde{q}^{*} Q_{\mathscr{V}}^{o^{}_{\mathbf{T}^{*}[{-}1]\boldsymbol{V}}}, \]

where the first map is induced by $\Xi _{\boldsymbol {q}}$. Recall that we have chosen trivializations of $Q_{\mathscr {U}}^{o^{}_{\mathbf {T}^{*}[-1]\boldsymbol {U}}}$ and $Q_{\mathscr {V}}^{o^{}_{\mathbf {T}^{*}[-1]\boldsymbol {V}}}$ in (2.25). Since we have

\[ \epsilon_{\dim N, r} / \epsilon_{\dim M, r} = \sqrt{-1}^{{\operatorname{vdim} \boldsymbol{q} \cdot (\operatorname{vdim} \boldsymbol{q} - 1) /2} + \operatorname{vdim}{\boldsymbol{V}} \cdot \operatorname{vdim}{\boldsymbol{q}}}, \]

these trivializations are identified by $\rho _{\tilde {q}}'$. This shows the commutativity of the diagram

where the two $\text {triv}$ in the right horizontal arrows denote the trivialization as above. Then the commutativity of diagram (

4.24

) follows from the commutativity of diagram (2.9).

5.1 Cohomological Donaldson–Thomas theory for local surfaces

Let $S$ be a smooth quasi-projective surface and denote by $p \colon X = \mathop {\mathrm {Tot}}\nolimits _S(\omega _S) \to S$ the projection from the total space of the canonical bundle. Denote by $\boldsymbol {\mathfrak {M}}_{S}$ (respectively, $\boldsymbol {\mathfrak {M}}_{X}$) the derived moduli stack of coherent sheaves on $S$ (respectively, $X$) with proper supports, and by $\boldsymbol {\pi }_{p} \colon \boldsymbol {\mathfrak {M}}_{X} \to \boldsymbol {\mathfrak {M}}_{S}$ the projection defined by $p_*$. By applying the main theorem of [Reference Brav and DyckerhoffBD19], $\boldsymbol {\mathfrak {M}}_{X}$ carries a canonical $-1$-shifted symplectic structure $\omega _{\boldsymbol {\mathfrak {M}}_{S}}$.

Theorem 5.1 There exists an equivalence of $-1$-shifted symplectic derived Artin stacks

(5.1)\begin{equation} \boldsymbol{\Psi} \colon (\boldsymbol{\mathfrak{M}}_{X}, \omega_{\boldsymbol{\mathfrak{M}}_{X}}) \simeq (\mathbf{T}^{*}[{-}1] \boldsymbol{\mathfrak{M}}_{S}, \omega_{\mathbf{T}^{*}[{-}1] \boldsymbol{\mathfrak{M}}_{S}}) \end{equation}

such that $\boldsymbol {\pi }_{p} \simeq \boldsymbol {\pi }_{\boldsymbol {\mathfrak {M}}_{S}} \circ \Phi$.