2 Motivic integration and stringy motifs

In this section, we review the theories of motivic integration over ordinary (untwisted) arcs and stringy invariants, mainly developed in [Reference KontsevichKon95, Reference Denef and LoeserDL99, Reference BatyrevBat98, Reference BatyrevBat99, Reference Denef and LoeserDL02, Reference SebagSeb04].

2.1 Centered log varieties

We call an integral $D$-scheme $X$ a $D$-variety if $X$ is flat, separated and of finite type over $D$, and $X$ is smooth over $D$ at the generic point of $X$. For a $D$-variety $X$, we denote the smooth locus of $X$ by $X_{\text{sm}}$ and the regular locus by $X_{\text{reg}}$.

Let $X$ be a normal $D$-variety. We can define the canonical sheaf ${\it\omega}_{X}={\it\omega}_{X/D}$ of $X$ over $D$ as in [Reference KollárKol13, page 8]. On $X_{\text{sm}}$, the canonical sheaf is isomorphic to $\wedge ^{d}{\rm\Omega}_{X/D}$, with $d$ the relative dimension of $X$ over $D$. Therefore, we can think of ${\it\omega}_{X}$ as a subsheaf of $(\wedge ^{d}{\rm\Omega}_{X/D})\otimes K(X)$. We define the canonical divisor of $X$, denoted by $K_{X}=K_{X/D}$, to be the linear equivalence class of Weil divisors corresponding to ${\it\omega}_{X}$.

A log $D$-variety is a pair $(X,{\rm\Delta})$ of a normal $D$-variety $X$ and a Weil $\mathbb{Q}$-divisor ${\rm\Delta}$ such that $K_{X}+{\rm\Delta}$ is $\mathbb{Q}$-Cartier. We call ${\rm\Delta}$ the boundary of the log variety. The canonical divisor of a log $D$-variety $(X,{\rm\Delta})$ is $K_{(X,{\rm\Delta})}:=K_{X}+{\rm\Delta}$.

A centered log $D$-variety is a triple $\mathfrak{X}=(X,{\rm\Delta},W)$ such that $(X,{\rm\Delta})$ is a log $D$-variety and $W$ is a closed subset of $X_{0}$, where $X_{0}$ denotes the special fiber of the structure morphism $X\rightarrow D$ with the reduced structure. We call $W$ the center of $\mathfrak{X}$. We also say that $\mathfrak{X}$ is a centered log structure on $X$. For a centered log variety $\mathfrak{X}=(X,{\rm\Delta},W)$, we define a canonical divisor of $\mathfrak{X}$ as the one of $(X,{\rm\Delta})$:

$$\begin{eqnarray}K_{\mathfrak{X}}:=K_{(X,{\rm\Delta})}=K_{X}+{\rm\Delta}.\end{eqnarray}$$

Sometimes, we identify a normal $\mathbb{Q}$-Gorenstein ($K_{X}$ is $\mathbb{Q}$-Cartier) $D$-variety $X$ with the log $D$-variety $(X,0)$, and identify a log $D$-variety $(X,{\rm\Delta})$ with the centered log $D$-variety $(X,{\rm\Delta},X_{0})$:

(2.1)$$\begin{eqnarray}\begin{array}{@{}c@{}}\end{array}\end{eqnarray}$$

2.2 Crepant morphisms

For centered log $D$-varieties $\mathfrak{X}=(X,{\rm\Delta},W)$ and $\mathfrak{X}^{\prime }=(X^{\prime },{\rm\Delta}^{\prime },W^{\prime })$, a morphism $f:\mathfrak{X}\rightarrow \mathfrak{X}^{\prime }$ is just a morphism $f:X\rightarrow X^{\prime }$ of the underlying varieties with $f(W)\subset W^{\prime }$. We say that a morphism $\mathfrak{X}\rightarrow \mathfrak{X}^{\prime }$ is proper or birational if it is so as the morphism $f:X\rightarrow X^{\prime }$ of the underlying varieties. We say that a morphism $f:\mathfrak{X}\rightarrow \mathfrak{X}^{\prime }$ is crepant if

$$\begin{eqnarray}f^{-1}(W^{\prime })=W\quad \text{and}\quad K_{\mathfrak{ X}}=f^{\ast }K_{\mathfrak{ X}^{\prime }}.\end{eqnarray}$$

The right equality should be understood as that for $r\in \mathbb{Z}_{{>}0}$ such that $r(K_{X}+{\rm\Delta})$ and $r(K_{X^{\prime }}+{\rm\Delta}^{\prime })$ are Cartier, we have a natural isomorphism

$$\begin{eqnarray}{\it\omega}_{X^{\prime }}^{[r]}(r{\rm\Delta}^{\prime })\cong f^{\ast }{\it\omega}_{X}^{[r]}(r{\rm\Delta}).\end{eqnarray}$$

Here, ${\it\omega}_{X}^{[r]}(r{\rm\Delta})$ is the invertible sheaf, which is identical to ${\it\omega}_{X}^{\otimes r}(r{\rm\Delta})$ on $X_{\text{reg}}$. We adopt this convention throughout the paper.

Given a generically étale morphism $f:X\rightarrow X^{\prime }$ of normal $D$-varieties, a centered log structure $\mathfrak{X}^{\prime }$ on $X^{\prime }$ induces a unique centered log structure $\mathfrak{X}$ on $X$ such that the morphism $f:\mathfrak{X}\rightarrow \mathfrak{X}^{\prime }$ is crepant. Conversely, if $f:X\rightarrow X^{\prime }$ is additionally proper, then for each centered log structure $\mathfrak{X}$ on $X$, there exists at most one centered log structure on $\mathfrak{X}^{\prime }$ such that $f:\mathfrak{X}\rightarrow \mathfrak{X}^{\prime }$ is crepant.

Remark 2.1. For our purposes, we may slightly weaken the assumptions in the definition of crepant morphisms. For instance, concerning the equality $f^{-1}(W^{\prime })=W$, we only need this equality outside $X_{\text{sm}}\setminus X_{\text{reg}}$. This is because the locus $X_{\text{sm}}\setminus X_{\text{reg}}$ does not contribute to stringy motifs at all, which are defined below. However, for simplicity, we will cling to our definition as above.

2.3 Motivic integration

Let $\mathfrak{X}=(X,{\rm\Delta},W)$ be a centered log $D$-variety. An arc of $\mathfrak{X}$ is a $D$-morphism $D\rightarrow X$ sending the closed point of $D$ into $W$. The arc space of $\mathfrak{X}$, denoted $J_{\infty }\mathfrak{X}$, is a $k$-scheme parameterizing the arcs of $\mathfrak{X}$. We put $D_{n}:=\text{Spec}\,{\mathcal{O}}_{D}/\mathfrak{m}_{D}^{n+1}$, with $\mathfrak{m}_{D}$ the maximal ideal of ${\mathcal{O}}_{D}$. An $n$-jet of $\mathfrak{X}$ is a $D$-morphism $D_{n}\rightarrow X$ sending the unique point of $D_{n}$ into $W$. For each $n$, there exists a $k$-scheme $J_{n}\mathfrak{X}$ parametrizing $n$-jets of $\mathfrak{X}$. For $n\geqslant m$, we have natural morphisms $J_{n}\mathfrak{X}\rightarrow J_{m}\mathfrak{X}$, and the arc space $J_{\infty }\mathfrak{X}$ is identified with the projective limit of $J_{n}\mathfrak{X}$, $n\geqslant 0$ with respect to these maps. We have the induced maps

$$\begin{eqnarray}{\it\pi}_{n}:J_{\infty }\mathfrak{X}\rightarrow J_{n}\mathfrak{X}.\end{eqnarray}$$

For $n<\infty$, $J_{n}\mathfrak{X}$ are of finite type over $k$. For a morphism $f:\mathfrak{Y}\rightarrow \mathfrak{X}$ and each $n\in \mathbb{Z}_{{\geqslant}0}\cup \{\infty \}$, there exists a natural map

$$\begin{eqnarray}f_{n}:J_{n}\mathfrak{Y}\rightarrow J_{n}\mathfrak{X}.\end{eqnarray}$$

The arc space $J_{\infty }\mathfrak{X}$ has the so-called motivic measure, denoted by ${\it\mu}_{J_{\infty }\mathfrak{X}}$. The measure takes values in some (semi-)ring, say ${\mathcal{R}}$, which is often a suitable modification of the Grothendieck (semi-)ring of $k$-varieties. In this paper, we fix ${\mathcal{R}}$ satisfying the following properties. Denoting by $[T]$ the class of a $k$-variety $T$ in ${\mathcal{R}}$,

• ∙ for a bijective morphism $S\rightarrow T$, we have $[S]=[T]$ in ${\mathcal{R}}$;

• ∙ putting $\mathbb{L}:=[\mathbb{A}_{k}^{1}]$, we have all fractional powers $\mathbb{L}^{a}$, $a\in \mathbb{Q}$ in ${\mathcal{R}}$;

• ∙ an infinite series $\sum _{i=1}^{\infty }[T_{i}]\mathbb{L}^{a_{i}}$ with $\lim _{i\rightarrow \infty }\dim T_{i}+a_{i}=-\infty$ converges;

• ∙ for a morphism $f:S\rightarrow T$, and for $n\in \mathbb{Z}_{{\geqslant}0}$, if every fiber of $f$ admits a homeomorphism from or to the quotient $\mathbb{A}_{k}^{n}/G$ for some linear action of a finite group $G$ on $\mathbb{A}_{k}^{n}$, then $[S]=[T]\mathbb{L}^{n}$.

One possible choice is the field of Puiseux series in $t^{-1}$,

$$\begin{eqnarray}{\mathcal{R}}:=\mathop{\bigcup }_{r=1}^{\infty }\mathbb{Z}((t^{-1/r})),\end{eqnarray}$$

where we put $[T]$ to be the Poincaré polynomial as in [Reference NicaiseNic11].

A subset $A\subset J_{\infty }\mathfrak{X}$ is called stable if there exists $n\in \mathbb{Z}_{{\geqslant}0}$ such that ${\it\pi}_{n}(A)\subset J_{n}\mathfrak{X}$ is a constructible subset and $A={\it\pi}_{n}^{-1}{\it\pi}_{n}(A)$, and for every $m\geqslant n$, every fiber of the map ${\it\pi}_{n+1}(A)\rightarrow {\it\pi}_{n}(A)$ is homeomorphic to $\mathbb{A}_{k}^{n}$. The measure of a stable subset $A$ is given by

$$\begin{eqnarray}{\it\mu}_{J_{\infty }\mathfrak{X}}(A):=[{\it\pi}_{n}(A)]\mathbb{L}^{-nd}\quad (n\gg 0).\end{eqnarray}$$

More generally, we can define the measure for measurable subsets, which are roughly the limits of stable subsets.

Let ${\rm\Phi}:C\rightarrow {\mathcal{R}}\cup \{\infty \}$ be a measurable function on a subset $C\subset J_{\infty }\mathfrak{X}$. That is, the image of ${\rm\Phi}$ is countable, all fibers ${\rm\Phi}^{-1}(a)$ are measurable and ${\it\mu}_{J_{\infty }\mathfrak{X}}({\rm\Phi}^{-1}(\infty ))=0$. We define

$$\begin{eqnarray}\int _{C}{\rm\Phi}\,{\it\mu}_{J_{\infty }\mathfrak{X}}:=\mathop{\sum }_{a\in {\mathcal{R}}}{\it\mu}_{J_{\infty }\mathfrak{X}}({\rm\Phi}^{-1}(a))\cdot a\in {\mathcal{R}}\cup \{\infty \}.\end{eqnarray}$$

2.4 Stringy invariants

We still suppose that $\mathfrak{X}=(X,{\rm\Delta},W)$ is a centered log $D$-variety.

Definition 2.2. To a coherent ideal sheaf $I\neq 0$ on $X$ defining a closed subscheme $Z\subsetneq X$, we associate the order function,

$$\begin{eqnarray}\text{ord}\,I=\text{ord}\,Z:J_{\infty }\mathfrak{X}\rightarrow \mathbb{Z}_{{\geqslant}0}\cup \{\infty \},\end{eqnarray}$$

as follows. For an arc ${\it\gamma}:D\rightarrow X$, the pullback ${\it\gamma}^{-1}I$ of $I$ is an ideal of ${\mathcal{O}}_{D}$ and of the form $\mathfrak{m}_{D}^{l}$ for some $l\in \mathbb{Z}_{{\geqslant}0}\cup \{\infty \}$, where we put $(0):=\mathfrak{m}_{D}^{\infty }$ by convention. For a fractional ideal $I$ (that is, a coherent ${\mathcal{O}}_{X}$-submodule of $K(X)$), if we write $I=I_{+}\cdot I_{-}^{-1}$ for ideal sheaves $I_{+}$ and $I_{-}$ with $I_{-}$ locally principal, then we put

$$\begin{eqnarray}\text{ord}\,I:=\text{ord}\,I_{+}-\text{ord}\,I_{-}.\end{eqnarray}$$

Here, we put $\text{ord}\,I=\infty$ if either $\text{ord}\,I_{+}=\infty$ or $\text{ord}\,I_{-}=\infty$. Similarly, for a $\mathbb{Q}$-linear combination $Z=\sum _{i=1}^{n}a_{i}Z_{i}$ of closed subschemes $Z_{i}\subsetneq X$, we define

$$\begin{eqnarray}\text{ord}\,Z:=\mathop{\sum }_{i=1}^{n}a_{i}\cdot \text{ord}\,Z_{i},\end{eqnarray}$$

taking values in $\mathbb{Q}\cup \{\infty \}$.

Remark 2.3. For a closed subscheme $Z\subsetneq X$, we expect that $(\text{ord}\,Z)^{-1}(\infty )$ has measure zero. The author does not know whether this has been proved, but this follows from the change of variables formula, if there exists a resolution of singularities $f:\tilde{X}\rightarrow X$ so that $\tilde{X}$ is regular and $\tilde{X}_{0}\cup f^{-1}(Z)$ is a simple normal crossing divisor. If the expectation is actually true, then order functions for fractional ideals and $\mathbb{Q}$-linear combination of closed subschemes are well-defined modulo measure zero subsets.

Let $r\in \mathbb{Z}_{{>}0}$ be such that $rK_{\mathfrak{X}}$ is Cartier. Since the sheaf ${\mathcal{O}}_{X}(rK_{\mathfrak{X}})={\it\omega}_{X}^{[r]}(r{\rm\Delta})$ is invertible and thought of as a subsheaf of the constant sheaf $(\wedge ^{d}{\rm\Omega}_{X/D})^{\otimes r}\otimes K(X)$, we can define a fractional ideal sheaf $I_{\mathfrak{X}}^{r}$ by the equality of subsheaves of $(\wedge ^{d}{\rm\Omega}_{X/D})^{\otimes r}\otimes K(X)$,

$$\begin{eqnarray}\left.\left(\mathop{d}^{\wedge }{\rm\Omega}_{X/D}\right)^{\otimes r}\!\right/\!\text{tors}=I_{\mathfrak{ X}}^{r}\cdot {\mathcal{O}}_{X}(rK_{\mathfrak{X}}).\end{eqnarray}$$

We then put a function $\mathbf{f}_{\mathfrak{X}}$ on $J_{\infty }\mathfrak{X}$ by

$$\begin{eqnarray}\mathbf{f}_{\mathfrak{X}}:=\frac{1}{r}\text{ord}\,I_{\mathfrak{X}}^{r}.\end{eqnarray}$$

Since $(I_{\mathfrak{X}}^{r})^{n}=I_{\mathfrak{X}}^{r\cdot n}$, the function $\mathbf{f}_{\mathfrak{X}}$ is independent of the choice of $r$. $$If $X$ is smooth, then we simply have $\mathbf{f}_{\mathfrak{X}}=\text{ord}\,{\rm\Delta}$.

Definition 2.4. The stringy motif of $\mathfrak{X}$ is defined to be

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{X}):=\int _{J_{\infty }\mathfrak{X}}\mathbb{L}^{\mathbf{f}_{\mathfrak{X}}}\,d{\it\mu}_{J_{\infty }\mathfrak{X}}.\end{eqnarray}$$

We also write $M_{\text{st}}(\mathfrak{X})=M_{\text{st}}(X,{\rm\Delta})_{W}$, and sometimes omit ${\rm\Delta}$ if ${\rm\Delta}=0$, and $W$ if $W=X_{0}$. When the integral above converges, we call $\mathfrak{X}$stringily log terminal. When it diverges, we put $M_{\text{st}}(\mathfrak{X}):=\infty$.

Conjecture 2.5. If a morphism $f:\mathfrak{X}\rightarrow \mathfrak{X}^{\prime }$ of centered log $D$-varieties is proper, birational and crepant, then

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{X})=M_{\text{st}}(\mathfrak{X}^{\prime }).\end{eqnarray}$$

Proposition 2.6. Conjecture 2.5 holds if there exists a proper birational morphism $Y\rightarrow X$ of $D$-varieties such that $Y\otimes _{{\mathcal{O}}_{D}}K(D)$ is smooth over $K(D)$. In particular, Conjecture 2.5 holds if $K(D)$ has characteristic zero.

Proof. Let $X_{{\it\eta}}$ be the generic fiber of $X\rightarrow D$. From the Hironaka theorem, there exists a coherent ideal sheaf $I_{{\it\eta}}\subset {\mathcal{O}}_{X_{{\it\eta}}}$ such that the blowup of $X_{{\it\eta}}$ along $I_{{\it\eta}}$ is smooth over $K(D)$. Let $I\subset {\mathcal{O}}_{X}$ be a coherent ideal sheaf such that $I|_{X_{{\it\eta}}}=I_{{\it\eta}}$. The blowup of $X$ along $I$ has a smooth generic fiber. Hence, the second assertion of the lemma follows from the first one.

To prove the first one, we can apply the version of the change of variables formula proved by Sebag [Reference SebagSeb04, Theorem 8.0.5] (see also [Reference Nicaise and SebagNS11]). If $\mathfrak{Y}$ is the centered log structure on $Y$ such that the induced morphism $f:\mathfrak{Y}\rightarrow \mathfrak{X}$ is crepant, then the change of variables formula shows that

$$\begin{eqnarray}\int _{J_{\infty }\mathfrak{X}}\mathbb{L}^{\mathbf{f}_{\mathfrak{X}}}\,d{\it\mu}_{J_{\infty }\mathfrak{X}}=\int _{J_{\infty }\mathfrak{Y}}\mathbb{L}^{\mathbf{f}_{\mathfrak{X}}\circ f_{\infty }-\text{ord}\,\text{jac}_{f}}\,d{\it\mu}_{J_{\infty }\mathfrak{Y}}.\end{eqnarray}$$

Here, $\text{ord}\,\text{jac}_{f}$ is the function of Jacobian orders as defined in [Reference SebagSeb04, page 29]. For $r\in \mathbb{Z}_{{>}0}$ such that $rK_{\mathfrak{X}}$ and $rK_{\mathfrak{Y}}$ are Cartier, we have

$$\begin{eqnarray}\displaystyle \left.\left(f^{\ast }\left(\mathop{d}^{\wedge }{\rm\Omega}_{X/D}\right)^{\otimes r}\right)\!\right/\!\text{tors} & = & \displaystyle f^{-1}I_{\mathfrak{X}}^{r}\cdot {\mathcal{O}}_{\mathfrak{Y}}(rK_{\mathfrak{Y}})\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle \left.\left(\mathop{d}^{\wedge }{\rm\Omega}_{Y/D}\right)^{\otimes r}\!\right/\!\text{tors} & = & \displaystyle I_{\mathfrak{Y}}^{r}\cdot {\mathcal{O}}_{\mathfrak{Y}}(rK_{\mathfrak{Y}}).\nonumber\end{eqnarray}$$

This shows that

$$\begin{eqnarray}\mathbf{f}_{\mathfrak{X}}\circ f_{\infty }-\text{ord}\,\text{jac}_{f}=\mathbf{f}_{\mathfrak{Y}}.\end{eqnarray}$$

We obtain $M_{\text{st}}(\mathfrak{X})=M_{\text{st}}(\mathfrak{Y})$, and similarly $M_{\text{st}}(\mathfrak{X}^{\prime })=M_{\text{st}}(\mathfrak{Y})$. We have proved the proposition.◻

Proposition 2.7. Let $\mathfrak{X}=(X,{\rm\Delta},W)$ be a centered log $D$-variety and write

$$\begin{eqnarray}{\rm\Delta}=\mathop{\sum }_{h=1}^{l}a_{h}A_{h}+\mathop{\sum }_{i=1}^{m}b_{i}B_{i}+\mathop{\sum }_{j=1}^{n}c_{j}C_{j}\quad (a_{h},b_{i}\in \mathbb{Q},\,c_{j}\in \mathbb{Q}\setminus \{0\})\end{eqnarray}$$

such that $A_{h}$ are the irreducible components of the closure of $X_{0}\cap X_{\text{sm}}$, $B_{i}$ are the irreducible components of $X_{0}\setminus X_{\text{sm}}$ and $C_{j}$ are prime divisors not contained in $X_{0}$. Let

$$\begin{eqnarray}\displaystyle A_{h}^{\circ }:=A_{h}\cap X_{\text{sm}}=A_{h}\setminus \left(\overline{X_{0}\setminus A_{h}}\right) & & \displaystyle \nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle C_{J}^{\circ }:=\mathop{\bigcap }_{j\in J}C_{j}\setminus \mathop{\bigcup }_{j\in \{1,\ldots ,n\}\setminus J}C_{j}, & & \displaystyle \nonumber\end{eqnarray}$$

with $\overline{X_{0}\setminus A_{h}}$ the closure of $X_{0}\setminus A_{h}$. We suppose that $X$ is regular and that $\bigcup _{j=1}^{n}C_{j}$ is simple normal crossing, that is, for any $J\subset \{1,\ldots ,n\},$ the scheme-theoretic intersection $\bigcap _{j\in J}C_{j}$ is smooth over $D$. Then $\mathfrak{X}$ is stringily log terminal if and only if $c_{j}<1$ for every $j$, with $C_{j}\cap W\cap X_{\text{sm}}\neq \emptyset$. Moreover, if this is the case,

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{X})=\mathop{\sum }_{h=1}^{l}\mathbb{L}^{a_{h}}\mathop{\sum }_{J\subset \{1,\ldots ,n\}}[W\cap A_{h}^{\circ }\cap C_{J}^{\circ }]\mathop{\prod }_{j\in J}\frac{\mathbb{L}-1}{\mathbb{L}^{1-c_{j}}-1}.\end{eqnarray}$$

Proof. We first note that the locus $X_{0}\setminus X_{\text{sm}}$ does not have any arc, and hence does not contribute to $M_{\text{st}}(\mathfrak{X})$. Since

$$\begin{eqnarray}X_{0}\cap X_{\text{sm}}=\mathop{\sqcup }_{h=1}^{l}A_{h}^{\circ },\end{eqnarray}$$

we can decompose $M_{\text{st}}(\mathfrak{X})$ into the sum of components corresponding to $A_{h}^{\circ }$, $h=1,\ldots ,l$. The divisor $a_{h}A_{h}$ contributes to the component corresponding to $A_{h}^{\circ }$ by the multiplication with $\mathbb{L}^{a_{h}}$. From all of these arguments, the proposition has been reduced to the formula

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{X})=\mathop{\sum }_{J\subset \{1,\ldots ,n\}}[W\cap C_{J}^{\circ }]\mathop{\prod }_{j\in J}\frac{\mathbb{L}-1}{\mathbb{L}^{1-c_{j}}-1}\end{eqnarray}$$

in the case where $X$ is smooth and ${\rm\Delta}=\sum _{j}c_{j}C_{j}$. This is the standard explicit formula (see for instance [Reference BatyrevBat98]).◻

2.5 Group actions

Definition 2.8. A centered log $G$-$D$-variety is a centered log $D$-variety $\mathfrak{X}=(X,{\rm\Delta},W)$ endowed with a faithful $G$-action on $X$ such that ${\rm\Delta}$ and $W$ are stable under the $G$-action. Given a variety $X$ with a faithful $G$-action, we say that a centered log structure $\mathfrak{X}$ on $X$ is $G$-equivariant if it is a centered log $G$-$D$-variety.

For a centered log $G$-$D$-variety $\mathfrak{X}$, the arc space $J_{\infty }\mathfrak{X}$ has a natural $G$-action. We define a motivic measure on $(J_{\infty }\mathfrak{X})/G$, denoted by ${\it\mu}_{(J_{\infty }\mathfrak{X})/G}$, in the same way as defining the motivic measure on $J_{\infty }\mathfrak{X}$, except that in the definition of stable subsets, say $A$, fibers of ${\it\pi}_{n+1}(A)\rightarrow {\it\pi}_{n}(A)$ are only assumed to be homeomorphic to the quotient $\mathbb{A}_{k}^{d}/H$ for some linear action of a finite group $H$ on $\mathbb{A}_{k}^{d}$.

The function $\mathbf{f}_{\mathfrak{X}}$ on $J_{\infty }\mathfrak{X}$ is $G$-invariant and gives a function on $(J_{\infty }\mathfrak{X})/G$, which we again denote by $\mathbf{f}_{\mathfrak{X}}$. We define

$$\begin{eqnarray}M_{\text{st},G}(\mathfrak{X}):=\int _{(J_{\infty }\mathfrak{X})/G}\mathbb{L}^{\mathbf{f}_{\mathfrak{X}}}\,d{\it\mu}_{(J_{\infty }\mathfrak{X})/G}.\end{eqnarray}$$

The reader should not confuse $M_{\text{st},G}(\mathfrak{X})$ with the orbifold stringy motif $M_{\text{st}}^{G}(\mathfrak{X})$, defined later.

Let us define a $G$-prime divisor as a divisor of the form $\sum _{i=1}^{l}D_{i}$, where $D_{i}$ are prime divisors permuted transitively by the $G$-actions.

Proposition 2.9. Let $\mathfrak{X}=(X,{\rm\Delta},W)$ be a centered log $G$-$D$-variety and write

$$\begin{eqnarray}{\rm\Delta}=\mathop{\sum }_{h=1}^{l}a_{h}A_{h}+\mathop{\sum }_{i=1}^{m}b_{i}B_{i}+\mathop{\sum }_{j=1}^{n}c_{j}C_{j}\quad (a_{h},b_{i}\in \mathbb{Q},\,c_{j}\in \mathbb{Q}\setminus \{0\})\end{eqnarray}$$

such that $A_{h}$ are the distinct $G$-prime divisors such that $\bigcup A_{h}$ is equal to the closure of $X_{0}\cap X_{\text{sm}}$, $B_{i}$ are the distinct $G$-prime divisors with $\bigcup _{i}B_{i}=X_{0}\setminus X_{\text{sm}}$ and $C_{j}$ are $G$-prime divisors not contained in $X_{0}$. We suppose that

• ∙ $X$ is regular,

• ∙ for any $J\subset \{1,\ldots ,n\},$ the scheme-theoretic intersection $\bigcap _{j\in J}C_{j}$ is smooth over $D$, and

• ∙ for every $j$ with $C_{j}\cap W\cap X_{\text{sm}}\neq \emptyset$, $c_{j}<1$.

With the same notation as in Proposition 2.7, we have

$$\begin{eqnarray}M_{\text{st},G}(\mathfrak{X})=\mathop{\sum }_{h=1}^{l}\mathbb{L}^{a_{h}}\mathop{\sum }_{J\subset \{1,\ldots ,n\}}\left[\frac{W\cap A_{h}^{\circ }\cap C_{J}^{\circ }}{G}\right]\mathop{\prod }_{j\in J}\frac{\mathbb{L}-1}{\mathbb{L}^{1-c_{j}}-1}.\end{eqnarray}$$

3 $G$-arcs

In the last subsection, we considered motivic integration over varieties endowed with finite group actions. However, we considered only ordinary (untwisted) arcs, which are not general enough to apply to the McKay correspondence. Suitably generalized arcs were introduced by Denef and Loeser [Reference Denef and LoeserDL02] in characteristic zero. The author [Reference YasudaYasa] further generalized them to arbitrary characteristics. We may use generalized arcs of orbifolds or Deligne–Mumford stacks as in [Reference Lupercio and PoddarLP04, Reference YasudaYas04, Reference YasudaYas06, Reference YasudaYasa], so that we can treat general orbifolds, having group actions only locally. We do not pursue generalization in this direction, however.

From now on, we fix a finite group $G$.

We now fix the following notation: $E$ is a $G$-cover of $D$, $F$ is a connected component of $E$ with a stabilizer $H$ so that $F$ is an $H$-cover of $D$.

Lemma 3.3. Let $\text{Aut}(E)$ be the automorphism group of $E$ as a $G$-cover of $D$. That is, it consists of $G$-equivariant $D$-automorphisms of $E$. We have a natural isomorphism

$$\begin{eqnarray}\text{Aut}(E)\cong C_{G}(H)^{\text{op}},\end{eqnarray}$$

where the right-hand side is the opposite group of the centralizer of $H$ in $G$.

Proof. If $E$ is the trivial $G$-cover $D\times G$ of $D$, then its automorphisms are nothing but the right $G$-action on $G$. Therefore, $\text{Aut}(E)=G^{\text{op}}$.

For the general case, let $E_{F}$ be the normalization of the fiber product $E\times _{D}F$. This is a trivial $G$-cover of $F$, and we have a natural injection

$$\begin{eqnarray}\text{Aut}(E)\rightarrow \text{Aut}(E_{F})=G^{\text{op}}.\end{eqnarray}$$

Its image is the automorphisms of $E_{F}$ compatible with the action of $\text{Gal}(F/D)=H$. This shows the lemma.◻

Let $V$ be a $D$-variety endowed with a faithful left $G$-action.

Obviously,

$$\begin{eqnarray}J_{\infty }^{G}V=\mathop{\sqcup }_{E\in G\text{-}\text{Cov}(D)}J_{\infty }^{G,E}V.\end{eqnarray}$$

Let $\text{Hom}_{D}^{G}(E,V)$ be the space of $G$-equivariant $D$-morphisms $E\rightarrow V$. We define a left action of $C_{G}(H)=\text{Aut}(E)^{\text{op}}$ on $\text{Hom}_{D}^{G}(E,V)$ as follows. For $a\in C_{G}(H)=\text{Aut}(E)^{\text{op}}$ and $f\in \text{Hom}_{D}^{G}(E,V)$,

(3.1)$$\begin{eqnarray}\displaystyle a\cdot (E\xleftarrow[]{f}V):=(V\xleftarrow[]{f}E\xleftarrow[]{a}E)=(V\xleftarrow[]{a}V\xleftarrow[]{f}E). & & \displaystyle\end{eqnarray}$$

By definition, we have

(3.2)$$\begin{eqnarray}J_{\infty }^{G,E}V=\text{Hom}_{D}^{G}(E,V)/C_{G}(H).\end{eqnarray}$$

For $n\in \mathbb{Z}_{{\geqslant}0}$, we put $F_{n}:=F/\mathfrak{m}_{F}^{n\cdot h+1}$, with $h:=\sharp H$, and define $E_{n}:=\bigcup _{g\in G}g(F_{n})$. In particular, $F_{0}\cong \text{Spec}\,k$, and $E_{0}$ consists of the closed points of $E$ with the reduced scheme structure.

Definition 3.5. We define an $E$-twisted $G$-n-jet of $V$ as a $G$-equivariant $D$-morphism $E_{n}\rightarrow V$, and put

$$\begin{eqnarray}\displaystyle J_{n}^{G,E}V:=\text{Hom}_{D}^{G}(E_{n},V)/C_{G}(H)\quad \text{and} & & \displaystyle \nonumber\\ \displaystyle J_{n}^{G}V=\mathop{\sqcup }_{E\in G\text{-}\text{Cov}(D)}J_{n}^{G,E}V. & & \displaystyle \nonumber\end{eqnarray}$$

Here, the $C_{G}(H)$-action on $\text{Hom}_{D}^{G}(E_{n},V)$ is similarly defined to (3.1).

Note that if $E\not \cong E^{\prime }$, then $E$-twisted and $E^{\prime }$-twisted $G$-$n$-jets never give the same point of $J_{n}^{G}V$. For each $n\in \mathbb{Z}_{{\geqslant}0}$ and $E\in G\text{-}\text{Cov}(D)$, we have natural maps

$$\begin{eqnarray}J_{\infty }^{G,E}V\rightarrow J_{n}^{G,E}V\quad \text{and}\quad J_{\infty }^{G}V\rightarrow J_{n}^{G}V,\end{eqnarray}$$

both of which we will denote by ${\it\pi}_{n}$. We have obtained the following commutative diagram:

Let $X$ be the quotient scheme $V/G$, writing the quotient morphism as

$$\begin{eqnarray}p:V\rightarrow X.\end{eqnarray}$$

Given a $G$-arc $E\rightarrow V$, we get an arc $D\rightarrow X$ by taking the $G$-quotients of the source and the target. This gives a natural map

$$\begin{eqnarray}p_{\infty }:J_{\infty }^{G}V\rightarrow J_{\infty }X.\end{eqnarray}$$

Let $T\subset V$ be the ramification locus of ${\it\pi}$, say, with the reduced scheme structure and $\bar{T}\subset X$ its image. The map $p_{\infty }$ restricts to the bijection

$$\begin{eqnarray}J_{\infty }^{G}V\setminus J_{\infty }^{G}T\rightarrow J_{\infty }X\setminus J_{\infty }\bar{T}.\end{eqnarray}$$

For $n<\infty$, we have a natural map

$$\begin{eqnarray}p_{n}:{\it\pi}_{n}(J_{\infty }^{G}V)\rightarrow J_{n}X,\end{eqnarray}$$

where ${\it\pi}_{n}$ denotes the natural map $J_{\infty }^{G}V\rightarrow J_{n}^{G}V$.

For a centered log $G$-$D$-variety $\mathfrak{V}$ and $n\in \mathbb{Z}_{{\geqslant}0}\cup \{\infty \}$, we define $J_{n}^{G}\mathfrak{V}$ and $J_{n}^{G,E}\mathfrak{V}$ as the subsets of $J_{n}^{G}V$ and $J_{n}^{G,E}V$ consisting of the morphisms $E_{n}\rightarrow V$ sending the closed points of $E_{n}$ into the center of $\mathfrak{V}$.

4 The untwisting technique revisited

In this section, we revisit the technique of untwisting, which was first used by Denef and Loeser [Reference Denef and LoeserDL02] in characteristic zero, and generalized to arbitrary characteristics by the author [Reference YasudaYasa]. Our constructions below are slightly different and refined from the ones in [Reference YasudaYasa].

Let us now turn to the case where $V$ is an affine space over $D$ and the given $G$-action is linear. We keep fixed a $G$-cover $E$ of $D$ and a connected component $F$ of $E$ with stabilizer $H$.

For a free ${\mathcal{O}}_{D}$-module $M$ of rank $d$, let ${\mathcal{O}}_{V}:=S_{{\mathcal{O}}_{D}}^{\bullet }M$ be its symmetric algebra, and put

$$\begin{eqnarray}V=\text{Spec}\,{\mathcal{O}}_{V}=\mathbb{A}_{D}^{d}.\end{eqnarray}$$

We suppose that the module $M$ and hence the ${\mathcal{O}}_{D}$-algebra ${\mathcal{O}}_{V}$ have faithful right $G$-actions. Then $V$ has the induced left $G$-action. The set $\text{Hom}_{D}^{G}(E,V)$ can be identified with the ${\mathcal{O}}_{D}$-module

$$\begin{eqnarray}\displaystyle {\rm\Xi}_{F}:=\text{Hom}_{{\mathcal{O}}_{D}}^{H}(M,{\mathcal{O}}_{F})=\text{Hom}_{{\mathcal{O}}_{D}}^{G}(M,{\mathcal{O}}_{E}). & & \displaystyle \nonumber\end{eqnarray}$$

We call ${\rm\Xi}_{F}$ the tuning module.

From (3.2),

$$\begin{eqnarray}J_{\infty }^{G,E}V={\rm\Xi}_{F}/C_{G}(H).\end{eqnarray}$$

Note that the $C_{G}(H)$-action on ${\rm\Xi}_{F}$ is ${\mathcal{O}}_{D}$-linear.

Lemma 4.3. The maps

$$\begin{eqnarray}{\it\pi}_{n+1}(J_{\infty }^{G}V)\rightarrow {\it\pi}_{n}(J_{\infty }^{G}V)\end{eqnarray}$$

have fibers homeomorphic to the quotient of $\mathbb{A}_{k}^{d}$ by a linear action of a finite group.

Proof. If we denote the map ${\rm\Xi}_{F}\rightarrow \text{Hom}_{D}^{H}(F_{n},V)$ again by ${\it\pi}_{n}$, the image ${\it\pi}_{n}({\rm\Xi}_{F})$ is isomorphic to $({\mathcal{O}}_{D}/\mathfrak{m}_{D}^{n+1})^{\oplus d}$. This shows that the fibers of

$$\begin{eqnarray}{\it\pi}_{n+1}({\rm\Xi}_{F})\rightarrow {\it\pi}_{n}({\rm\Xi}_{F})\end{eqnarray}$$

are isomorphic to $\mathbb{A}_{k}^{d}$, proving the lemma.◻

Definition 4.6. We define modules,

$$\begin{eqnarray}\displaystyle & M^{|F|}:=\text{Hom}_{{\mathcal{O}}_{D}}({\rm\Xi}_{F},{\mathcal{O}}_{D})\quad \text{and} & \displaystyle \nonumber\\ \displaystyle & M^{\left\langle F\right\rangle }:=M^{|F|}\otimes _{{\mathcal{O}}_{D}}{\mathcal{O}}_{F}=\text{Hom}_{{\mathcal{O}}_{D}}({\rm\Xi}_{F},{\mathcal{O}}_{F}), & \displaystyle \nonumber\end{eqnarray}$$

which are free modules of rank $d$ over ${\mathcal{O}}_{D}$ and ${\mathcal{O}}_{F}$ respectively. We define an ${\mathcal{O}}_{D}$-linear map

$$\begin{eqnarray}\displaystyle u^{\ast }=u_{F}^{\ast }:M & \rightarrow & \displaystyle M^{\left\langle F\right\rangle }\nonumber\\ \displaystyle m & \mapsto & \displaystyle ({\rm\Xi}\ni f\mapsto f(m)\in {\mathcal{O}}_{F}),\nonumber\end{eqnarray}$$

identifying ${\rm\Xi}_{F}$ with $\text{Hom}_{{\mathcal{O}}_{D}}^{H}(M,{\mathcal{O}}_{F})$ rather than $\text{Hom}_{{\mathcal{O}}_{D}}^{G}(M,{\mathcal{O}}_{E})$.

Note that the $H$- and $C_{G}(H)$-actions above on $M^{\left\langle F\right\rangle }$ commute.

Definition 4.8. We define the untwisting variety (resp. pre-untwisting variety) of $V$ with respect to $F$ as

$$\begin{eqnarray}V^{|F|}:=\text{Spec}\,S_{{\mathcal{O}}_{D}}^{\bullet }M^{|F|}=\mathbb{A}_{D}^{d}~(\text{resp. }V^{\left\langle F\right\rangle }:=\text{Spec}\,S_{{\mathcal{O}}_{F}}^{\bullet }M^{\left\langle F\right\rangle }=\mathbb{A}_{F}^{d}).\end{eqnarray}$$

We denote the projection $V^{\langle F\rangle }\rightarrow V^{|F|}$ by $r=r_{F}$, where $r$ stands for the restriction of scalars (see diagram (4.1) below). The map $u^{\ast }$ defines a $D$-morphism

$$\begin{eqnarray}u:V^{\left\langle F\right\rangle }\rightarrow V,\end{eqnarray}$$

which is both $H$- and $C_{G}(H)$-equivariant. We call the pair of $r$ and $u$ the untwisting correspondence of $V$ with respect to $F$.

Let $X:=V/G$, and identify ${\mathcal{O}}_{X}$ with $({\mathcal{O}}_{V})^{G}$. Since the $H$-invariant subring of ${\mathcal{O}}_{V^{\left\langle F\right\rangle }}$ is

$$\begin{eqnarray}({\mathcal{O}}_{V^{\left\langle F\right\rangle }})^{H}={\mathcal{O}}_{V^{|F|}},\end{eqnarray}$$

we have

$$\begin{eqnarray}u^{\ast }({\mathcal{O}}_{X})\subset {\mathcal{O}}_{V^{|F|}}.\end{eqnarray}$$

We denote the induced morphism $V^{|F|}\rightarrow X$ by $p^{|F|}$. We have the following commutative diagram:

(4.1)$$\begin{eqnarray}\begin{array}{@{}c@{}}\end{array}\end{eqnarray}$$

In summary, we have a one-to-one correspondence between ${\rm\Xi}_{F}$ and $J_{\infty }V^{|F|}$, induced from the untwisting correspondence. From Lemma 4.7, the correspondence is compatible with the $C_{G}(H)$-actions on both sides. Therefore, it descends to a one-to-one correspondence between $J_{\infty }^{G,E}V$ and $(J_{\infty }V^{|F|})/C_{G}(H)$. We obtain the following commutative diagram:

(4.2)$$\begin{eqnarray}\begin{array}{@{}c@{}}\end{array}\end{eqnarray}$$

For $n<\infty$, we have a similar diagram:

(4.3)$$\begin{eqnarray}\begin{array}{@{}c@{}}\end{array}\end{eqnarray}$$

Note that the arrow ${\it\beta}$ is no longer bijective. When $n=0$, the diagram is represented as

(4.4)$$\begin{eqnarray}\begin{array}{@{}c@{}}\end{array}\end{eqnarray}$$

Here, $(V_{0})^{H}$ is the fixed-point locus of the $H$-action on $V_{0}$.

5 The change of variables formula

The untwisting technique, discussed in the last section, enables us to deduce a conjectural change of variables formula for the map $p_{\infty }:J_{\infty }^{G}V\rightarrow J_{\infty }X$. In turn, it derives the McKay correspondence for linear actions in the next section.

We keep the notation from the last section.

Definition 5.1. Let $f:T\rightarrow S$ be a morphism of $D$-varieties which is generically étale. The Jacobian ideal (sheaf)

$$\begin{eqnarray}\text{Jac}_{f}=\text{Jac}_{T/S}\subset {\mathcal{O}}_{T}\end{eqnarray}$$

is defined as the zeroth Fitting ideal (sheaf) of ${\rm\Omega}_{T/S}$, the sheaf of Kähler differentials. We denote by $\mathbf{j}_{f}$ the order function of $\text{Jac}_{f}$ on $J_{\infty }T$, $(J_{\infty }T)/G$ or $J_{\infty }^{G}T$ if $T$ has a faithful action of a finite group $G$. The ambiguity of the domain will not cause confusion.

Conjecture 5.3. Let the assumption be as in Section 4. Let ${\rm\Phi}:J_{\infty }X\supset A\rightarrow {\mathcal{R}}\cup \{\infty \}$ be a measurable function, with $A\subset p_{\infty }(J_{\infty }^{G,E}V)$, and let $p_{(\infty )}^{|F|}$ be the natural map $(J_{\infty }V^{|F|})/C_{G}(H)\rightarrow J_{\infty }X$. We have

$$\begin{eqnarray}\displaystyle \int _{A}{\rm\Phi}\,d{\it\mu}_{J_{\infty }X}=\int _{(p_{(\infty )}^{|F|})^{-1}(A)}({\rm\Phi}\circ p_{(\infty )}^{|F|})\mathbb{L}^{-\mathbf{j}_{p^{|F|}}}\,d{\it\mu}_{(J_{\infty }V^{|F|})/C_{G}(H)}. & & \displaystyle \nonumber\end{eqnarray}$$

Although there is no written proof of the conjecture in this general form in the literature as far as the author knows, the conjecture is likely to be proved by using existing techniques and arguments from [Reference Denef and LoeserDL02] and [Reference SebagSeb04].

Definition 5.4. [Reference YasudaYasa]

For $E\in G\text{-}\text{Cov}(D)$ with a connected component $F$, we define the weights of $E$ and $F$ with respect to $V$ as

$$\begin{eqnarray}\displaystyle \mathbf{w}_{V}(E)=\mathbf{w}_{V}(F):=\text{codim}((V_{0})^{H},V_{0})-\boldsymbol{v}_{V}(E), & & \displaystyle \nonumber\end{eqnarray}$$

with

$$\begin{eqnarray}\displaystyle \boldsymbol{v}_{V}(E)=\boldsymbol{v}_{V}(F) & := & \displaystyle \frac{1}{\sharp G}\cdot \text{length}\frac{\text{Hom}_{{\mathcal{O}}_{D}}(M,{\mathcal{O}}_{E})}{{\mathcal{O}}_{E}\cdot {\rm\Xi}_{F}}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{\sharp H}\cdot \text{length}\frac{\text{Hom}_{{\mathcal{O}}_{D}}(M,{\mathcal{O}}_{F})}{{\mathcal{O}}_{F}\cdot {\rm\Xi}_{F}}.\nonumber\end{eqnarray}$$

For the generalization to the case where $k$ is only perfect, see [Reference Wood and YasudaWY15].

The definition above gives the weight function,

$$\begin{eqnarray}\mathbf{w}_{V}:G\text{-}\text{Cov}(D)\rightarrow \frac{1}{\sharp G}\mathbb{Z}.\end{eqnarray}$$

We will denote the composition

$$\begin{eqnarray}J_{\infty }^{G}V\rightarrow G\text{-}\text{Cov}(D)\rightarrow \frac{1}{\sharp G}\mathbb{Z}\end{eqnarray}$$

again by $\mathbf{w}_{V}$.

Definition 5.5. For an ideal $I\subset {\mathcal{O}}_{V}$ stable under the $G$-action and a $G$-arc ${\it\gamma}:E\rightarrow V$, we define a function

$$\begin{eqnarray}\text{ord}\,I:J_{\infty }^{G}V\rightarrow \frac{1}{\sharp G}\mathbb{Z}\cup \{\infty \}\end{eqnarray}$$

by

$$\begin{eqnarray}(\text{ord}\,I)({\it\gamma}):=\frac{1}{\sharp G}\text{length}\frac{{\mathcal{O}}_{E}}{{\it\gamma}^{-1}I}=\frac{1}{\sharp H}\text{length}\frac{{\mathcal{O}}_{F}}{({\it\gamma}|_{F})^{-1}I}.\end{eqnarray}$$

We then extend this to $G$-stable fractional ideals and $G$-stable $\mathbb{Q}$-linear combinations of closed subschemes as in Definition 2.2.

The conjectural change of variables formula is stated as follows.

Conjecture 5.6. [Reference YasudaYasa]

For a measurable function ${\rm\Phi}:J_{\infty }X\supset C\rightarrow {\mathcal{R}}\cup \{\infty \}$, we have

$$\begin{eqnarray}\int _{C}{\rm\Phi}\,d{\it\mu}_{J_{\infty }X}=\int _{p_{\infty }^{-1}(C)}({\rm\Phi}\circ p_{\infty })\mathbb{L}^{-\mathbf{j}_{p}+\mathbf{w}_{V}}\,d{\it\mu}_{J_{\infty }^{G}V}.\end{eqnarray}$$

To explain where the formula comes from, we first show a lemma.

Lemma 5.7. We have

$$\begin{eqnarray}\text{Jac}_{V^{\left\langle F\right\rangle }/V\times _{D}F}=\mathfrak{m}_{F}^{\sharp H\cdot \boldsymbol{v}_{V}(F)}{\mathcal{O}}_{V^{\left\langle F\right\rangle }}.\end{eqnarray}$$

Proof. Let $u^{\prime }:V^{\left\langle F\right\rangle }\rightarrow V\times _{D}F$ be the natural map. We have the standard exact sequence

$$\begin{eqnarray}(u^{\prime })^{\ast }{\rm\Omega}_{V\times _{D}F/F}\rightarrow {\rm\Omega}_{V^{\left\langle F\right\rangle }/F}\rightarrow {\rm\Omega}_{V^{\left\langle F\right\rangle }/V\times _{D}F}\rightarrow 0.\end{eqnarray}$$

The left map is identical to the map

$$\begin{eqnarray}M\otimes _{{\mathcal{O}}_{D}}{\mathcal{O}}_{V^{\left\langle F\right\rangle }}\rightarrow M^{\left\langle F\right\rangle }\otimes _{{\mathcal{O}}_{F}}{\mathcal{O}}_{V^{\left\langle F\right\rangle }}.\end{eqnarray}$$

Since the Fitting ideal is compatible with base changes (for instance, see [Reference EisenbudEis95, Corollary 20.5]), if $I$ denotes the zeroth Fitting ideal of

$$\begin{eqnarray}\text{coker}(M\otimes _{{\mathcal{O}}_{D}}{\mathcal{O}}_{F}\rightarrow M^{\left\langle F\right\rangle }),\end{eqnarray}$$

we have $\text{Jac}_{V^{\left\langle F\right\rangle }/V\times _{D}F}=I\cdot {\mathcal{O}}_{V^{\left\langle F\right\rangle }}$. It is now easy to see that $I=\mathfrak{m}_{F}^{\sharp H\cdot \boldsymbol{v}_{V}(F)}$, for instance, by considering a triangular matrix representing the map $M\otimes _{{\mathcal{O}}_{D}}{\mathcal{O}}_{F}\rightarrow M^{\left\langle F\right\rangle }$ for suitable bases.◻

Conjecture 5.6 can be guessed from the following conjecture.

Conjecture 5.8. For ${\it\gamma}\in J_{\infty }^{G}V$ and $n\gg 0$, the fiber of the map

$$\begin{eqnarray}p_{n}:{\it\pi}_{n}(J_{\infty }^{G}V)\rightarrow J_{n}X\end{eqnarray}$$

over the image of ${\it\gamma}$ is homeomorphic to a quotient of the affine space

$$\begin{eqnarray}\mathbb{A}_{k}^{(\mathbf{j}_{p}-\mathbf{w}_{V})({\it\gamma})}\end{eqnarray}$$

by a linear finite group action.

To see this, we first note that since two $G$-arcs $E\rightarrow V$ and $E^{\prime }\rightarrow V$ with $E\not \cong E^{\prime }$ have distinct images in $J_{n}X$ for $n\gg 0$, we can focus on $J_{\infty }^{G,E}V$ for fixed $E$. Fixing a $G$-arc ${\it\gamma}:E\rightarrow V$, we consider the map

$$\begin{eqnarray}(J_{n}V^{|F|})/C_{G}(H)\rightarrow J_{n}X.\end{eqnarray}$$

The fiber of this map over the image of ${\it\gamma}$ should be homeomorphic to

$$\begin{eqnarray}\mathbb{A}_{k}^{\boldsymbol{ j}_{p^{|F|}}({\it\gamma}^{\prime })}/A,\end{eqnarray}$$

where ${\it\gamma}^{\prime }$ is an arc of $V^{|F|}$ corresponding to ${\it\gamma}$, and $A$ is a certain subgroup of $C_{G}(H)$ acting linearly on the affine space. This fact would be proved in the course of proving Conjecture 5.3. On the other hand, the map

$$\begin{eqnarray}{\it\pi}_{n}(\text{Hom}_{F}^{H}(F,V^{\left\langle F\right\rangle }))/C_{G}(H)\rightarrow {\it\pi}_{n}(J_{\infty }^{G}V)\end{eqnarray}$$

induced by $u$ has fibers homeomorphic to

$$\begin{eqnarray}\mathbb{A}_{k}^{\text{codim}((V_{0})^{H},V_{0})}/B\end{eqnarray}$$

for some finite group $B$, which can be seen by looking at diagrams (4.2)–(4.4). From Lemma 5.7,

$$\begin{eqnarray}\displaystyle \mathbf{j}_{p^{|F|}}\hspace{-1.0pt}-\hspace{-1.0pt}\text{codim}((V_{0})^{H},V_{0}) & = & \displaystyle \mathbf{j}_{V^{\left\langle F\right\rangle }/X\times _{D}F}-\text{codim}((V_{0})^{H},V_{0})\nonumber\\ \displaystyle & = & \displaystyle (\mathbf{j}_{V\times _{D}F/X\times _{D}F}+\mathbf{j}_{V^{\left\langle F\right\rangle }/V\times _{D}F})\hspace{-2.0pt}-\hspace{-2.0pt}\text{codim}((V_{0})^{H},V_{0})\nonumber\\ \displaystyle & = & \displaystyle \mathbf{j}_{p}-\mathbf{w}_{V},\nonumber\end{eqnarray}$$

concluding Conjecture 5.8.

6 The McKay correspondence for linear actions

To state the McKay correspondence conjecture for linear actions, we first define the notion of orbifold stringy motifs. Keeping the notation from the last section, let $\mathfrak{X}$, $\mathfrak{V}$, $\mathfrak{V}^{\left\langle F\right\rangle }$ and $\mathfrak{V}^{|F|}$ be centered log structures on $X$, $V$, $V^{\left\langle F\right\rangle }$ and $V^{|F|}$ respectively, so that the following morphisms are all crepant:

Since $X$ is $\mathbb{Q}$-factorial, either $\mathfrak{X}$ or $\mathfrak{V}$ determines the other centered log structures. The centered log structure $\mathfrak{V}$ is $G$-equivariant and $\mathfrak{V}^{|F|}$$C_{G}(H)$-equivariant.

Definition 6.1. We define the orbifold stringy motif of the centered log $G$-$D$-variety $\mathfrak{V}$ to be

$$\begin{eqnarray}M_{\text{st}}^{G}(\mathfrak{V}):=\int _{J_{\infty }^{G}\mathfrak{V}}\mathbb{L}^{\mathbf{f}_{\mathfrak{V}}+\mathbf{w}_{V}}\,d{\it\mu}_{\mathfrak{V}}^{G}.\end{eqnarray}$$

Note that since $\mathfrak{V}$ is smooth over $D$, we have $\mathbf{f}_{\mathfrak{V}}=\text{ord}\,{\rm\Delta}$ for the boundary ${\rm\Delta}$ of $\mathfrak{V}$.

Arguments as in the proof of Proposition 2.6 deduce the following conjecture from Conjecture 5.6.

Conjecture 6.2. (The motivic McKay correspondence for linear actions I)

We have

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{X})=M_{\text{st}}^{G}(\mathfrak{V}).\end{eqnarray}$$

We next formulate a conjecture presented in a slightly different way so that we are able to generalize it to the nonlinear case easily.

Definition 6.3. For $E\in G\text{-}\text{Cov}(D)$, we define the $E$-parts of $M_{\text{st}}^{G}(\mathfrak{V})$ and $M_{\text{st}}(\mathfrak{X})$ respectively by

$$\begin{eqnarray}\displaystyle M_{\text{st}}^{G,E}(\mathfrak{V}): & = & \displaystyle \int _{J_{\infty }^{G,E}\mathfrak{V}}\mathbb{L}^{\mathbf{f}_{\mathfrak{V}}+\mathbf{w}_{V}}\,d{\it\mu}_{J_{\infty }^{G}\mathfrak{V}}\quad \text{and}\nonumber\\ \displaystyle M_{\text{st}}^{E}(\mathfrak{X}) & := & \displaystyle \int _{p_{\infty }(J_{\infty }^{G,E}\mathfrak{V})}\mathbb{L}^{\mathbf{f}_{\mathfrak{X}}}\,d{\it\mu}_{J_{\infty }\mathfrak{X}}.\nonumber\end{eqnarray}$$

By the same reasoning as that for the last conjecture, we would have

(6.1)$$\begin{eqnarray}M_{\text{st}}^{G,E}(\mathfrak{V})=M_{\text{st}}^{E}(\mathfrak{X}).\end{eqnarray}$$

On the other hand, from Conjecture 5.3, we would have

(6.2)$$\begin{eqnarray}M_{\text{st}}^{E}(\mathfrak{X})=M_{\text{st},C_{G}(H)}(\mathfrak{V}^{|F|}).\end{eqnarray}$$

Let $G\text{-}\text{Cov}(D)=\sqcup _{i=0}^{\infty }A_{i}$ be a conjectural stratification with finite-dimensional strata $A_{i}$ (see Remark 3.6). The author [Reference YasudaYasa] conjectures also that each stratum $A_{i}$ may not be of finite type over $k$, but the limit of a family

$$\begin{eqnarray}X_{1}\xrightarrow[]{f_{1}}X_{2}\xrightarrow[]{f_{2}}\cdots\end{eqnarray}$$

such that $X_{j}$ are of finite type and $f_{i}$ are homeomorphisms. We then define a constructible subset of $G\text{-}\text{Cov}(D)$ as a constructible subset of $\sqcup _{i=0}^{n}A_{i}$ for some $n<\infty$, which would be well defined thanks to this conjecture. For a constructible subset $C$ of $G\text{-}\text{Cov}(D)$, its class $[C]$ in ${\mathcal{R}}$ is well defined. Let ${\it\tau}$ denote the tautological motivic measure on $G\text{-}\text{Cov}(D)$ given by ${\it\tau}(C):=[C]$ for a constructible subset $C$. If a function ${\rm\Phi}:G\text{-}\text{Cov}(D)\rightarrow {\mathcal{R}}\cup \{\infty \}$ is constructible, that is, its image is countable and all fibers ${\rm\Phi}^{-1}(a)$, $a\in {\mathcal{R}}$ are constructible, then the integral $\int _{G\text{-}\text{Cov}(D)}{\rm\Phi}\,d{\it\tau}$ is defined by

$$\begin{eqnarray}\int _{G\text{-}\text{Cov}(D)}{\rm\Phi}\,d{\it\tau}:=\mathop{\sum }_{a\in {\mathcal{R}}}{\it\tau}({\rm\Phi}^{-1}(a))\cdot a\in {\mathcal{R}}\cup \{\infty \}.\end{eqnarray}$$

From Conjecture 6.2 and conjectural equations (6.1) and (6.2), it seems natural to expect

$$\begin{eqnarray}M_{\text{st}}^{G}(\mathfrak{V})=\int _{G\text{-}\text{Cov}(D)}M_{\text{st},C_{G}(H)}(\mathfrak{V}^{|F|})\,d{\it\tau},\end{eqnarray}$$

and hence we have the following conjecture.

Conjecture 6.4. (The motivic McKay correspondence for linear actions II)

We have

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{X})=\int _{G\text{-}\text{Cov}(D)}M_{\text{st},C_{G}(H)}(\mathfrak{V}^{|F|})\,d{\it\tau}.\end{eqnarray}$$

This formulation of the McKay correspondence is what we generalize to the nonlinear case.

To make this conjecture more meaningful, it would be nice if we could compute $M_{\text{st},C_{G}(H)}(\mathfrak{V}^{|F|})$ explicitly. For this purpose, next we see how to determine the centered log structures $\mathfrak{V}^{\left\langle F\right\rangle }$ and $\mathfrak{V}^{|F|}$ from $\mathfrak{V}$. Let us write $\mathfrak{V}=(V,{\rm\Delta},W)$, $\mathfrak{V}^{\left\langle F\right\rangle }=(V,{\rm\Delta}^{\left\langle F\right\rangle },W^{\left\langle F\right\rangle })$ and $\mathfrak{V}^{|F|}=(V,{\rm\Delta}^{|F|},W^{|F|})$. The centers $W^{\left\langle F\right\rangle }$ and $W^{|F|}$ are simply determined by

$$\begin{eqnarray}W^{\left\langle F\right\rangle }=u^{-1}(W)\quad \text{and}\quad W^{|F|}=r(W^{\left\langle F\right\rangle }).\end{eqnarray}$$

The boundaries ${\rm\Delta}^{\left\langle F\right\rangle }$ and ${\rm\Delta}^{|F|}$ are determined as follows.

Lemma 6.5. Regarding $V_{0}^{\left\langle F\right\rangle }$ and $V_{0}^{|F|}$ prime divisors on $V^{\left\langle F\right\rangle }$ and $V^{|F|}$, we have

$$\begin{eqnarray}\displaystyle & {\rm\Delta}^{\left\langle F\right\rangle }=u^{\ast }{\rm\Delta}-(\sharp H\cdot \boldsymbol{v}_{V}(E)+d_{F/D})\cdot V_{0}^{\left\langle F\right\rangle }, & \displaystyle \nonumber\\ \displaystyle & {\rm\Delta}^{|F|}=\displaystyle \frac{1}{\sharp H}\cdot r_{\ast }u^{\ast }{\rm\Delta}-\boldsymbol{v}_{V}(E)\cdot V_{0}^{|F|}. & \displaystyle \nonumber\end{eqnarray}$$

Here, $d_{F/D}$ is the different exponent of $F/D$, characterized by ${\rm\Omega}_{F/D}\cong {\mathcal{O}}_{F}/\mathfrak{m}_{F}^{d_{F/D}}$.

Proof. For the first equality, we have

$$\begin{eqnarray}\displaystyle u^{\ast }(K_{V}+{\rm\Delta}) & = & \displaystyle K_{V^{\left\langle F\right\rangle }}-K_{V^{\left\langle F\right\rangle }/V}+u^{\ast }{\rm\Delta}\nonumber\\ \displaystyle & = & \displaystyle K_{V^{\left\langle F\right\rangle }}-K_{V^{\left\langle F\right\rangle }/V\times _{D}F}-(u^{\prime })^{\ast }K_{V\times _{D}F/V}+u^{\ast }{\rm\Delta}.\nonumber\end{eqnarray}$$

Here, $K_{V^{\left\langle F\right\rangle }}$ is the canonical divisor of $V^{\left\langle F\right\rangle }$ as a $D$-variety rather than an $F$-variety, and $u^{\prime }$ denotes the natural morphism $V^{\left\langle F\right\rangle }\rightarrow V\times _{D}F$. From Lemma 5.7,

$$\begin{eqnarray}K_{V^{\left\langle F\right\rangle }/V\times _{D}F}=\sharp H\cdot \boldsymbol{v}_{V}(E)\cdot V_{0}^{\left\langle F\right\rangle }.\end{eqnarray}$$

Since $(u^{\prime })^{\ast }K_{V\times _{D}F/V}$ is the pullback of $K_{F/D}$, we have

$$\begin{eqnarray}(u^{\prime })^{\ast }K_{V\times _{D}F/V}=d_{F/D}\cdot V_{0}^{\left\langle F\right\rangle }.\end{eqnarray}$$

These equalities show the first equality of the lemma.

The second one follows from

$$\begin{eqnarray}\displaystyle & & \displaystyle r^{\ast }\left(K_{V^{|F|}}+\frac{1}{\sharp H}\cdot r_{\ast }u^{\ast }{\rm\Delta}-\boldsymbol{v}(E)\cdot V_{0}^{|F|}\right)\nonumber\\ \displaystyle & & \displaystyle \quad =K_{V^{\left\langle F\right\rangle }}-K_{V^{\left\langle F\right\rangle }/V^{|F|}}+u^{\ast }{\rm\Delta}-\sharp H\cdot \boldsymbol{v}_{V}(E)\cdot V_{0}^{\left\langle F\right\rangle }\nonumber\\ \displaystyle & & \displaystyle \quad =K_{V^{\left\langle F\right\rangle }}+u^{\ast }{\rm\Delta}-(\sharp H\cdot \boldsymbol{v}_{V}(E)+d_{F/E})\cdot V_{0}^{\left\langle F\right\rangle }\nonumber\\ \displaystyle & & \displaystyle \quad =K_{V^{\left\langle F\right\rangle }}+{\rm\Delta}^{\left\langle F\right\rangle }.\square\nonumber\end{eqnarray}$$

Example 6.6. Suppose that ${\rm\Delta}=0$ and $W=\{o\}$, with $o\in V_{0}$ the origin. Then ${\rm\Delta}^{|F|}=-\boldsymbol{v}_{V}(E)\cdot V_{0}^{|F|}$ and $W^{|F|}\cong \mathbb{A}_{k}^{\text{codim}((V_{0})^{H},V_{0})}$. Hence,

$$\begin{eqnarray}M_{\text{st}}^{G,E}(\mathfrak{V})=M_{\text{st},C_{G}(H)}(\mathfrak{V}^{|F|})=\mathbb{L}^{\mathbf{w}_{V}(E)}.\end{eqnarray}$$

Conjecture 6.4 is reduced to the form

(6.3)$$\begin{eqnarray}M_{\text{st}}(\mathfrak{X})=\int _{G\text{-}\text{Cov}(D)}\mathbb{L}^{\mathbf{w}_{V}}\,d{\it\tau}.\end{eqnarray}$$

If $p:V\rightarrow X$ is étale in codimension one, and if we denote $p(o)$ again by $o$, then $M_{\text{st}}(\mathfrak{X})=M_{\text{st}}(X)_{o}$, and the last equality is exactly what was conjectured in [Reference YasudaYasa].

Remark 6.7. If $\sharp G$ is prime to the characteristic of $k$, then $G\text{-}\text{Cov}(D)$ is identified with the set of conjugacy classes of $G$, denoted by $\text{Conj}(G)$. Equality (6.3) in the last example is then written as

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{X})=\mathop{\sum }_{[g]\in \text{Conj}(G)}\mathbb{L}^{\mathbf{w}_{V}(g)}.\end{eqnarray}$$

Expressing the weights $\mathbf{w}_{V}(g)$ in terms of eigenvalues, we recover results by Batyrev [Reference BatyrevBat99] and Denef and Loeser [Reference Denef and LoeserDL02].

7 The McKay correspondence for nonlinear actions

In this section, we generalize Conjecture 6.4 to the nonlinear case. This is rather easy, once we have formulated the conjecture as it is.

Let us consider an affine $D$-variety $\mathsf{v}=\text{Spec}\,{\mathcal{O}}_{\mathsf{v}}$ endowed with a faithful $G$-action. We fix a $G$-equivariant (locally closed) immersion

$$\begin{eqnarray}\mathsf{v}{\hookrightarrow}V\end{eqnarray}$$

into an affine space $V\cong \mathbb{A}_{D}^{d}$ endowed with a linear $G$-action. Identifying $G$-arcs of $\mathsf{v}$ with those of $V$ factoring through $\mathsf{v}$, we regard $J_{\infty }^{G}\mathsf{v}$ as a subset of $J_{\infty }^{G}V$.

Remark 7.1. Such an immersion always exists. Indeed, let $f_{1},\ldots ,f_{n}$ be generators of ${\mathcal{O}}_{\mathsf{v}}$ as an ${\mathcal{O}}_{D}$-algebra, let $A:=\bigcup _{i}f_{i}G$, the union of their orbits, and let ${\mathcal{O}}_{D}[x_{f}\mid f\in A]$ be the polynomial ring with variables $x_{f}$, $f\in A$ over ${\mathcal{O}}_{D}$. The ring has a natural $G$-action by permutations of variables. The ${\mathcal{O}}_{D}$-algebra homomorphism

$$\begin{eqnarray}{\mathcal{O}}_{D}[x_{f}\mid f\in A]\rightarrow {\mathcal{O}}_{\mathsf{v}},\quad x_{f}\mapsto f\end{eqnarray}$$

defines a desired immersion. Moreover, this construction gives a closed immersion into $V$ on which $G$ acts by permutations. In this case, our weight function $\mathbf{w}_{V}$ is closely related to the Artin and Swan conductors [Reference Wood and YasudaWY15], although we do not use this fact in this paper.

Let $\mathsf{x}:=\mathsf{v}/G$. The following diagram shows natural morphisms of relevant varieties, and the symbols $t$, $s$ and $q$ denote morphisms as indicated:

(7.1)$$\begin{eqnarray}\begin{array}{@{}c@{}}\end{array}\end{eqnarray}$$

The one-to-one correspondence obtained in the last section,

$$\begin{eqnarray}J_{\infty }^{G,E}V\leftrightarrow (J_{\infty }V^{|F|})/C_{G}(H),\end{eqnarray}$$

induces a one-to-one correspondence

$$\begin{eqnarray}J_{\infty }^{G,E}\mathsf{v}\leftrightarrow (J_{\infty }\mathsf{v}^{|F|})/C_{G}(H).\end{eqnarray}$$

We obtain the following diagram:

If we put $J_{\infty }^{E}\mathsf{x}$ to be the image of $J_{\infty }^{G,E}\mathsf{v}$ in $J_{\infty }\mathsf{x}$, then we can naturally expect that $J_{\infty }^{E}\mathsf{x}$ coincides with the images of $J_{\infty }\mathsf{v}^{|F|}$ and $J_{\infty }\mathsf{v}^{|F|,{\it\nu}}$ modulo measure zero subsets.

From now on, we suppose that $\mathsf{v}$ is normal. Let $\mathfrak{v}$, $\mathfrak{v}^{\left\langle F\right\rangle ,{\it\nu}}$, $\mathfrak{v}^{|F|,{\it\nu}}$ and $\mathfrak{x}$ be centered log structures on $\mathsf{v}$, $\mathsf{v}^{\left\langle F\right\rangle ,{\it\nu}}$ and $\mathsf{v}^{|F|,{\it\nu}}$ respectively such that the morphisms

are all crepant. The centered log $D$-varieties $\mathfrak{v}$ and $\mathfrak{v}^{|F|,{\it\nu}}$ are $G$- and $C_{G}(H)$-equivariant respectively. If we define the $E$-part $M_{\text{st}}^{E}(\mathfrak{x})$ of $M_{\text{st}}(\mathfrak{x})$, we can expect

$$\begin{eqnarray}M_{\text{st}}^{E}(\mathfrak{x})=M_{\text{st},C_{G}(H)}(\mathfrak{v}^{|F|,{\it\nu}}),\end{eqnarray}$$

similarly to the linear case. The equality is a slight generalization of Conjecture 2.5 and would follow from the change of variables formula generalized along the line of [Reference Denef and LoeserDL02], applied to the almost bijection

$$\begin{eqnarray}J_{\infty }\mathfrak{v}^{|F|,{\it\nu}}\rightarrow J_{\infty }^{E}\mathfrak{x}.\end{eqnarray}$$

It is then natural to expect the following conjecture.

Conjecture 7.3. (The McKay correspondence for nonlinear actions)

We have

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{x})=\int _{G\text{-}\text{Cov}(D)}M_{\text{st},C_{G}(H)}(\mathfrak{v}^{|F|,{\it\nu}})\,d{\it\tau}.\end{eqnarray}$$

Definition 7.4. We define the $E$-part of the orbifold stringy motif of $\mathfrak{v}$ as

$$\begin{eqnarray}\displaystyle M_{\text{st}}^{G,E}(\mathfrak{v}) & :=M_{\text{st},C_{G}(H)}(\mathfrak{v}^{|F|,{\it\nu}}), & \displaystyle \nonumber\end{eqnarray}$$

and the orbifold stringy motif of $\mathfrak{v}$ as

$$\begin{eqnarray}M_{\text{st}}^{G}(\mathfrak{v}):=\int _{G\text{-}\text{Cov}(D)}M_{\text{st}}^{G,E}(\mathfrak{v})\,d{\it\tau}.\end{eqnarray}$$

With this definition, the last conjecture simply says

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{x})=M_{\text{st}}^{G}(\mathfrak{v}).\end{eqnarray}$$

Remark 7.5. The reader may wonder why we do not define $M_{\text{st}}^{G}(\mathfrak{v})$ as a motivic integral on $J_{\infty }^{G}\mathfrak{v}$, which appears to be more natural. It is because the author does not know whether one can define a motivic measure on $J_{\infty }^{G}\mathfrak{v}$, as he does not know how to compute dimensions of fibers of

$$\begin{eqnarray}{\it\pi}_{n}(\text{Hom}_{F}^{H}(F,\mathsf{v}^{\left\langle F\right\rangle ,{\it\nu}}))/C_{G}(H)\rightarrow {\it\pi}_{n}(J_{\infty }^{G}\mathsf{v}).\end{eqnarray}$$

Knowing it was, in the linear case, a key in formulating the change of variables formula (Conjecture 5.6) and determining the integrand $\mathbb{L}^{\mathbf{f}_{\mathfrak{V}}+\mathbf{w}_{V}}$ in the definition of $M_{\text{st}}^{G}(\mathfrak{V})$.

8 Computing boundaries of untwisting varieties

To compute examples of the wild McKay correspondence, we need to determine centered log varieties $\mathfrak{v}^{|F|,{\it\nu}}$. It is easy to determine the center. In this section, supposing that $\mathsf{v}$ and $\mathsf{v}^{|F|}$ are both normal and complete intersections in $V$ and $V^{|F|}$ respectively, we compute the boundary of $\mathfrak{v}^{|F|}$.

Let

$$\begin{eqnarray}t:\mathsf{v}^{\left\langle F\right\rangle }\rightarrow \mathsf{v}\quad \text{and}\quad s:\mathsf{v}^{\left\langle F\right\rangle }\rightarrow \mathsf{v}^{|F|}\end{eqnarray}$$

be the natural morphisms, although they are different from the morphisms denoted by the same symbols in diagram (7.1) unless $\mathsf{v}^{\left\langle F\right\rangle }$ is also normal. The subvariety $\mathsf{v}^{\left\langle F\right\rangle }\subset V^{\left\langle F\right\rangle }$ is a complete intersection. To see this, first note that if $s^{-1}(\mathsf{v}^{|F|})$ denotes the scheme-theoretic preimage, then $\left(s^{-1}(\mathsf{v}^{|F|})\right)_{\text{red}}=\mathsf{v}^{\left\langle F\right\rangle }$. The subscheme $s^{-1}(\mathsf{v}^{|F|})\subset V^{\left\langle F\right\rangle }$ is a complete intersection, hence Cohen–Macaulay, and generically reduced. From [Reference EisenbudEis95, Theorem 18.15], $s^{-1}(\mathsf{v}^{|F|})$ is actually reduced and

$$\begin{eqnarray}s^{-1}(\mathsf{v}^{|F|})=\mathsf{v}^{\left\langle F\right\rangle }.\end{eqnarray}$$

In general, for a complete intersection subvariety $Y\subset X$, its conormal sheaf ${\mathcal{C}}_{Y/X}$ is defined as $I_{Y}/I_{Y}^{2}$, with $I_{Y}\subset {\mathcal{O}}_{X}$ the defining ideal sheaf of $Y$. We put

$$\begin{eqnarray}\det {\mathcal{C}}_{Y/X}:=\mathop{\text{codim}(Y,X)}^{\wedge }{\mathcal{C}}_{Y/X}.\end{eqnarray}$$

There exists a unique effective $H$-stable Cartier divisor $A_{F}$ on $\mathsf{v}^{\left\langle F\right\rangle }$ such that

$$\begin{eqnarray}t^{\ast }(\det {\mathcal{C}}_{\mathsf{v}/V})=(\det {\mathcal{C}}_{\mathsf{v}^{\left\langle F\right\rangle }/V^{\left\langle F\right\rangle }})(-A_{F}).\end{eqnarray}$$

Proposition 8.1. Let ${\it\delta}$ and ${\it\delta}^{|F|}$ be the boundaries of $\mathfrak{v}$ and $\mathfrak{v}^{|F|}$ respectively, and $C:=V_{0}^{|F|}|_{\mathsf{v}^{|F|}}$, the restriction of the prime divisor $V_{0}^{|F|}$ on $V^{|F|}$ to $\mathsf{v}^{|F|}$. Then

$$\begin{eqnarray}{\it\delta}^{|F|}=\frac{1}{\sharp H}\cdot s_{\ast }\left(t^{\ast }{\it\delta}+A_{F}\right)-\boldsymbol{v}_{V}(E)\cdot C.\end{eqnarray}$$

Proof. Let ${\it\epsilon}^{|F|}$ be the right-hand side of the equality. As in the proof of Lemma 6.5, it suffices to show that the pullbacks of divisors $K_{\mathsf{v}}+{\it\delta}$ and $K_{\mathsf{v}^{|F|}}+{\it\epsilon}^{|F|}$ to $\mathsf{v}^{\left\langle F\right\rangle }$ coincide. Since

$$\begin{eqnarray}s^{\ast }\left(\frac{1}{\sharp H}s_{\ast }t^{\ast }{\it\delta}\right)=t^{\ast }{\it\delta},\end{eqnarray}$$

we may suppose ${\it\delta}=0$ and hence

$$\begin{eqnarray}{\it\epsilon}^{|F|}=\frac{1}{\sharp H}\cdot s_{\ast }A_{F}-\boldsymbol{v}_{V}(E)C.\end{eqnarray}$$

By abuse of notation, identifying a divisor corresponding to an invertible sheaf, from the adjunction formula, we have

$$\begin{eqnarray}\displaystyle t^{\ast }K_{\mathsf{v}} & = & \displaystyle t^{\ast }\left(K_{V}|_{\mathsf{v}}-\det {\mathcal{C}}_{\mathsf{v}/V}\right)\nonumber\\ \displaystyle & = & \displaystyle \left(u^{\ast }K_{V}\right)|_{\mathsf{v}^{\left\langle F\right\rangle }}-\det {\mathcal{C}}_{\mathsf{v}^{\left\langle F\right\rangle }/V^{\left\langle F\right\rangle }}+A_{F}.\nonumber\end{eqnarray}$$

On the other hand, since $s^{\ast }(\det {\mathcal{C}}_{\mathsf{v}^{|F|}/V^{|F|}})=\det {\mathcal{C}}_{\mathsf{v}^{\left\langle F\right\rangle }/V^{\left\langle F\right\rangle }}$,

$$\begin{eqnarray}\displaystyle s^{\ast }(K_{\mathsf{v}^{|F|}}+{\it\epsilon}^{|F|}) & = & \displaystyle s^{\ast }(K_{V^{|F|}}|_{\mathsf{v}^{|F|}}\hspace{-2.0pt}-\hspace{-2.0pt}\det {\mathcal{C}}_{\mathsf{v}^{|F|}/V^{|F|}})\hspace{-2.0pt}-\hspace{-2.0pt}\sharp H\cdot \boldsymbol{v}_{V}(E)V_{0}^{\left\langle F\right\rangle }|_{\mathsf{v}^{\left\langle F\right\rangle }}\hspace{-1.0pt}+\hspace{-1.0pt}A_{F}\nonumber\\ \displaystyle & = & \displaystyle (r^{\ast }K_{V^{|F|}}-\sharp H\cdot \boldsymbol{v}_{V}(E)V_{0}^{\left\langle F\right\rangle })|_{\mathsf{v}^{\left\langle F\right\rangle }}-\det {\mathcal{C}}_{\mathsf{v}^{\left\langle F\right\rangle }/V^{\left\langle F\right\rangle }}+A_{F}.\nonumber\end{eqnarray}$$

From Lemma 6.5,

$$\begin{eqnarray}\displaystyle r^{\ast }K_{V^{|F|}}-\sharp H\cdot \boldsymbol{v}_{V}(E)V_{0}^{\left\langle F\right\rangle } & = & \displaystyle K_{V^{\left\langle F\right\rangle }/D}-d_{F/D}V_{0}^{\left\langle F\right\rangle }-\sharp H\cdot \boldsymbol{v}_{V}(E)V_{0}^{\left\langle F\right\rangle }\nonumber\\ \displaystyle & = & \displaystyle u^{\ast }K_{V},\nonumber\end{eqnarray}$$

which shows the proposition. ◻

It is handy to rewrite the proposition in the case of hypersurfaces as follows.

Corollary 8.2. Suppose that $\mathsf{v}\subset V$ is a hypersurface defined by a polynomial $f\in {\mathcal{O}}_{V}$, and write

$$\begin{eqnarray}u_{F}^{\ast }f={\it\pi}_{F}^{b}{\it\phi},\end{eqnarray}$$

where ${\it\pi}_{F}$ is a uniformizer of ${\mathcal{O}}_{F}$, $b$ is an integer $b\geqslant 0$, and ${\it\phi}\in {\mathcal{O}}_{V^{\left\langle F\right\rangle }}$, with ${\it\pi}_{F}\nmid {\it\phi}$. Then, with the notation as above, we have

$$\begin{eqnarray}{\it\delta}^{|F|}=\frac{1}{\sharp H}s_{\ast }t^{\ast }{\it\delta}+\left(\frac{b}{\sharp H}-\boldsymbol{v}_{V}(E)\right)\cdot C.\end{eqnarray}$$

Proof. The corollary follows from

$$\begin{eqnarray}A_{F}=b\cdot (V_{0}^{\left\langle F\right\rangle }|_{\mathsf{v}_{0}^{\left\langle F\right\rangle }})\quad \text{and}\quad s_{\ast }A_{F}=b\cdot C.\square\end{eqnarray}$$

9 A tame singular example

In this section, we verify Conjecture 7.3 for an example of the tame case where $\mathsf{v}$ is not regular.

Suppose that $k$ has characteristic $\neq 2$. Let $D:=\text{Spec}\,k[[{\it\pi}]]$, $V:=\text{Spec}\,k[[{\it\pi}]][x,y,z]$ and $\mathsf{v}=\text{Spec}\,k[[{\it\pi}]][x,y,z]/(xz-y^{2})$, the trivial family of the $A_{1}$-singularity over $\text{Spec}\,k[[{\it\pi}]]$. We suppose that $G=\mathbb{Z}/2\mathbb{Z}=\{1,g\}$ acts on $V$ by

$$\begin{eqnarray}xg=-x,\qquad yg=y,\qquad zg=-z.\end{eqnarray}$$

The subvariety $\mathsf{v}$ is stable under the $G$-action, and the quotient variety $\mathsf{x}=\mathsf{v}/G$ can be embedded into $\mathbb{A}_{k[[{\it\pi}]]}^{3}=\text{Spec}\,k[[{\it\pi}]][u,v,w]$ and gives the hypersurface defined by the equation $uv-w^{4}=0$. Thus, $\mathsf{x}$ is the trivial family of the $A_{3}$-singularity over $\text{Spec}\,k[[{\it\pi}]]$.

Since the morphism $\mathsf{v}\rightarrow \mathsf{x}$ is étale in codimension one, it is crepant (with the identification (2.1)). Let $\tilde{\mathsf{x}}_{0}\rightarrow \mathsf{x}_{0}$ be the minimal resolution, and let $\tilde{\mathsf{x}}:=\mathsf{x}\otimes _{k}k[[{\it\pi}]]$. The natural morphism $\tilde{\mathsf{x}}\rightarrow \mathsf{x}$ is crepant. From Proposition 2.7,

$$\begin{eqnarray}M_{\text{st}}(\mathsf{x})=M_{\text{st}}(\tilde{\mathsf{x}})=[\tilde{\mathsf{x}}_{0}]=\mathbb{L}^{2}+3\mathbb{L}.\end{eqnarray}$$

Next, we will compute $M_{\text{st}}^{G}(\mathsf{v})$ and verify that it coincides with $M_{\text{st}}(\mathsf{x})$. There are exactly two $G$-covers of $D$ up to isomorphism: the trivial one $E_{1}=D\sqcup D\rightarrow D$ and the nontrivial one

$$\begin{eqnarray}E_{2}=\text{Spec}\,k[[{\it\pi}^{1/2}]]\rightarrow D=\text{Spec}\,k[[{\it\pi}]],\end{eqnarray}$$

and hence

$$\begin{eqnarray}M_{\text{st}}^{G}(\mathsf{v})=M_{\text{st}}^{G,E_{1}}(\mathsf{v})+M_{\text{st}}^{G,E_{2}}(\mathsf{v}).\end{eqnarray}$$

As for the first term $M_{\text{st}}^{G,E_{1}}(\mathsf{v})$, we have $\mathsf{v}^{|D|}=\mathsf{v}$. Consider the minimal resolution $\tilde{\mathsf{v}}_{0}\rightarrow \mathsf{v}_{0}$ and put $\tilde{\mathsf{v}}:=\tilde{\mathsf{v}}_{0}\otimes _{k}k[[{\it\pi}]]$. Then the morphism $\tilde{\mathsf{v}}\rightarrow \mathsf{v}$ is crepant. Since the $G$-action on the exceptional locus is trivial, from Proposition 2.9,

$$\begin{eqnarray}M_{\text{st}}^{G,E_{1}}(\mathsf{v})=M_{\text{st},G}(\tilde{\mathsf{v}})=\mathbb{L}^{2}+\mathbb{L}.\end{eqnarray}$$

Next, we compute $M_{\text{st}}^{G,E_{2}}(\mathsf{v})$. For $F=E_{2}$, the tuning module ${\rm\Xi}_{F}$ has a basis

(9.1)$$\begin{eqnarray}{\it\pi}^{1/2}x^{\ast },y^{\ast },{\it\pi}^{1/2}z^{\ast },\end{eqnarray}$$

with $x^{\ast },y^{\ast },z^{\ast }$ the dual basis of $x$, $y$, $z$. If we denote the dual basis of (9.1) by $\text{x}$, $\text{y}$, $\text{z}$, then we can write $u^{\ast }$ as

$$\begin{eqnarray}\displaystyle u^{\ast }:k[[{\it\pi}]][x,y,z] & \rightarrow & \displaystyle k[[{\it\pi}^{1/2}]][\text{x},\text{y},\text{z}]\nonumber\\ \displaystyle x & \mapsto & \displaystyle {\it\pi}^{1/2}\text{x}\nonumber\\ \displaystyle y & \mapsto & \displaystyle \text{y}\nonumber\\ \displaystyle z & \mapsto & \displaystyle {\it\pi}^{1/2}\text{z}.\nonumber\end{eqnarray}$$

We see that $\mathsf{v}^{|F|}$ is given by

$$\begin{eqnarray}{\it\pi}\text{x}\text{z}-\text{y}^{2}=0.\end{eqnarray}$$

Since the nonregular locus of $\mathsf{v}^{|F|}$ has dimension one, the variety $\mathsf{v}^{|F|}$ is normal. From Corollary 8.2, the boundary ${\it\delta}^{|F|}$ of $\mathfrak{v}^{|F|}$ is given by

$$\begin{eqnarray}{\it\delta}^{|F|}=-V_{0}^{|F|}|_{\mathsf{v}^{|F|}}.\end{eqnarray}$$

Hence,

$$\begin{eqnarray}M_{\text{st}}^{G,E_{2}}(\mathsf{v})=M_{\text{st},G}(\mathfrak{v}^{|F|})=M_{\text{st},G}(\mathsf{v}^{|F|})\mathbb{L}^{-1}.\end{eqnarray}$$

The $G$-action on $\mathsf{v}^{|F|}$ is given by

$$\begin{eqnarray}\text{x}g=-\text{x},\qquad \text{y}g=\text{y},\qquad \text{z}g=-\text{z}.\end{eqnarray}$$

The nonregular locus of $\mathsf{v}^{|F|}$ consists of three irreducible components

$$\begin{eqnarray}\displaystyle & C_{1}=\{\text{x}=\text{y}=\text{z}=0\},\qquad C_{2}=\{\text{x}=\text{y}={\it\pi}=0\}, & \displaystyle \nonumber\\ \displaystyle & C_{3}=\{\text{y}=\text{z}={\it\pi}=0\}. & \displaystyle \nonumber\end{eqnarray}$$

Let $\mathsf{v}_{1}\rightarrow \mathsf{v}^{|F|}$ be the blowup along $C_{1}$. Then the nonregular locus of $\mathsf{v}_{1}$ is exactly the union of the strict transforms $C_{2}^{\prime }$ and $C_{3}^{\prime }$ of $C_{2}$ and $C_{3}$. Moreover, the singularities of $\mathsf{v}_{1}$ are two trivial families of the $A_{1}$-singularity over $\mathbb{A}_{k}^{1}$. Let $\mathsf{v}_{2}\rightarrow \mathsf{v}_{1}$ be the blowup along $C_{2}^{\prime }$ and $C_{3}^{\prime }$. Then $\mathsf{v}_{2}$ is regular. If $A_{2}$ and $A_{3}$ are the exceptional prime divisors over $C_{2}^{\prime }$ and $C_{3}^{\prime }$ respectively, then the smooth locus of $\mathsf{v}_{2}\rightarrow D$ in the special fiber is the disjoint union of open subsets $A_{2}^{\prime }\subset E_{2}$ and $A_{3}^{\prime }\subset E_{3}$ with $A_{2}^{\prime }\cong A_{3}^{\prime }\cong \mathbb{A}_{k}^{2}$. Since the morphism $\mathsf{v}_{2}\rightarrow \mathsf{v}^{|F|}$ is crepant and the $G$-action on its exceptional locus is trivial,

$$\begin{eqnarray}M_{\text{st},G}(\mathsf{v}^{|F|})=M_{\text{st},G}(\mathsf{v}_{2})=[A_{2}^{\prime }\sqcup A_{3}^{\prime }]=2\mathbb{L}^{2}\end{eqnarray}$$

and

$$\begin{eqnarray}M_{\text{st}}^{G}(\mathsf{v})=M_{\text{st}}^{G,E_{1}}(\mathsf{v})+M_{\text{st}}^{G,E_{2}}(\mathsf{v})=\mathbb{L}^{2}+3\mathbb{L},\end{eqnarray}$$

as desired.

10 A wild nonlinear example

In this section, we verify Conjecture 7.3 for an example of wild nonlinear actions.

Suppose that $k$ has characteristic two. Let $V:=\text{Spec}\,k[[{\it\pi}]][x,y]$, on which the group $G=\{1,g\}\cong \mathbb{Z}/2\mathbb{Z}$ acts by the transposition of $x$ and $y$, and $\mathsf{v}:=\text{Spec}\,k[[{\it\pi}]][x,y]/(x+y+xy)$. The completion of $\mathsf{v}$ at the origin $o\in \mathsf{v}_{0}\subset V_{0}$ gives

$$\begin{eqnarray}\text{Spec}\,k[[{\it\pi},x]],\end{eqnarray}$$

with the $G$-action by

$$\begin{eqnarray}xg=\frac{x}{1+x}=x+x^{2}+x^{3}+\cdots \,.\end{eqnarray}$$

The invariant subring of $k[[{\it\pi},x]]$ is

$$\begin{eqnarray}k[[{\it\pi},x]]^{g}=k\left[\!\left[{\it\pi},\frac{x^{2}}{1+x}\right]\!\right].\end{eqnarray}$$

Since

$$\begin{eqnarray}k[[x]]=\frac{k[[\frac{x^{2}}{1+x}]][X]}{\left\langle F(X)\right\rangle },\qquad F(X):=X^{2}+\frac{x^{2}}{1+x}X+\frac{x^{2}}{1+x},\end{eqnarray}$$

the different of $k[[x]]/k[[\frac{x^{2}}{1+x}]]$ is

$$\begin{eqnarray}\left\langle F^{\prime }(x)\right\rangle =\left\langle x^{2}\right\rangle\end{eqnarray}$$

(see [Reference SerreSer79, page 56, Corollary 2]). Let $Z\subset V$ be the zero section of $V\rightarrow D$, defined by the ideal $\left\langle x,y\right\rangle \subset k[[{\it\pi}]][x,y]$. We regard $Z$ as a prime divisor on $\mathsf{v}$. Note that $2Z$ is defined by $x+y=0$.

If we put $\mathfrak{v}=(\mathsf{v},{\it\delta}=-2Z,o)$ and $\mathfrak{x}=(\mathsf{x},0,\bar{o})$, with $\bar{o}$ the image of $o$, then the quotient morphism $q:\mathfrak{v}\rightarrow \mathfrak{x}$ is crepant. We obviously have

$$\begin{eqnarray}M_{\text{st}}(\mathfrak{x})=1.\end{eqnarray}$$

Next, we will verify that $M_{\text{st}}^{G}(\mathfrak{v})=1$. For the trivial $G$-cover $E_{1}=D\sqcup D\rightarrow D$, since $\mathsf{v}^{|D|}=\mathsf{v}$, from Proposition 2.9, we have

$$\begin{eqnarray}M_{\text{st}}^{G,E_{1}}(\mathfrak{v})=\frac{\mathbb{L}-1}{\mathbb{L}^{3}-1}=\frac{1}{\mathbb{L}^{2}+\mathbb{L}+1}.\end{eqnarray}$$

Let $E=F=\text{Spec}\,k[[{\it\rho}]]$ be any nontrivial $G$-cover of $D=\text{Spec}\,k[[{\it\pi}]]$. The associated tuning module ${\rm\Xi}_{F}$ is generated by two elements ${\it\alpha}_{1}$ and ${\it\alpha}_{2}$ given by

$$\begin{eqnarray}{\it\alpha}_{1}:x\mapsto 1,\,y\mapsto 1\end{eqnarray}$$

and

$$\begin{eqnarray}{\it\alpha}_{2}:x\mapsto {\it\rho},\,y\mapsto {\it\rho}g.\end{eqnarray}$$

Let $\text{x}$ and $\text{y}$ be the dual basis of ${\it\alpha}_{1}$ and ${\it\alpha}_{2}$. Then $u^{\ast }$ is given by

$$\begin{eqnarray}\displaystyle k[[{\it\pi}]][x,y] & \rightarrow & \displaystyle k[[{\it\rho}]][\text{x},\text{y}]\nonumber\\ \displaystyle x & \mapsto & \displaystyle \text{x}+{\it\rho}\text{y}\nonumber\\ \displaystyle y & \mapsto & \displaystyle \text{x}+({\it\rho}g)\text{y}.\nonumber\end{eqnarray}$$

Therefore, $\mathsf{v}^{\left\langle F\right\rangle }$ and $\mathsf{v}^{|F|}$ are defined by

$$\begin{eqnarray}\displaystyle & & \displaystyle (\text{x}+{\it\rho}\text{y})(\text{x}+({\it\rho}g)\text{y})+(\text{x}+{\it\rho}\text{y})+(\text{x}+({\it\rho}g)\text{y})\nonumber\\ \displaystyle & & \displaystyle \quad =\text{x}^{2}+\text{Nr}({\it\rho})\text{y}^{2}+\text{Tr}({\it\rho})\text{y}(1+\text{x})\nonumber\\ \displaystyle & & \displaystyle \quad =0.\nonumber\end{eqnarray}$$

The $G$-action on $k[[{\it\rho}]][\text{x},\text{y}]$ is given by

(10.1)$$\begin{eqnarray}\text{x}g=\text{x},\qquad \text{y}g=\frac{{\it\rho}g}{{\it\rho}}\text{y}.\end{eqnarray}$$

The pullback of $2Z$ to $\mathsf{v}^{\left\langle F\right\rangle }$ is defined by $\text{Tr}({\it\rho})\text{y}$. Let $S:=\mathsf{v}_{0}^{|F|}$, regarded as a prime divisor on $\mathsf{v}^{|F|}$, and let $B$ be the prime divisor on $\mathsf{v}^{|F|}$ such that $2B$ is defined by $\text{y}=0$. From Corollary 8.2, the boundary ${\it\delta}^{|F|}$ of $\mathfrak{v}^{|F|}$ is

$$\begin{eqnarray}-4nS-2B,\end{eqnarray}$$

with $n\in \mathbb{Z}_{{>}0}$ given by $\left\langle \text{Tr}({\it\rho})\right\rangle =\left\langle {\it\pi}^{n}\right\rangle$. The center of $\mathfrak{v}^{|F|}$ is $\mathsf{v}_{0}^{|F|}$. Hence,

$$\begin{eqnarray}M_{\text{st},G}(\mathfrak{v}^{|F|})=M_{\text{st},G}(\mathsf{v}^{|F|},-2B)\mathbb{L}^{-2n}.\end{eqnarray}$$

Let us now consider the case $n=1$. The variety $\mathsf{v}^{|F|}$ has two $A_{1}$-singularities at

$$\begin{eqnarray}(\text{x},\text{y},{\it\pi})=(0,0,0),\,(0,1,0).\end{eqnarray}$$

Blowing them up, we get a crepant morphism $\tilde{\mathsf{v}}^{|F|}\rightarrow \mathsf{v}^{|F|}$. Let $N_{0}$ and $N_{1}$ be the exceptional prime divisors over $(0,0,0)$ and $(0,1,0)$ respectively. The $G$-action on $N_{0}$ is trivial, and the one on $N_{1}$ is linear. Let $\tilde{B}\subset \tilde{\mathsf{v}}^{|F|}$ be the strict transform of $B$. The morphism $(\tilde{\mathsf{v}}^{|F|},-2\tilde{B}-N_{0})\rightarrow (\mathsf{v}^{|F|},-2B)$ is crepant. Since $\tilde{\mathsf{v}}^{|F|}$ is regular, the smooth locus of $\tilde{\mathsf{v}}^{|F|}\rightarrow D$ in the special fiber is