Stokes structure of mild difference modules

We set

$\displaystyle\widetilde{\mathbb{C}}=\{(t,e^{{\tt{i}}\theta})\in\mathbb{C}\times S^{1}\mid t=|t|e^{{\tt{i}}\theta}\},$

which is called the real blowing up of $\mathbb{C}$ at the origin. When we distinguish a variable such as $t$, we use the notation $\widetilde{\mathbb{C}}_{t}$. There are maps $\varpi\colon\widetilde{\mathbb{C}}\longrightarrow\mathbb{C}$, $\widetilde{\jmath}\colon\mathbb{C}^{*}\hookrightarrow\widetilde{\mathbb{C}}$, and $\widetilde{\imath}\colon S^{1}\hookrightarrow\widetilde{\mathbb{C}}$ defined by $\varpi(t,e^{{\tt{i}}\theta})=t$, $\widetilde{\jmath}(t)=(t,t/|t|)$, and $\widetilde{\imath}(e^{{\tt{i}}\theta})=(0,e^{{\tt{i}}\theta})$, respectively. We sometimes denote the boundary of $\widetilde{\mathbb{C}}_{t}$ by $S^{1}_{t}$ to distinguish a variable such as $t$.

Let $\Delta\coloneqq\{t\in\mathbb{C}\mid|t|<1\}$ be a unit open disc. We set $\Delta^{*}\coloneqq\Delta\setminus\{0\}$ and $\widetilde{\Delta}\coloneqq\{(t,e^{{\tt{i}}\theta})\in\widetilde{\mathbb{C}}\mid t\in\Delta\}.$ Let $\varpi_{\Delta}\colon\widetilde{\Delta}\to\Delta$ be the projection. When we distinguish a variable such as $t$, we use the notations $\Delta_{t}$, $\Delta_{t}^{*}$ and $\widetilde{\Delta}_{t}$. The natural inclusions are denoted by $\jmath_{\Delta}\colon\Delta^{*}\to\Delta$, $\widetilde{\imath}_{\Delta}\colon S^{1}\to\widetilde{\Delta}$, and $\widetilde{\jmath}_{\Delta}\colon\Delta^{*}\to\widetilde{\Delta}$. Let $\varphi_{t}\colon\Delta_{t}\to\mathbb{C}_{t}$ be a holomorphic function defined as

$\displaystyle\varphi_{t}(t)\coloneqq\frac{t}{1+t}.$

The map uniquely extends to a continuous map $\widetilde{\varphi}_{t}\colon\widetilde{\Delta}_{t}\longrightarrow\widetilde{\mathbb{C}}_{t}$.

Let $\mathbb{C}_{s}$ be a complex plane with a coordinate $s$. When we use the two complex variables $s$ and $t$, we implicitly assume the relation

(1)

$\displaystyle s=t^{-1}.$

In other words, we consider the complex projective line $\mathbb{P}^{1}$ covered by two open subsets $\mathbb{C}_{s}$ and $\mathbb{C}_{t}$ with the relation (1). Let $\varphi_{s}\colon\mathbb{C}_{s}\to\mathbb{C}_{s}$ be the map defined as

$\displaystyle\varphi_{s}(s)=s+1.$

Under the relation (1), the map $\varphi_{s}$ coincides with the map $\varphi_{t}$ on the domain $\Delta_{t}^{*}=\{s\in\mathbb{C}_{s}\mid|s|>1\}$. Hence the maps $\varphi_{s}$ and $\varphi_{t}$ are glued to an automorphism on $\mathbb{P}^{1}$.

Using the notations in §2.1.1, set

$\displaystyle\widetilde{\mathscr{O}}\coloneqq\widetilde{\imath}^{-1}\widetilde{\jmath}_{*}\mathscr{O}_{\mathbb{C}^{*}},$

which is a sheaf on $S^{1}$. There are subsheaves $\mathscr{A}^{\leqslant 0}$, and $\mathscr{A}^{<0}$ in $\widetilde{\mathscr{O}}$ characterized by their asymptotic behavior as follows (see [Sabbah] for precise definitions):

• •

  $\mathscr{A}^{\leqslant 0}$ is the sheaf of holomorphic functions which are of moderate growth.

• •

  $\mathscr{A}^{<0}$ is the sheaf of holomorphic functions which are of rapid decay.

To emphasize the coordinate function such as $t$, we use the notation $\widetilde{\mathscr{O}}_{t}$, $\mathscr{A}_{t}^{\leqslant 0}$, e.t.c.

The map $\widetilde{\varphi}_{t}\colon\widetilde{\Delta}_{t}\to\widetilde{\mathbb{C}}_{t}$ defined in §2.1.2 naturally induces a morphism $\widetilde{\varphi}_{t}^{*}\colon\widetilde{\jmath}_{*}\mathscr{O}_{\mathbb{C}^{*}}\longrightarrow\widetilde{\varphi}_{t*}\widetilde{\jmath}_{\Delta*}\mathscr{O}_{\Delta^{*}}$ defined as the composition with the map $\varphi$ restricted to a suitable open subset. We then set

$\displaystyle\widetilde{\phi}_{t}\coloneqq\widetilde{\imath}^{-1}(\widetilde{\varphi}_{t}^{*})\colon\widetilde{\mathscr{O}}_{t}\to\widetilde{\mathscr{O}}_{t},$

where we used the relation $\widetilde{\varphi}_{t}\circ\widetilde{\imath}_{\Delta}=\widetilde{\imath}$. Note that the subsheaves $\mathscr{A}^{\leqslant 0}_{t}$ and $\mathscr{A}^{<0}_{t}$ are invariant under the automorphism $\widetilde{\phi}_{t}$. We then set $\nabla_{\widetilde{\phi}_{t}}\coloneqq\widetilde{\phi}_{t}-\mathrm{id}_{\widetilde{\mathscr{O}}}.$ The restrictions of $\nabla_{\widetilde{\phi}_{t}}$ to $\mathscr{A}^{\leqslant 0}_{t}$ and $\mathscr{A}^{<0}_{t}$ are also denoted by the same symbol. Set

(2)

$\displaystyle u\coloneqq\exp(2\pi{\tt{i}}s)=\exp(2\pi{\tt{i}}t^{-1})$

and $v=u^{-1}$. Then, we have the equality $\nabla_{\widetilde{\phi}_{t}}(u)=0$.

Let $\mathbb{C}[u^{\pm 1}]$ denote the ring of Laurent polynomials in $u$.

Fix $m\in\mathbb{Z}_{>0}$. Recall the notations in §2.3 such as $\tau$ and $S^{1}_{\tau}$ corresponding to $m$. We recall that there is a sheaf $\widetilde{\mathscr{O}}_{\tau}$ on ${S}^{1}_{\tau}$ defined in §2.2.1.

Let $\rho\colon S^{1}_{\tau}\to S^{1}_{t}$ be the map defined in §2.3. Let $\rho^{*}\colon\widetilde{\mathscr{O}}_{t}\to\rho_{*}\widetilde{\mathscr{O}}_{\tau}$ be the adjoint morphism. We regard $\widetilde{\mathscr{O}}_{t}$ as a subsheaf of $\rho_{*}\widetilde{\mathscr{O}}_{\tau}$ by this $\rho^{*}$.

We shall define partial orders on the space of sections of sheaves introduced above.

We shall define the notion of a pre-Stokes filtration on a $\mathscr{A}_{{\mathrm{per}}}$-module as follows.

A pair $(\mathscr{L},\mathscr{L}_{\bullet})$ of a $\mathscr{A}_{{\mathrm{per}}}$-module and a pre-Stokes filtration on it is called a pre-Stokes filtered $\mathscr{A}_{{\mathrm{per}}}$-module. A morphism between two pre-Stokes filtered $\mathscr{A}_{{\mathrm{per}}}$-modules $(\mathscr{L},\mathscr{L}_{\bullet})$, and $(\mathscr{L}^{\prime},\mathscr{L}_{\bullet}^{\prime})$ is a morphism $\lambda\colon\mathscr{L}\to\mathscr{L}^{\prime}$ of $\mathscr{A}_{{\mathrm{per}}}$-modules such that $\lambda_{|U}(\mathscr{L}_{\leqslant\mathfrak{a}})\subset\mathscr{L}^{\prime}_{\leqslant\mathfrak{a}}$ for any $\mathfrak{a}\in\mathscr{I}{(U)}$.

For each $\mathfrak{a}\in\mathscr{I}(U)$ and a pre-Stokes filtered $\mathscr{A}_{{\mathrm{per}}}$-module $\mathscr{L}=(\mathscr{L},\mathscr{L}_{\bullet})$, we set

$\displaystyle\mathscr{L}_{<\mathfrak{a}}\coloneqq\sum_{\mathfrak{b}<_{U}\mathfrak{a}}\mathscr{L}_{\leqslant\mathfrak{b}}$

and ${\mathrm{gr}}_{\mathfrak{a}}\mathscr{L}\coloneqq\mathscr{L}_{\leqslant\mathfrak{a}}/\mathscr{L}_{<\mathfrak{a}}$. By condition (3) in the Definition, we have the inclusion $\mathscr{A}^{<0}_{|U}\cdot\mathscr{L}_{\leqslant\mathfrak{a}}\subset\mathscr{L}_{<\mathfrak{a}}.$ Hence it is natural to regard ${\mathrm{gr}}_{\mathfrak{a}}\mathscr{L}$ as a module over $\mathscr{A}^{\leqslant 0}_{{\mathrm{per}}|U}/\mathscr{A}_{{\mathrm{per}}|U}^{<0}=\mathbb{C}_{U}$. Again by condition (3), we have an isomorphism

(5)

$\displaystyle u^{n}\colon{\mathrm{gr}}_{\mathfrak{a}}\mathscr{L}\xrightarrow{\sim}{\mathrm{gr}}_{\mathfrak{a}+2\pi{\tt{i}}sn}\mathscr{L}$

for any $n\in\mathbb{Z}$. Hence $\bigoplus_{\mathfrak{a}\in\mathscr{I}(U)}{\mathrm{gr}}_{\mathfrak{a}}\mathscr{L}$ is naturally equipped with the structure of a sheaf of $\mathbb{C}[u^{\pm 1}]$-modules over $U$. Then the sheaf $\Sigma(\mathscr{L})$ of sets defined as

$\displaystyle\Sigma(\mathscr{L})(U)\coloneqq\{\mathfrak{a}\in\mathscr{I}(U)\mid{\mathrm{gr}}_{\mathfrak{a}}\mathscr{L}\neq 0\},$

is equipped with the the action of the additive group $2\pi{\tt{i}}s\mathbb{Z}_{S^{1}}$ in a natural way.

For two (pre-)Stokes filtered $\mathscr{A}_{\mathrm{per}}$-modules $(\mathscr{L}_{1},\mathscr{L}_{1,\bullet})$ and $(\mathscr{L}_{2},\mathscr{L}_{2,\bullet})$ we can define the (pre-)Stokes filtrations on the tensor product $\mathscr{L}_{1}\otimes\mathscr{L}_{2}$ and the sheaf of internal-homs ${\mathscr{H}\!\!om}(\mathscr{L}_{1},\mathscr{L}_{2})$ as follows:

	$\displaystyle(\mathscr{L}_{1}\otimes\mathscr{L}_{2})_{\leqslant\mathfrak{a}}$	$\displaystyle\coloneqq\sum_{\mathfrak{b}\in\mathscr{I}(U)}\mathscr{L}_{1,\leqslant\mathfrak{b}}\otimes_{\mathscr{A}_{\mathrm{per}}^{\leqslant 0}}\mathscr{L}_{2,\leqslant\mathfrak{a}-\mathfrak{b}},$
	$\displaystyle{\mathscr{H}\!\!om}(\mathscr{L}_{1},\mathscr{L}_{2})_{\leqslant\mathfrak{a}}$	$\displaystyle\coloneqq\sum_{\mathfrak{b}\in\mathscr{I}(U)}{\mathscr{H}\!\!om}_{\mathscr{A}_{\mathrm{per}}^{\leqslant 0}}(\mathscr{L}_{1,\leqslant\mathfrak{b}},\mathscr{L}_{2,\leqslant\mathfrak{a}+\mathfrak{b}}).$

We also use the notation $\mathscr{E}nd(\mathscr{L})\coloneqq{\mathscr{H}\!\!om}(\mathscr{L},\mathscr{L})$.

Recall that a difference ring is a pair $(R,\Phi)$ of a commutative ring $R$ with a unit $1$ and a ring automorphism $\Phi$ on $R$. If $R$ is a field, then $(R,\Phi)$ is called a difference field. Let $\mathscr{O}_{t}(*0)=\mathscr{O}_{\mathbb{C}_{t}}(*0)_{0}=\mathbb{C}(\!\{t\}\!)$ be the field of convergent Laurent series in $t$. Let $\phi_{t}\colon\mathscr{O}_{t}(*0)\to\mathscr{O}_{t}(*0)$ be the automorphism defined by $\phi_{t}(f)(t)\coloneqq f(t(1+t)^{-1})$. Then the pair $(\mathscr{O}_{t}(*0),\phi_{t})$ is an example of a difference field. The pair $(\widehat{\mathscr{O}}_{t}(*0),\widehat{\phi}_{t})$ of the formal completion $\widehat{\mathscr{O}}_{t}(*0)\coloneqq\mathbb{C}(\!(t)\!)$ of $\mathscr{O}_{t}(*0)$ and the automorphism $\widehat{\phi}_{t}$ defined in a similar way as $\phi_{t}$ is also an instance of a difference field.

A(n invertible) difference module over a difference ring $(R,\Phi)$ is a pair $(M,\Psi)$ of a finitely generated $R$-module $M$ and an automorphism $\Psi\colon M\to M$ of abelian groups such that $\Psi(rm)=\Phi(r)\Psi(m)$ for any $r\in R$ and $m\in M$. We abbreviate $(M,\Psi)$ by omitting $\Psi$ if there is no fear of confusion. The category of difference modules over $(\mathscr{O}_{t}(*0),\phi_{t})$ is denoted by ${\mathsf{Diffc}}$, which is an abelian category.

For a difference module $(\mathscr{M},\psi)$ over $(\mathscr{O}_{t}(*0),\phi_{t})$, the automorphism $\psi$ naturally extends to an automorphism $\widehat{\psi}$ on $\widehat{\mathscr{M}}\coloneqq\widehat{\mathscr{O}}_{t}(*0)\otimes_{\mathscr{O}_{t}{(*0)}}\mathscr{M}$. The pair $(\widehat{\mathscr{M}},\widehat{\psi}{})$ is called the formal completion of $(\mathscr{M},\psi{})$, which is a difference module over $\widehat{\mathscr{O}}_{t}(*0)\coloneqq\mathbb{C}(\!(t)\!)$.

For a constant matrix $G\in\mathrm{End}(\mathbb{C}^{r})$, we let $\mathscr{R}_{G}=(\mathscr{O}_{t}(*0)^{\oplus r},\psi_{G})$ be a difference module defined by $\psi_{G}=(1+t)^{-G}\phi_{t}^{\oplus r}$, where we set $(1+t)^{-G}=\exp(-G\log(1+t))$ with $\log(1+t)=-\sum_{n=1}^{\infty}n^{-1}(-t)^{n}$. The formal completion of $\mathscr{R}_{G}$ is denoted by $\widehat{\mathscr{R}}_{G}$.

Fix a positive integer $m$. We use the notations in §2.3. There is a natural inclusion $\rho^{*}\colon\mathscr{O}_{t}(*0)\to\mathscr{O}_{\tau}(*0)$. Regard $t=\tau^{m}$ as an element of $\mathscr{O}_{\tau}(*0)$. The pair of the field $\mathscr{O}_{\tau}(*0)$ and an automorphism $\phi_{\tau}\colon\mathscr{O}_{\tau}(*0)\to\mathscr{O}_{\tau}(*0)$ defined as $\phi_{\tau}(\tau)\coloneqq\tau(1+t)^{-1/m}$ forms a difference field. The notion of formal completion is defined analogously as in the case of $(\mathscr{O}_{t}(*0),\phi_{t})$. For a difference module $\mathscr{M}$ over $(\mathscr{O}_{t}{(*0)},\phi_{t})$, the pull back $\rho^{*}\mathscr{M}\coloneqq\mathscr{O}_{\tau}(*0)\otimes\mathscr{M}$ is a difference module over $(\mathscr{O}_{\tau}(*0),\phi_{\tau})$ in a natural way.

Fix a positive integer $m$ as in §4.1.3. Let $\mathfrak{a}=\mathfrak{a}(\tau)$ be as (4) in Definition 3.1. We note that $\exp(\phi_{\tau}(\mathfrak{a})-\mathfrak{a})$ is a well defined element in $\mathscr{O}_{\tau}$. We define a difference module $\mathscr{E}^{\mathfrak{a}}\coloneqq(\mathscr{O}_{\tau}{(*0)},\psi_{\mathfrak{a}})$ by $\psi_{\mathfrak{a}}(1)\coloneqq\exp(\phi_{\tau}(\mathfrak{a})-\mathfrak{a})$. The formal completion of $\mathscr{E}^{\mathfrak{a}}$ will be denoted by $\widehat{\mathscr{E}}^{\mathfrak{a}}{}$. For two such difference modules $\mathscr{E}^{\mathfrak{a}}$ and $\mathscr{E}^{\mathfrak{b}}$, we have $\mathscr{E}^{\mathfrak{a}}\simeq\mathscr{E}^{\mathfrak{b}}$ if and only if $\mathfrak{a}-\mathfrak{b}\in 2\pi{\tt{i}}s\mathbb{Z}$.

We recall the definition of mild difference modules: