Wick rotation for D-modules

TL;DR

This paper extends the classical Wick rotation technique to the setting of D-modules and higher codimensional submanifolds, broadening its applicability in mathematical physics and algebraic geometry.

Contribution

It introduces a novel generalization of Wick rotation applicable to D-modules and complex submanifolds of higher codimension.

Findings

Extended Wick rotation to D-modules.

Applied Wick rotation to higher codimensional submanifolds.

Provided new tools for mathematical physics and algebraic geometry.

Abstract

We extend the classical Wick rotation to D-modules and higher codimensional submanifolds.

Equations

$${\tau }_X:TX\to X,{\pi }_X:T^{\ast }X\to X$$

$$\displaystyle\gamma^{\circ}=\{(x;\xi)\in E^{*};\langle\xi,v\rangle\geq 0\mbox{ for all $x\in M,v\in\gamma_{x}$}\}.$$

$$\displaystyle\nu_{M}\colon\mathrm{D}^{\mathrm{b}}({\bf k}_{X})\xrightarrow[]{}\mathrm{D}^{\mathrm{b}}_{{\mathbb{R}}^{+}}({\bf k}_{T_{M}X})\mbox{ specialization along $M$},$$

$$\displaystyle\mu_{M}\colon\mathrm{D}^{\mathrm{b}}({\bf k}_{X})\xrightarrow[]{}\mathrm{D}^{\mathrm{b}}_{{\mathbb{R}}^{+}}({\bf k}_{T^{*}_{M}X})\mbox{ microlocalization along $M$},$$

$$\mu hom:{\mathrm{D}}^{\mathrm{b}}({\mathbf{k}}_X)\times {\mathrm{D}}^{\mathrm{b}}({\mathbf{k}}_X{)}^{\mathrm{op}}\to {\mathrm{D}}_{{\mathbb{R}}^{+}}^{\mathrm{b}}({\mathbf{k}}_{T^{\ast }X}).$$

$$\mathrm{R}{{\pi }_M}_{\ast }\mathrm{R}{\Gamma }_{\lambda }({\mu }_M(F)){\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\omega }_{X/M}\simeq \mathrm{R}{{\tau }_M}_{\ast }\mathrm{R}{\Gamma }_{\mathrm{Int}({\lambda }^{\circ a})}({\nu }_M(F)).$$

$$T^{\ast }M\hookrightarrow T^{\ast }{T}_M^{\ast }X\simeq T_{T_M^{\ast }X}{T}^{\ast }X.$$

$$\mathrm{R}{{\pi }_N}_{\ast }\mathrm{R}{g_{Nd}}_{\ast }(g_{N\pi }^{!}{\mu }_M(F))$$

$$\mathrm{R}{{\pi }_N}_{\ast }{\mu }_N(g^{!}F)$$

$$M=\{y^{\prime }=y^{\prime \prime }=0)\},$$

$$L=\{x^{\prime \prime }=y^{\prime \prime }=0\},$$

$$N=\{x^{\prime \prime }=y^{\prime }=y^{\prime \prime }=0)\},$$

$$g_d:(x^{\prime },y^{\prime };{\xi }^{\prime },{\xi }^{\prime \prime },{\eta }^{\prime },{\eta }^{\prime \prime })\mapsto (x^{\prime },y^{\prime };{\xi }^{\prime },{\eta }^{\prime }),$$

$${\mathcal{A}}_M:={\mathcal{O}}_X{\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\mathbb{C}}_M,{\mathcal{B}}_M:={\mathrm{R}\mathcal{H}\text{mspace}-3.0muom}_{\text{raise}4.52083ptto0.70004pt}({\mathrm{D}}_X^{\prime }{\mathbb{C}}_M,{\mathcal{O}}_X),{\mathcal{C}}_M:=\mu hom({\mathrm{D}}_X^{\prime }{\mathbb{C}}_M,{\mathcal{O}}_X).$$

$${\mathcal{B}}_M\simeq \mathrm{R}{\Gamma }_M({\mathcal{O}}_X){\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\omega }_M\simeq H_M^n({\mathcal{O}}_X){\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\mathrm{or}}_M,$$

$${\mathcal{C}}_M\simeq {\mu }_M({\mathcal{O}}_X){\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\omega }_M\simeq H^n({\mu }_M({\mathcal{O}}_X)){\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\mathrm{or}}_M.$$

$$\displaystyle\rm spec\colon\Gamma(M;\mathscr{B}_{M})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}\Gamma(T^{*}_{M}X;\mathscr{C}_{M}).$$

$$\mathcal{M}\simeq {\mathcal{D}}_X^{N_0}/{\mathcal{D}}_X^{N_1}\cdot P_0$$

$${\mathcal{M}}^{\bullet }:=0\to {\mathcal{D}}_X^{N_r}\to \cdots \to {\mathcal{D}}_X^{N_1}\stackrel{\cdot P_0}{\to }{\mathcal{D}}_X^{N_0}\to 0.$$

$$\mathcal{S}\text{mspace}-2.5muol(\mathcal{M}):={\mathrm{R}\mathcal{H}\text{mspace}-3.0muom}_{\text{raise}4.52083ptto0.70004pt{\mathcal{D}}_X}(\mathcal{M},{\mathcal{O}}_X).$$

$$\mathcal{S}\text{mspace}-2.5muol(\mathcal{M})\simeq 0\to {\mathcal{O}}_X^{N_0}\stackrel{P_0\cdot }{\to }{\mathcal{O}}_X^{N_1}\to \cdots {\mathcal{O}}_X^{N_r}\to 0,$$

$$\displaystyle{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{A}_{M})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{B}_{M}).$$

$$\displaystyle f^{-1}{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{O}_{X})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathrm{R}\Gamma_{Y}\mathscr{O}_{X})\,[2d].$$

$${\mathrm{hypchar}}_M(\mathcal{M})=T^{\ast }M\cap C_{T_M^{\ast }X}(\mathrm{char}(\mathcal{M}))$$

$$T_N^{\ast }M\cap {\mathrm{hypchar}}_M(\mathcal{M})\subset T_M^{\ast }M,$$

$$\displaystyle\left\{\parbox{279.86023pt}{ there exist an open neighborhood $U$ of $x_{0}$ in $M$ and an open conic neighborhood $\gamma$ of $\theta_{0}\in{\mathbb{R}}^{n}$ such that $\sigma(P)(x;\theta+\sqrt{-1}\eta)\not=0$ for all $\eta\in{\mathbb{R}}^{n}$, $x\in U$ and $\theta\in\gamma$. }\right.$$

$$\displaystyle\left\{\parbox{279.86023pt}{ there exists an open neighborhood $U$ of $x_{0}$ in $M$ such that $\sigma(P)(x;\theta_{0}+\sqrt{-1}\eta)\not=0$ for all $\eta\in{\mathbb{R}}^{n}$, and $x\in U$. }\right.$$

$$\displaystyle\parbox{301.38794pt}{$Y$ is non characteristic for $\mathscr{M}$}.$$

$$\mathrm{R}{g_{Nd}}_{!}({\omega }_{N/M}{\otimes }_{\text{raise}4.52083ptto-0.70004pt}{g}_{N\pi }^{-1}{\mathrm{R}\mathcal{H}\text{mspace}-3.0muom}_{\text{raise}4.52083ptto0.70004pt{\mathcal{D}}_X}(\mathcal{M},{\mu }_M({\mathcal{O}}_X)))\simeq {\mu }_N({\omega }_{L/X}{\otimes }_{\text{raise}4.52083ptto-0.70004pt}{g}^{-1}{\mathrm{R}\mathcal{H}\text{mspace}-3.0muom}_{\text{raise}4.52083ptto0.70004pt{\mathcal{D}}_X}(\mathcal{M},{\mathcal{O}}_X)).$$

$$\mathrm{R}{g_{Nd}}_{!}{g}_{N\pi }^{-1}({\omega }_{X/M}{\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\mathrm{R}\mathcal{H}\text{mspace}-3.0muom}_{\text{raise}4.52083ptto0.70004pt{\mathcal{D}}_X}(\mathcal{M},{\mu }_M({\mathcal{O}}_X)))\simeq {\mu }_N({\omega }_{L/N}{\otimes }_{\text{raise}4.52083ptto-0.70004pt}{g}^{-1}{\mathrm{R}\mathcal{H}\text{mspace}-3.0muom}_{\text{raise}4.52083ptto0.70004pt{\mathcal{D}}_X}(\mathcal{M},{\mathcal{O}}_X)).$$

$${f_{Nd}}_{!}{f}_{N\pi }^{-1}{\mathrm{R}\mathcal{H}\text{mspace}-3.0muom}_{\text{raise}4.52083ptto0.70004pt{\mathcal{D}}_X}(\mathcal{M},{\mathcal{C}}_M)\simeq {\mathrm{R}\mathcal{H}\text{mspace}-3.0muom}_{\text{raise}4.52083ptto0.70004pt{\mathcal{D}}_Y}(f^D\mathcal{M},{\mathcal{C}}_N).$$

$${\tilde{\mathcal{B}}}_N=\mathrm{R}{\Gamma }_N({\mathcal{O}}_X){\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\mathrm{or}}_N[n]\simeq H_N^n({\mathcal{O}}_X){\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\mathrm{or}}_N.$$

$${\tilde{\mathcal{B}}}_N\simeq {\Gamma }_N{\mathcal{B}}_M{\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\mathrm{or}}_{N/M}\simeq {\Gamma }_N{\mathcal{B}}_M{\otimes }_{\text{raise}4.52083ptto-0.70004pt}{\omega }_{M/N}[-d].$$

Full text

Key words: causal manifolds, microlocal sheaf theory, hyperbolic D-modules, hyperfunctions, Wick rotationMSC: 35A27, 58J15, 58J45, 81T20Research supported by the ANR-15-CE40-0007 “MICROLOCAL”.

1 Introduction

Let $M$ be a real analytic manifold of the type $N\times{\mathbb{R}}$ and let $X=Y\times{\mathbb{C}}$ be a complexification of $M$. Consider a differential operator $P$ on $X$ such that $P$ is hyperbolic on $M$ with respect to the direction $N\times\{0\}$, a typical example being the wave operator on a spacetime. Denote by $L$ the real manifold $N\times\sqrt{-1}{\mathbb{R}}$. It may happen, and it happens for the wave operator, that $P$ is elliptic on $L$. Passing from $M$ to $L$ is called the Wick rotation by physicists who deduce interesting properties of $P$ on $M$ from the study of $P$ on $L$.

In the situation above, we had $codim_{M}N=codim_{L}N=1$. In this paper, we treat the general case of two real analytic manifolds $M$ and $L$ in $X$, $X$ being a complexification of both $M$ and $L$, such that the intersection $N\mathbin{:=}M\cap L$ is clean, and we consider a coherent $\mathscr{D}_{X}$-module $\mathscr{M}$ which is hyperbolic with respect to $M$ on $N$ and elliptic on $L$. The main result is Theorem 3.10 which describes an isomorphism in a neighborhood of $N$ between the complex of hyperfunction solutions of $\mathscr{M}$ on $L$ defined in a given cone $\gamma\subset T_{N}L$ and the complex of hyperfunction solutions of $\mathscr{M}$ on $M$ with wave front set in a cone $\lambda\subset T^{*}_{M}X$ associated with $\gamma$. It is also proved that this isomorphism is compatible with the boundary values morphism from $M$ to $N$ and from $L$ to $N$.

Aknowledgements This paper was initiated by a series of discussions with Christian Gérard who kindly explained us some problems associated with the classical Wick rotation (see [GW17]). We sincerely thank him for his patience and his explanations.

2 Sheaves, D-modules and wave front sets

2.1 Sheaves

We shall use the microlocal theory of sheaves of [KS90] and mainly follow its terminology. For the reader’s convenience, we recall a few notations and results.

Geometry

Let $X$ be a real manifold of class $C^{\infty}$. For a subset $A\subset X$, we denote by $\overline{A}$ its closure and by ${\rm Int}(A)$ its interior. We denote by

[TABLE]

the tangent bundle and the cotangent bundle to $X$. For a closed submanifold $M$ of $X$, we denote by $\tau_{M}\colon T_{M}X\xrightarrow[]{}M$ and $\pi_{M}\colon T^{*}_{M}X\xrightarrow[]{}M$ the normal bundle and the conormal bundle to $M$ in $X$. In particular, $T^{*}_{X}X$ is the zero-section of $T^{*}X$, that we identify with $X$.

For a vector bundle $\pi\colon E\xrightarrow[]{}X$, we identify $X$ with the zero-section, we denote by $E_{x}$ the fiber of $E$ at $x\in X$, we set $\dot{E}=E\setminus X$ and we denote by $\dot{\pi}\colon\dot{E}\xrightarrow[]{}X$ the projection. For a cone $\gamma$ in a vector bundle $E\xrightarrow[]{}X$, we set $\gamma_{x}=\gamma\cap E_{x}$, we denote by $\gamma^{a}=-\gamma$ the opposite cone and by $\gamma^{\circ}$ the polar cone in the dual vector bundle $E^{*}$,

[TABLE]

For $A\subset X$, the Whitney normal cone of $A$ along $M$, $C_{M}(A)\subset T_{M}X$, is defined in [KS90]*Def. 4.1.1.

To a morphism of manifolds $f\colon Y\xrightarrow[]{}X$, one associates the maps:

[TABLE]

where $f_{d}$ is the transpose of the tangent map $Tf\colon TY\xrightarrow[]{}Y\times_{X}TX$.

Definition 2.1**.**

Let $\Lambda$ be a closed conic subset of $T^{*}X$. One says that $f$ is non characteristic for $\Lambda$ if the map $f_{d}$ is proper on $f_{\pi}^{-1}(\Lambda)$.

Sheaves

Let ${\bf k}$ be a field. One denotes by $\mathrm{D}^{\mathrm{b}}({\bf k}_{X})$ the bounded derived category of sheaves of ${\bf k}$-vector spaces on $X$. We simply call an object of this category “a sheaf”. For a closed subset $A$ of a manifold we denote by ${\bf k}_{A}$ the constant sheaf on $A$ with stalk ${\bf k}$ extended by [math] outside of $A$. More generally, we shall identify a sheaf on $A$ and its extension by [math] outside of $A$. If $A$ is locally closed, we keep the notation ${\bf k}_{A}$ as far as there is no risk of confusion. We denote by $\omega_{X}$ the dualizing complex on $X$. Recall that $\omega_{X}\simeq\operatorname{or}_{X}\,[\dim X]$ where $\operatorname{or}_{X}$ is the orientation sheaf and $\dim X$ is the dimension of $X$. More generally, we consider the relative dualizing complex associated with a morphism $f\colon Y\xrightarrow[]{}X$, $\omega_{Y/X}=\omega_{Y}\mathbin{\otimes_{\raise 4.52083pt\hbox to-0.70004pt{}{}}}f^{-1}(\omega_{X}^{\otimes-1})$ and its inverse, $\omega_{X/Y}=\omega_{Y/X}^{\otimes-1}$. We denote by {\rm D}^{\prime}_{X}({\,\raise 1.72218pt\hbox{\scriptscriptstyle\bullet}\,})={\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}}({\,\raise 1.72218pt\hbox{\scriptscriptstyle\bullet}\,},{\bf k}_{X}) the duality functor on $X$.

We shall use freely the six Grothendieck operations on sheaves.

Microlocalization

For a closed submanifold $M$ of $X$, we have the functors

[TABLE]

Here, for a vector bundle $E\xrightarrow[]{}M$ or $E\xrightarrow[]{}X$, $\mathrm{D}^{\mathrm{b}}_{{\mathbb{R}}^{+}}({\bf k}_{E})$ is the full subcategory of $\mathrm{D}^{\mathrm{b}}({\bf k}_{E})$ consisting of conic sheaves, that is, sheaves locally constant under the ${\mathbb{R}}^{+}$-action.

The functor $\mu_{M}$, called Sato’s microlocalization functor, is the Fourier–Sato transform of the specialization functor $\nu_{M}$. The bifunctor $\mu hom$ of [KS90] is a slight generalization of $\mu_{M}$. Recall that \mu_{M}({\,\raise 1.72218pt\hbox{\scriptscriptstyle\bullet}\,})=\mu hom({\bf k}_{M},{\,\raise 1.72218pt\hbox{\scriptscriptstyle\bullet}\,}).

Let $\lambda$ be a closed convex proper cone of $T^{*}_{M}X$ containing the zero-section $M$. For $F\in\mathrm{D}^{\mathrm{b}}({\bf k}_{X})$, we have an isomorphism (see [KS90]*Th. 4.3.2):

[TABLE]

(Recall that $\lambda^{\circ a}$ is the opposite of the polar cone $\lambda^{\circ}$.)

Microsupport

To a sheaf $F$ is associated its microsupport $\operatorname{\mu supp}(F)$444$\operatorname{\mu supp}(F)$ was denoted $\operatorname{SS}(F)$ in [KS90]., a closed ${\mathbb{R}}^{+}$-conic co-isotropic subset of $T^{*}X$.

Let us recall some results that we shall use.

Theorem 2.2**.**

Let $f\colon Y\xrightarrow[]{}X$ be a morphism of real manifolds and let $F\in\mathrm{D}^{\mathrm{b}}({\bf k}_{X})$. Assume that $f$ is non characteristic for $F$, that is, for $\operatorname{\mu supp}(F)$. Then the morphism $f^{-1}F\mathbin{\otimes_{\raise 4.52083pt\hbox to-0.70004pt{}{}}}\,\omega_{Y/X}\xrightarrow[]{}f^{!}F$ is an isomorphim.

As a particular case of this result, we get a kind of Petrowski theorem for sheaves (see Theorem 2.27 below):

Corollary 2.3**.**

Let $M$ be a closed submanifold of $X$ and let $F\in\mathrm{D}^{\mathrm{b}}({\bf k}_{X})$. Assume that $T^{*}_{M}X\cap\operatorname{\mu supp}(F)\subset T^{*}_{X}X$. Then $F\mathbin{\otimes_{\raise 4.52083pt\hbox to-0.70004pt{}{}}}{\bf k}_{M}\simeq\mathrm{R}\Gamma_{M}F\mathbin{\otimes_{\raise 4.52083pt\hbox to-0.70004pt{}{}}}\operatorname{or}_{M/X}\,[\operatorname{codim}_{X}M]$.

Let $M$ be a closed submanifold of $X$. If $\Lambda\subset T^{*}X$ is a closed conic subset, its Whitney normal cone along $T^{*}_{M}X$ is a closed biconic subset of $T_{T^{*}_{M}X}T^{*}X\simeq T^{*}T^{*}_{M}X$. Moreover, there exists a natural embedding

[TABLE]

Now we consider a morphism of manifolds $g\colon L\xrightarrow[]{}X$ and let $M\subset X$ and $N\subset L$ be two closed submanifolds such that the map $g$ induces a closed embedding $g|_{N}\colon N\hookrightarrow M$. One gets the maps

[TABLE]

The next result is a particular case of [KS90]*Th. 6.7.1 in which we choose $V=T^{*}_{N}L$ and write $g\colon L\xrightarrow[]{}X$ instead of $f\colon Y\xrightarrow[]{}X$. (The reason of this change of notations is that we need to consider the complexification of the embedding $N\hookrightarrow M$ that we shall denote by $f\colon Y\hookrightarrow X$.)

Theorem 2.4**.**

Let $F\in\mathrm{D}^{\mathrm{b}}({\bf k}_{X})$ and assume

1. (a)

$g$* is non characteristic for $F$,* 2. (b)

the map $g_{N\pi}$ is non characteristic for $C_{T^{*}_{M}X}(\operatorname{\mu supp}(F))$, 3. (c)

$g_{d}^{-1}T_{N}^{*}L\cap g_{\pi}^{-1}\operatorname{\mu supp}(F)\subset L\times_{X}T^{*}_{M}X$.

Then one has the commutative diagram of natural isomorphisms on $T^{*}_{L}X$:

[TABLE]

Notation 2.5**.**

As usual, we have simply writen $\omega_{M}$ instead of $\pi^{-1}\omega_{M}$ and similarly with other locally constant sheaves.

Consider the projections

[TABLE]

One has the isomorphisms

[TABLE]

and

[TABLE]

Moreover, one easily proves:

Lemma 2.6**.**

The isomorphisms (2.23) and (2.24) are compatible with the morphisms obtained by applying $\mathrm{R}{\pi_{N}}_{*}$ to (2.17).

Lemma 2.7**.**

In the situation of Theorem 2.4 assume moreover that $g\colon L\xrightarrow[]{}X$ is a closed embedding, $N=L\cap M$ and the intersection is clean (that is, $TN=N\times_{M}TM\cap N\times_{L}TL$). Then condition (c) follows from (b).

Proof.

Let us choose a local coordinate system $(x^{\prime},x^{\prime\prime},y^{\prime},y^{\prime\prime})$ on $X$ such that $M=\{y^{\prime}=y^{\prime\prime}=0)\}$ and $L=\{x^{\prime\prime}=y^{\prime\prime}=0\}$. Denote by $(x^{\prime},x^{\prime\prime},y^{\prime},y^{\prime\prime};\xi^{\prime},\xi^{\prime\prime},\eta^{\prime},\eta^{\prime\prime})$ the coordinates on $T^{*}X$ and by $(x^{\prime},x^{\prime\prime};\xi^{\prime},\xi^{\prime\prime})$ the coordinates on $T^{*}M$. Then

[TABLE]

Therefore $g_{d}^{-1}T_{N}^{*}L=\{(x^{\prime},y^{\prime};\xi^{\prime},\xi^{\prime\prime},\eta^{\prime},\eta^{\prime\prime})\in L\times_{X}T^{*}X;y^{\prime}=\xi^{\prime}=0\}=T^{*}_{N}X$. Let $\theta\in T_{T^{*}_{M}X}T^{*}X$ with $\theta\notin C_{T^{*}_{M}X}(\operatorname{\mu supp}(F))$. Then $(x^{\prime},x^{\prime\prime};\eta^{\prime},\eta^{\prime\prime})+\theta\notin\operatorname{\mu supp}(F)$. Choosing $\theta\in T^{*}_{N}M$, $\theta\neq 0$, we get that $(x^{\prime},0;0,\xi^{\prime\prime},\eta^{\prime},\eta^{\prime\prime})\in\operatorname{\mu supp}(F)$ implies $\xi^{\prime\prime}=0$. Q.E.D.

2.2 Analytic wave front set

From now on and until the end of this paper, unless otherwise specified, all manifolds are (real or complex) analytic and the base field ${\bf k}$ is ${\mathbb{C}}$.

Let $M$ be a real manifold of dimension $n$ and let $X$ be a complexification of $M$. One denotes by $\mathscr{A}_{M}$ the sheaf of complex valued real analytic functions on $M$, that is, $\mathscr{A}_{M}=\mathscr{O}_{X}|_{M}$.

One denotes by $\mathscr{B}_{M}$ and $\mathscr{C}_{M}$ the sheaves on $M$ and $T^{*}_{M}X$ of Sato’s hyperfunctions and microfunctions, respectively. Recall that these sheaves are defined by

[TABLE]

In particular, ${\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}}({\rm D}^{\prime}_{X}{\mathbb{C}}_{M},\mathscr{O}_{X})$ and $\mu hom({\rm D}^{\prime}_{X}{\mathbb{C}}_{M},\mathscr{O}_{X})$ are concentrated in degree [math]. Since ${\rm D}^{\prime}_{X}{\mathbb{C}}_{M}\simeq\operatorname{or}_{M}\,[-n]\simeq\omega_{M/X}\simeq\omega_{M}^{\otimes-1}$, we get that

[TABLE]

The sheaf $\mathscr{B}_{M}$ is flabby and the sheaf $\mathscr{C}_{M}$ is conically flabby.

Moreover, since $\mathrm{R}{\pi}_{*}\circ\mu hom\simeq{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}}$, we have the isomorphism \mathscr{B}_{M}\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{1.0mu}\sim\mspace{2.0mu}}}}{\pi}_{*}\mathscr{C}_{M}. One deduces the isomorphism:

[TABLE]

Definition 2.8** ([Sa70]).**

The analytic wave front set of a hyperfunction $u\in\Gamma(M;\mathscr{B}_{M})$, denoted $\rm WF(u)$, is the support of $\rm spec(u)$, a closed conic subset of $T^{*}_{M}X$.

The next result is well-known to the specialists. Let $M$ be a real analytic manifold, $X$ a complexification of $M$ and let $\lambda$ be a closed convex proper cone in $T^{*}_{M}X$.

Theorem 2.9**.**

Let $u\in\Gamma(M;\mathscr{B}_{M})$ with ${\rm WF}(u)\subset\lambda$. Assume that $M$ is connected and that $u\equiv 0$ on an open subset $U\subset M$, $U\neq\varnothing$. Then $u\equiv 0$ on $M$.

Proof.

Let $S=\operatorname{supp}(u)$ and let $x\in\partial S$. Choosing a local chart in a neighborhood of $x$, we may assume from the beginning that $M$ is open in ${\mathbb{R}}^{n}$ and that $\lambda\subset M\times\sqrt{-1}\gamma^{\circ}$ where $\gamma$ is a non empty open convex cone of ${\mathbb{R}}^{n}$. Then there exists a holomorphic function $f\in\Gamma((M\times\sqrt{-1}\gamma)\cap W;\mathscr{O}_{X})$, where $W$ is a connected open neighborhood of $M$ in $X$, such that $u=b(f)$, that is, $u$ is the boundary value of $f$. If $b(f)$ is analytic on $U$, then $f$ extends holomorphically in a neighborhood of $U$ in $X$. If moreover $f=0$ on $U$, then $f\equiv 0$ on $(M\times\sqrt{-1}\gamma)\cap W$ and thus $u\equiv 0$. Q.E.D.

2.3 D-modules

Let $(X,\mathscr{O}_{X})$ be a complex manifold. One denotes by $\mathscr{D}_{X}$ the sheaf of rings of finite order holomorphic differential operators on $X$. In the sequel, a $\mathscr{D}_{X}$-module means a left $\mathscr{D}_{X}$-module. Let $\mathscr{M}$ be a coherent $\mathscr{D}_{X}$-module. Locally on $X$, $\mathscr{M}$ may be represented as the cokernel of a matrix $\cdot P_{0}$ of differential operators acting on the right:

[TABLE]

and one shows that $\mathscr{M}$ is locally isomorphic to the cohomology of a bounded complex

[TABLE]

Clearly, $\mathscr{O}_{X}$ is a left $\mathscr{D}_{X}$-module. It is indeed coherent since $\mathscr{O}_{X}\simeq\mathscr{D}_{X}/\mathscr{I}$ where $\mathscr{I}$ is the left ideal generated by the vector fields. For a coherent $\mathscr{D}_{X}$-module $\mathscr{M}$, one sets for short

[TABLE]

Representing (locally) $\mathscr{M}$ by a bounded complex $\mathscr{M}^{\bullet}$ as above, we get

[TABLE]

where now $P_{0}\cdot$ operates on the left.

Hence a coherent $\mathscr{D}_{X}$-module is nothing but a system of linear partial differential equations.

To a coherent $\mathscr{D}_{X}$-module $\mathscr{M}$ is associated its characteristic variety $\operatorname{char}(\mathscr{M})$, a closed analytic ${\mathbb{C}}^{\times}$-conic co-isotropic subset of $T^{*}X$.

Theorem 2.10** (see [KS90]*Th. 11.3.3).**

Let $\mathscr{M}$ be a coherent $\mathscr{D}_{X}$-module. Then $\operatorname{\mu supp}({\mathscr{S}\mspace{-2.5mu}\mathit{ol}}(\mathscr{M}))=\operatorname{char}(\mathscr{M})$.

Let $f\colon Y\xrightarrow[]{}X$ be a morphism of complex manifolds. One can define the inverse image $f^{D}\mathscr{M}$, an object of $\mathrm{D}^{\mathrm{b}}(\mathscr{D}_{Y})$. The Cauchy-Kowalevska theorem has been extended to D-modules in Kashiwara’s thesis of 1970.

Theorem 2.11** (see [Ka70, Ka03]).**

Let $\mathscr{M}$ be a coherent $\mathscr{D}_{X}$-module and assume that $f$ is non characteristic for $\mathscr{M}$, that is, for $\operatorname{char}(\mathscr{M})$. Then

1. (i)

$f^{D}(\mathscr{M})$* is concentrated in degree [math] and is a coherent $\mathscr{D}_{Y}$-module,* 2. (ii)

$\operatorname{char}(f^{D}(\mathscr{M}))=f_{d}f_{\pi}^{-1}\operatorname{char}(\mathscr{M})$, 3. (iii)

one has a natural isomorphism f^{-1}{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{O}_{X})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{1.0mu}\sim\mspace{2.0mu}}}}{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{Y}}(f^{D}\mathscr{M},\mathscr{O}_{Y}).

Example 2.12**.**

Assume $\mathscr{M}=\mathscr{D}_{X}/\mathscr{D}_{X}\cdot P$ for a differential operator $P$ of order $m$ and $Y$ is a smooth hypersurface, non characteristic for $P$. Let $s=0$ be a reduced equation of $Y$. Then, $f^{D}(\mathscr{M})\simeq\mathscr{D}_{Y}/(s\cdot\mathscr{D}_{Y}+\mathscr{D}_{X}\cdot P)$ and it follows from the Weierstrass division theorem that, locally, $f^{D}\mathscr{M}\simeq\mathscr{D}_{Y}^{m}$. In this case, isomorphism (iii) in the above theorem is nothing but the Cauchy-Kowalevska theorem.

Definition 2.13**.**

Let $\mathscr{M}$ be a coherent $\mathscr{D}_{X}$-module and let $L\subset X$ be a real submanifold. One says that the pair $(L,\mathscr{M})$ is elliptic if $\operatorname{char}(\mathscr{M})\cap T^{*}_{L}X\subset T^{*}_{X}X$.

If $X$ is a complexification of a real manifold $M$, the pair $(M,\mathscr{M})$ is elliptic if and only if $\mathscr{M}$ is elliptic in the usual sense and Corollary 2.3 gives, for $F={\mathscr{S}\mspace{-2.5mu}\mathit{ol}}(\mathscr{M})$, the isomorphism

[TABLE]

In particular, the hyperfunction solutions of the system $\mathscr{M}$ are real analytic. More generally, we have

Theorem 2.14** ([Sa70]).**

Let $\mathscr{M}$ be a coherent $\mathscr{D}_{X}$-module and let $u\in\Gamma(M;{\mathscr{H}\mspace{-4.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{B}_{M}))$. Then $\rm WF(u)\subset T^{*}_{M}X\cap\operatorname{char}(\mathscr{M})$.

When $L=Y$ is a complex submanifold of complex codimension $d$, $(Y,\mathscr{M})$ is elliptic if and only if the embedding $Y\hookrightarrow X$ is non-characteristic for $\mathscr{M}$. In this case, Corollary 2.3 gives the isomorphism

[TABLE]

3 Wick rotation for D-modules

3.1 Hyperbolic D-modules

Let $M$ be a real manifold and let $X$ be a complexification of $M$. Recall the embedding $T^{*}M\hookrightarrow T^{*}T^{*}_{M}X$ of (2.7) and recall that for $S\subset T^{*}X$, the Whitney cone $C_{T^{*}_{M}X}(S)$ is contained in $T_{T^{*}_{M}X}T^{*}X\simeq T^{*}T^{*}_{M}X$. The next definition is extracted form [KS90]. See [Sc13] for details.

Definition 3.1**.**

Let $\mathscr{M}$ be a coherent left $\mathscr{D}_{X}$-module.

1. (a)

We set

[TABLE]

and call $\operatorname{hypchar}_{M}(\mathscr{M})$ the hyperbolic characteristic variety of $\mathscr{M}$ along $M$. 2. (b)

*A vector $\theta\in T^{*}M$ such that $\theta\notin\operatorname{hypchar}_{M}(\mathscr{M})$ is called *hyperbolic with respect to $\mathscr{M}$. 3. (c)

A submanifold $N$ of $M$ is called hyperbolic for $\mathscr{M}$ if

[TABLE]

that is, any nonzero vector of $T^{*}_{N}M$ is hyperbolic for $\mathscr{M}$. 4. (d)

For a differential operator $P$, we set $\operatorname{hypchar}(P)=\operatorname{hypchar}_{M}(\mathscr{D}_{X}/\mathscr{D}_{X}\cdot P)$.

Example 3.2**.**

Assume we have a local coordinate system $(x+\sqrt{-1}y)$ on $X$ with $M=\{y=0\}$ and let $(x+\sqrt{-1}y;\xi+\sqrt{-1}\eta)$ be the coordinates on $T^{*}X$ so that $T^{*}_{M}X=\{y=\xi=0\}$. Let $(x_{0};\theta_{0})\in T^{*}M$ with $\theta_{0}\not=0$. Let $P$ be a differential operator with principal symbol $\sigma(P)$. Then $(x_{0};\theta_{0})$ is hyperbolic for $P$ if and only if

[TABLE]

As noticed by M. Kashiwara, it follows from the local Bochner’s tube theorem that Condition (3.3) can be simplified: $(x_{0};\theta_{0})$ is hyperbolic for $P$ if and only if

[TABLE]

Hence, one recovers the classical notion of a (weakly) hyperbolic operator.

Notation 3.3**.**

As usual, we shall write ${\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{C}_{M})$ instead of ${\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\pi^{-1}\mathscr{D}_{X}}(\pi^{-1}\mathscr{M},\mathscr{C}_{M})$ and similarly with other sheaves on cotangent bundles.

3.2 Main tool

Consider as above a real manifold $M$ and a complexification $X$ of $M$, a closed submanifold $N$ of $M$, and $Y$ a complexification of $N$ in $X$. Denote as above by $f\colon Y\hookrightarrow X$ the embedding. Consider also another closed real submanifold $L\subset X$ such that $L\cap M=N$ and the intersection is clean. Denote by $g\colon L\hookrightarrow X$ the embedding and consider the Diagram 2.12.

Let $\mathscr{M}$ be a coherent $\mathscr{D}_{X}$-module and consider the hypotheses:

[TABLE]

Set $F={\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{O}_{X})$. Then hypothesis (a) of Theorem 2.4 is translated as hypothesis (3.5) and hypothesis (b) is translated as hypothesis (3.6).

We shall constantly use the next result.

Lemma 3.4** (see [JS16]*Lem. 3.5).**

Hypothesis (3.6) implies hypothesis (3.7).

Theorem 3.5**.**

Let $\mathscr{M}$ be a coherent left $\mathscr{D}_{X}$-module. Assume (3.5) and (3.6). Then one has the natural isomorphism

[TABLE]

Proof.

Apply Theorem 2.4 together with Lemma 2.7 to the sheaf $F={\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{O}_{X})$. We get:

[TABLE]

Equivalently, we have

[TABLE]

Finally $\omega_{X/M}\mathbin{\otimes_{\raise 4.52083pt\hbox to-0.70004pt{}{}}}\mu_{M}(\mathscr{O}_{X})\simeq\mathscr{C}_{M}$. Q.E.D.

Example 1: Cauchy problem for microfunctions

Let $M$, $X$, $L$, $N$ and $f$ be as above and assume that $L=Y$, hence $f=g$.

Corollary 3.6**.**

Let $\mathscr{M}$ be a coherent left $\mathscr{D}_{X}$-module. Assume (3.6). Then one has the natural isomorphism

[TABLE]

Proof.

Applying Theorem 2.11, we get $f^{-1}{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{O}_{X})\simeq{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{Y}}(f^{D}\mathscr{M},\mathscr{O}_{Y})$. (Recall that (3.6) implies (3.7).) Moreover, $\omega_{Y/N}\mathbin{\otimes_{\raise 4.52083pt\hbox to-0.70004pt{}{}}}\mu_{N}(\mathscr{O}_{Y})\simeq\mathscr{C}_{N}$. Finally, since $f_{Nd}$ is finite on $\operatorname{char}(\mathscr{M})$, we may replace $\mathrm{R}{f_{Nd}}_{!}$ with ${f_{Nd}}_{!}$. Q.E.D.

3.3 Boundary values

Let $M$ be a real $n$-dimensional manifold, $N$ a closed submanifod of codimesnion $d$, $X$ a complexification of $M$ and $Y$ a complexification of $N$ in $X$. We denote by $f\colon Y\hookrightarrow X$ the embedding.

Notation 3.7**.**

We set

[TABLE]

We shall not confuse the sheaf $\widetilde{\mathscr{B}}_{N}$ with the sheaf $\mathscr{B}_{N}$ of hyperfunctions on $N$. We have an isomorphism

[TABLE]

Let $\mathscr{M}$ be a coherent $\mathscr{D}_{X}$-module. Applying the functor \mathrm{R}\Gamma_{N}({\,\raise 1.72218pt\hbox{\scriptscriptstyle\bullet}\,})\mathbin{\otimes_{\raise 4.52083pt\hbox to-0.70004pt{}{}}}\operatorname{or}_{N}\,[n-d] to the isomorphism (iii) in Theorem 2.11 together with isomorphism (2.28) one recovers a well known result:

Lemma 3.8**.**

Assume (3.7). One has a natural isomorphism

[TABLE]

Appying the functor ${\rm D}^{\prime}_{X}$ to the morphism ${\mathbb{C}}_{M}\xrightarrow[]{}{\mathbb{C}}_{N}$, we get the morphism ${\rm D}^{\prime}_{X}({\mathbb{C}}_{N})\xrightarrow[]{}{\rm D}^{\prime}_{X}({\mathbb{C}}_{M})$, that is, the morphism $\operatorname{or}_{N}\,[d+n]\xrightarrow[]{}\operatorname{or}_{M}\,[n]$. Applying the functor {\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}}({\,\raise 1.72218pt\hbox{\scriptscriptstyle\bullet}\,},\mathscr{O}_{X}) we get the “restriction” morphism

[TABLE]

For a closed cone $\lambda\subset T^{*}_{M}X$, we set for short

[TABLE]

For an open cone $\gamma\subset T_{N}M$, we set for short :

[TABLE]

(In the sequel, we shall use this notation for another real manifold $L$ instead of $M$.)

Hence, for a closed convex proper cone $\delta\subset T^{*}_{N}M$ with $\delta\supset N$, setting $\gamma={\rm Int}(\delta^{\circ a})$, we have by (2.6):

[TABLE]

One can use (3.12) and the morphism ${\pi_{N}}_{*}\Gamma_{\delta}(\mu_{N}\mathscr{B}_{M})\xrightarrow[]{}{\pi_{N}}_{*}\mu_{N}\mathscr{B}_{M}\simeq\Gamma_{N}\mathscr{B}_{M}$ to obtain the morphism

[TABLE]

One can also construct (3.13) directly as follows. Let $U$ be an open subset of $M$ such that $\overline{U}\supset N$, $U$ is locally cohomologically trivial (see [KS90]*Exe. III.4). Then the morphism ${\mathbb{C}}_{\overline{U}}\xrightarrow[]{}{\mathbb{C}}_{N}$ gives by duality the morphism $\operatorname{or}_{N}\,[d+n]\xrightarrow[]{}\operatorname{or}_{U}\,[n]$ and one gets the morphism $\Gamma_{U}\mathscr{B}_{M}\xrightarrow[]{}\Gamma_{N}\mathscr{B}_{M}\mathbin{\otimes_{\raise 4.52083pt\hbox to-0.70004pt{}{}}}\omega_{M/N}$ by applying {\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}}({\,\raise 1.72218pt\hbox{\scriptscriptstyle\bullet}\,},\mathscr{O}_{X}) similarly as for $\rho_{MN}$. Taking the inductive limit with respect to the family of open sets $U$ such that $C_{M}(X\setminus U)\cap\gamma=\varnothing$ (see [KS90]*Th. 4.2.3), we recover the morphism (3.13).

In particular, for a coherent $\mathscr{D}_{X}$-module $\mathscr{M}$ we get the morphisms

[TABLE]

3.4 Wick rotation

Let $M$, $X$, $Y$, $N$, $L$, $f$ and $g$ be as above. Now, we also assume that $L$ is a real manifold of the same dimension than $M$ and $X$ is a complexification of $L$. We still consider diagram (2.12).

Consider the hypothesis

[TABLE]

(Here, $M\times_{X}T^{*}_{X}X$ stands for the zero-section of $T^{*}_{M}X$.)

Lemma 3.9**.**

Assume (3.14).Then we have the natural isomorphism

[TABLE]

Proof.

(i) Set for short $F={\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{C}_{M})$, $j=g_{N\pi}$ , $A=\lambda$, $B=j^{-1}A$. With these new notations, we have to prove the morphism

[TABLE]

is an isomorphism.

(ii) The morphism (3.16) is an isomorphism outside of the zero-section of $T^{*}_{M}X$ since then $\operatorname{supp}(F)=A\sqcup C$ with $A$ and $C$ closed and $A\cap C=\varnothing$, by the hypothesis (3.14).

(iii) Consider the diagram in which $s_{N}$ and $s_{M}$ denote the embeddings of the zero-sections:

[TABLE]

Since $\mathrm{R}{\pi_{N}}_{*}\simeq s_{N}^{-1}$, when applied to conic sheaves, it remains to show that (3.16) is an isomorphism after applying the functor $\mathrm{R}{\pi_{N}}_{*}$.

(iv) Consider the morphism of Sato’s distinguished triangles:

[TABLE]

It follows from (ii) that the vertical arrow $w$ on the right is an isomorphism. We are thus reduced to prove the isomorphism

[TABLE]

(v) Using the fact that $A\supset M$ and $B\supset N$ and that Diagram (3.21) with the arrows going down is Cartesian, we get

[TABLE]

Q.E.D.

Consider

[TABLE]

and recall notations (3.10) and (3.11).

Theorem 3.10** (Wick isomorphism Theorem).**

Let $\mathscr{M}$ be a coherent left $\mathscr{D}_{X}$-module and let $\gamma$ be as in (3.24). Assume (3.5), (3.6), (3.14) and also

[TABLE]

Then one has the commutative diagram in which the horizontal arrow is an isomorphism:

[TABLE]

Proof.

(i) As a particular case of Theorem 3.5 and using the fact that $g^{-1}{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{O}_{X})\simeq{\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{B}_{L})$, we get the isomorphism

[TABLE]

(ii) Set for short $F={\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{C}_{M})$. Using Lemma 3.9 and the fact that $g_{Nd}$ is proper on $\operatorname{supp}F$, we have the isomorphism

[TABLE]

Therefore, we have proved the isomorphism

[TABLE]

(iii) Let us apply the functor $\mathrm{R}{\pi_{N}}_{*}$ to (3.31). Since $g_{Nd}$ is proper on $\operatorname{supp}F$, setting $G={\mathrm{R}\mathscr{H}\mspace{-3.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\Gamma_{\lambda}\mathscr{C}_{M})$, we have (see Diagram 2.22)

[TABLE]

Hence, we have proved the isomorphism

[TABLE]

and the result follows from (3.12). Q.E.D.

3.5 The classical Wick rotation

Let us treat the classical Wick rotation. Hence, we assume that $M=N\times{\mathbb{R}}$ and $L=N\times\sqrt{-1}{\mathbb{R}}$. As usual, $Y$ is a complexification of $N$ and $X=Y\times{\mathbb{C}}$. We denote by $t+is$ the holomorphic coordinate on ${\mathbb{C}}$, by $(t+is;\tau+i\sigma)$ the symplectic coordinates on $T^{*}{\mathbb{C}}$ and by $(x;i\eta)$ a point of $T^{*}_{N}Y$. We identify $N$ and $N\times\{0\}\subset X$.

Let $P$ is a differential operator of order $m$, elliptic on $L$ and (weakly) hyperbolic on $M$ in the $\pm dt$ codirections. A typical example is the wave operator on a globally hyperbolic spacetime $N\times{\mathbb{R}}_{t}$. Set

[TABLE]

The map $g_{Nd}\colon N\times_{M}T^{*}_{M}X\xrightarrow[]{}T^{*}_{N}L$ is given by

[TABLE]

We shall apply the preceding result with $\gamma=L^{+}$. In that case, $\gamma^{\circ a}=\lambda$ and (3.25) is satisfied.

Let $\mathscr{M}=\mathscr{D}_{X}/\mathscr{D}_{X}\cdot P$. In the sequel we write for short $\mathscr{B}_{M}^{P}$ instead of ${\mathscr{H}\mspace{-4.0mu}om}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}(\mathscr{M},\mathscr{B}_{M})$ and similarly with other sheaves. Note that ${\mathscr{E}\mspace{-2.0mu}xt}_{\raise 4.52083pt\hbox to0.70004pt{}\mathscr{D}_{X}}^{1}(\mathscr{M},\widetilde{\mathscr{B}}_{N})\simeq\widetilde{\mathscr{B}}_{N}/P\cdot\widetilde{\mathscr{B}}_{N}$.

As a particular case of Theorem 3.10, we get:

Corollary 3.11**.**

We have a commutative diagram in which the horizontal arrow is an isomorphism:

[TABLE]