119

Relative Dolbeault cohomology
and Sato hyperfunctions
Tatsuo Suwa*

Department of Mathematics

Hokkaido University
This is a slightly extended version of the talk I gave at the RILIS Joint Research“Microlocal analysis and asymptotic analysis“. The contents are taken from [7] and [16]and we refer to these for details and more material. See also [6] and [18] for further

development and applications.

1

Introduction

Čech‐de Rham cohomology together with its integration theory has been effectively usedin various problems related to localization of characteristic classes. Likewise we may

develop the Čech‐Dolbeault cohomology theory and on the way we naturally come up with

the notion of relative Dolbeault cohomology. This cohomology turns out to be canonicallyisomorphic with the local (relative) cohomology of A. Grothendieck and M. Sato withcoefficients in the sheaf of holomorphic forms so that it provides a handy way of expressing
the latter.

In this article we present the theory of relative Dolbeault cohomology and give, asapplications, simple explicit expressions of Sato hyperfunctions, some fundamental oper‐
ations on them and related local duality theorems. Particularly noteworthy is that the
integration of hyperfunctions in our framework, which is a descendant of the integration

theory on the Čech‐de Rham cohomology, is simply given as the usual integration of dif‐ferential forms. Also the Thom class in relative de Rham cohomology plays an essential
role in the scene of interaction between topology and analysis.

2

Relative Dolbeault cohomology

2.1

Relative cohomology

Let \mathscr{S} be a sheaf of Abelian groups on a topological space X . For an open set V in
X , we denote by \mathscr{S}(V) the group of sections on V . Also for an open subset V'\subset V
we denote by \mathscr{S}(V_{:}V') the group of sections on V that vanish on V' . As reference

cohomology theory we adopt the one via flabby resolution (cf. [2], [9]). Thus for an open
set X' in X , H^{q}(X_{:}X';\mathscr{S}) denotes the q‐the cohomology of the complex \mathscr{F}(X, X')*

Supported by JSPS Grant

16K05116.

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with 0arrow \mathscr{S}arrow \mathscr{F} a flabby resolution. It is uniquely determined modulo canonicalisomorphisms, independently of the flabby resolution. Setting S=X\backslash X' , it will also
be denoted by H_{S}^{q}(X;\mathscr{S}) . This cohomology in the first expression is referred to as therelative cohomology of \mathscr{S} on (X, X') (cf. [11]) and in the second expression the local
cohomology of \mathscr{S} on X with support in S (cf. [4]).

2.2
Let

Čech‐Dolbeault cohomology
X be a complex manifold of dimensionn. We denote by \mathscr{E}_{X}^{(p,q)} and \mathscr{O}_{X}^{(p)} the sheavesof C^{\infty}(p, q) ‐forms and holomorphic p ‐forms on X . We denote \mathscr{O}_{X}^{(0)} by \mathscr{O}_{X} . We also omit
the suffix X if there is no fear of confusion. Recall that the Dolbeault complex (\mathscr{E}^{(p,.)_{4} .\overline{\partial})
gives a fine resolution of \mathscr{O}^{(p)}0arrow \mathscr{O}^{(p)}arrow \mathscr{E}^{(p0)}arrow^{\partial}\mathscr{E} ^{(p,1)}arrow^{\partial}.

.

arrow^{\partial}\mathscr{E}^{(p,n)}arrow 0.

.

Dolbeault cohomology: The Dolbeault cohomology H_{\partial}^{pq}(X) of X of type (p, q) is
the q‐th cohomology of the complex (\mathscr{E}^{(p. )}(X),\overline{\partial}) . The Dolbeault theorem says that
there is an isomorphism

H_{\partial}^{p.q}(X)\simeq H^{q}(X_{\grave{i} \cdot 0^{(p)}) .

(2.1)

Note that among the isomorphisms, there is a canonical one (cf. [16], [17]).

Čech‐Dolbeault cohomology:

The Čech‐Dolbeault cohomology may be defined foran arbitrary covering of a complex manifold. Here we recall the case of coverings con‐sisting of two open sets and refer to [15] and [16] for the general case and details.
Let \mathcal{V}=\{V_{0}, V_{1}\} be an open covering of X and set V_{01}=V_{0}\cap V_{1} . We set\mathscr{E}^{(p,q)}(\mathcal{V})=\mathscr{E}^{(p,q)}(V_{0})\oplus \mathscr{E}^{ (p,q)}(V_{1})\oplus \mathscr{E}^{(p,q-1)}(V_{01})Thus an element in \mathscr{E}^{(pq)}(\mathcal{V}) is expressed by a triple.

\xi=(\xi_{0}, \xi_{1}, \xi_{01}) . We define thedifferential\overline{\vartheta}:\mathscr{E}^{(p,q)}(\mathcal{V})ar ow \mathscr{E}^{(pq+1)} \backslash (\mathcal{V})by\overline{\vartheta}(\xi_{0:}\xi_{1}, \xi_{01})=(\overline{\partial}\xi_{0}, \overline{\partial}\xi_{1}, \xi{\imath} -\xi_{0}-\overline{\partial}\xi_{01}) .

Then we see that \overline{\vartheta}\circ\overline{\vartheta}=0.

Definition 2.2 The Čech‐Dolbeault cohomology H_{\theta}^{p,q}(\mathcal{V}) of \mathcal{V} of type (p_{7}q) is the q‐thcohomology of the complex (\mathscr{E}^{(p\cdot)}(\mathcal{V}), \overline{\vartheta}) .
Theorem 2.3 The inclusion \mathscr{E}^{(p,q)}(X)arrow \mathscr{E}^{(pq)}(\mathcal{V}) given by \omega\mapsto(\omega|_{V_{0\dot{\tau}} \omega|_{V_{1} ,0) induces
an isomorphism

H_{\frac{p}{\partial'} ^{q}(X)ar ow^{\sim}H_{\frac{p}{\vartheta'} ^{q}(\mathcal {V}).

Note that the inverse is given by assigning to the class of (\xi_{0}.\xi_{1\backslash }\xi_{01}) the class of\rho_{0}\xi_{0}+\rho_{1}\xi_{1}-\overline{0}\rho_{0}\wedge\xi_{01} . where { \rho_{0} , \rho ı} is a C^{\infty} partition of un ity subordinate to \mathcal{V}.

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121

2.3

Relative Dolbeault cohomology

Let X be as above and S a closed set in X . Letting V_{0}=X\backslash S and V_{1} a neighborhoodof S in X , we consider the coverings \mathcal{V}=\{V_{0}, V_{1}\} and \mathcal{V}'=\{V_{0}\} of X and X\backslash S . We set\mathscr{E}^{(p,q)}(\mathcal{V}_{\backslash }. \mathcal{V}')=\{\xi\in \mathscr{E}^{(p,q)}(\mathcal{V})|\xi_{0}=0\}=\mathscr{E}^{(p,q)}(V_{1})\oplus \mathscr{E}^{(p,q-1)}(V_{01}).

Then we see that (\mathscr{E}^{(p,.)}(\mathcal{V}, \mathcal{V}'), \vartheta) is a subcomplex of (\mathscr{E}^{(p,.)}(\mathcal{V})_{:}\overline{\vartheta}) .

Definition 2.4 The relative Dolbeault cohomology H_{\vartheta}^{p.q}(\mathcal{V}, \mathcal{V}') of (\mathcal{V}, \mathcal{V}') of type (p, q) is
the q‐th cohomology of the complex (\mathscr{E}^{(p. )}(\mathcal{V}, \mathcal{V}')\overline{\vartheta}) .
From the exact sequence of complexes

0arrow S_{Q}^{p\prime}.(\mathcal{V}, \mathcal{V}')ar ow^{j^{*} \mathscr{P} ^{\bullet}(\mathcal{V})ar ow^{i^{*} \mathscr{E}^{p},.(V_{0})ar ow 0,
where j^{*}(\xi_{1}.\xi_{01})=(0, \xi_{1}, \xi_{0]}) and i^{*} ( \xi_{0}, \xi_{1} , \xi0ı) =\xi_{0} , we have the following exact se‐quence:. .ar ow H_{\partial}^{p,q-1}(V_{0})ar ow\delta^{*}H_{\vartheta}^{p,q}(\mathcal {V},V')ar ow^{j^{*} H_{\vartheta}^{p,q} (v) arrow i^{*}H_{\frac{p}{\partial} ^{q}(V_{0})arrow..

(2.5)

where \delta^{*} assigns to the class of \theta the class of (0, -\theta) . From the above and Theorem 2.3,
we have:

Proposition 2.6 The cohomology H_{\frac{p}{\vartheta'} ^{q}(\mathcal{V}, \mathcal{V}') is determined uniquely modulo canonicalisomorphvsms, indepcndcntly of the ch_{oL}ce of V_{1}In view of the above we denote H_{\vartheta}^{pq}(\mathcal{V}, \mathcal{V}') also by H_{\vartheta}^{p.q}(X, X\backslash S) .

Proposition 2.7 (Excision) For any open set

Vcontaining S , there is a canonicalisomorphism

H_{\frac{p}{\vartheta} ^{q}(X, X\backslash S)\simeq H_{\vartheta}^{p,q}(V_{:} V\backslash S).

The relative Dolbeault cohomology share all the fundamental properties with therelative (local) cohomology of X with coefficients in \mathscr{O}^{(p)} . In fact we have (cf. [16]):

Theorem 2.8 (Relative Dolbeault theorem) There is a cononical isomorphism

H_{\vartheta}^{p.q}(X, X\backslash S)\simeq H_{S}^{q}(X;\mathscr{O}^{(p)}).

We have the cup product and integration theory in Čech‐Dolbcault cohomology, foi
which we come back in a special case,122
3Sato hyperfunctions

3.1

Hyperfunctions and hyperforms

Let M be a real analytic manifold of dimension n and X its complexification. For a sheaf\mathscr{S} on X , we denote by \mathscr{H}_{M}^{q}(\mathscr{S}) the sheaf defined by the presheaf
V\mapsto H_{M\cap V}^{q}(V, \mathscr{S}) . In

fact it is supported on M and may be thought of as a sheaf on M . We recall (cf. [8],[12]) that the sheaf of Sato hyperfunctions on M is defined by\mathscr{B}_{I1I}=\mathscr{H}_{M}^{n}(\mathscr{O}_{X})\otimes_{z_{M} or_{\Lambda l/X},
where or_{\mathfrak{h}I/X}=\mathscr{R}(\mathbb{Z}_{X}) is the relative orientation sheaf, i.e., the orientation sheaf of
the normal bundle T_{M}X . More generally we introduce the following:

Definition 3.1 The sheaf of p ‐hyperforms on M is defined by\mathscr{B}_{AI}^{(p)}=\mathscr{R}(\mathscr{O}_{X}^{(p)}) \otimes_{Z_{1\backslash f} or_{M/X}.
It is what is referred to as the sheaf of p‐forms with coefficients in hyperfunctions.
Since X is a complex manifold, it is always orientable. However the orientation we
consider is not necessarily the “usual one: Here we say an orientation of X is usual if(x_{1}, y_{1}, \ldots, x_{n}, y_{n}) is a positive coordinate system when (z_{1}, \ldots : z_{n}), z_{i}=x_{i}+\sqrt{-1}y_{i} , isa coordinate system on X . If M is orientable, so is T_{M}X . Thus in this case, for any open
set U\subset M , we have\mathscr{B}_{f1I}^{(p)}(U)=H_{U}^{n}(V;\mathscr{O}_{X}^{(p)})8_{Z_{\lambda I} (U)}H_{U}^{n}(V;\mathbb{Z}_{X}) ,(3.2)

where V is an open set in X containing U as a close set. We refer to such a V a complex
neighborhood of U in X.

Remark 3.3 In the above we used the fact that I1I is purely n ‐codimensional in X withrespect to \mathscr{O}_{X}^{(p)} and \mathbb{Z}_{X} . For the latter, this can be seen from the Thom isomorphism (cf.
Subsection 5.1 below),

When we specify various orientations, we adopt the convention that the orientationof T_{M}X followed by that of M gives the orientation of X . Thus if we specify orientationsof X and M , the orientation of T_{I_{1}I}X is determined and we have a canonical isomorphism
or_{J\downarrow T/X}\simeq \mathbb{Z}_{X} so that we have canonical isomorphisms\mathscr{B}_{M}^{(p)}\simeq \mathscr{H}_{M}^{n}(\mathscr{O}_{X}^{(p)})

and\mathscr{B}_{Y1I}^{(p)}(U)\simeq H_{U}^{n}(V, \mathscr{O}_{X}^{(p)}) for any open set U\subset M .

(3.4)

In the sequel, at some point the cohomology H_{U}^{n}(V;\mathbb{Z}_{X}) is embedded in H_{U}^{n}(V_{\grave{\tau}}\cdot \mathbb{C}_{X}) ,which is expressed by the relative de Rham cohomology, while H_{U}^{n}(V;\mathscr{O}_{X}^{(p)}) will be ex‐pressed by the relative Dolbeault cohomology.

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3.2

Hyperforms via relative Dolbeault cohomology

For simplicity we let M=\mathbb{R}^{n}\subset \mathbb{C}^{n}=X . We also orient \mathbb{R}^{n} and \mathbb{C}^{n} so that (x_{1} . . . , x_{n})and (y_{1\backslash }\ldots, y_{n:}x_{1}, \ldots, x_{n}) are positive coordinate systems. Thus (y_{1}, \ldots y_{n}) is a positive
coordinate system in the normal direction. Then for an open set U\subset \mathbb{R}^{n} the space ofp ‐hyperforms is

isomorphism

given by (3.4). On the other hand, by Theorem 2.8 there is a canonical\mathscr{B}^{(p)}(U)\simeq H_{\frac{p}{\vartheta'} ^{n}(V, V\backslash U).

In the sequel we identify \mathscr{B}^{(p)}(U) with H_{\vartheta}^{p.n}(V, V\backslash U) by the above isomorphism and give
explicit expressions of hyperforms and some of the fundamental operations on them.
Letting V_{0}=V\backslash U and V_{1} a neighborhood of U in V , we consider the open coverings\mathcal{V}=\{V_{0}, V_{1}\} and \mathcal{V}'=\{V_{0}\} of

V and V\backslash U .

Then

H_{\vartheta}^{pn}(V, V\backslash U)=H_{\frac{p}{\vartheta'} ^{n}(\mathcal{V}, \mathcal{V}')and a p ‐hyperfom is represented by a pair (\xi_{1}, \xi_{01}) with \xi_{1}a(p_{1}\backslash n) ‐form on Vı, which is
automatically \overline{\partial}‐closed. and \xi_{01}a(p.n-1) ‐form on V_{01} such that \xi_{1}=\overline{\partial}\xi_{0{\imath} on V_{01} . We

have the exact sequence (cf. (2.5))

H_{\frac{p}{\partial'} ^{n-1}(V)ar ow H_{\frac{p}{\vartheta'} ^{n-1} (V\backslash U)ar ow^{\delta^{*} H_{\frac{p}{\vartheta'} ^{n}(V. V\backslash U) arrow^{j^{*} H_{\frac{p}{\partial} ^{n}(V).

By a theorem of Grauert [3], we may take as V a Stein open set and, if we do this, we have

H_{\frac{p}{\partial'} ^{n}(V)\simeq H^{n}(V, \mathscr{O}^{(p)})=0 . Thus \delta^{*} is surjective and every element in H_{\frac{p}{\vartheta'} ^{n}(V_{\backslash }V\backslash U) is
represented by a cocycle of the form (0, -\theta) with \theta a\overline{\partial} ‐closed (p, n-1) ‐form on V\backslash U,
In the case n>1, H_{\partial}^{p,n-1}(V)\simeq H^{n-1}(V, \mathscr{O}^{(p)})=0 and \delta^{*} is an isomorphism. In thecase n=1 , we have the exact sequence

H_{\partial}^{p0}(V)ar ow H_{\frac{p}{\vartheta'} ^{0}(V\backslash U) arrow^{\delta^{*} H_{\frac{p}{\vartheta'} ^{1}(V, V\backslash U)ar ow 0,
where H_{\vartheta}^{p,0}(V\backslash U)\simeq H^{0}(V\backslash U, \mathscr{O}^{(p)} ) and H_{\frac{p}{\partial} ^{0}(V)\simeq H^{0}(V, \mathscr{O}^{(p)}) . Thus, for p=0 , werecover the original expression of hyperfunctions by Sato in one dimensional case.

Remark 3.5 Although a hyperform may be represented by a single differential form inmost of the cases, it is important to keep in mind that it is represented by a pair (\xi_{1}, \xi_{01})in general.

4

Some fundamental operations

Let U be an open set in \mathbb{R}^{n} and V a complex neighborhood of U in \mathbb{C}^{n} , as in Subsection 3.2.
Multiplication by real analytic functions: Let \mathscr{A}(U) denote the space of real
analytic functions on U . We define the multiplication\mathscr{A}(U)\cross H_{\vartheta}^{p,n}(V, V\backslash U)arrow H_{\vartheta} ^{p,n}(V_{:}V\backslash U)
by assigning to (f_{\backslash }[\xi]) the class of (\overline{f}\xi_{1}.\overline{f}\xi_{01}) with \overline{f} a holomorphic extension of f . Thenthe following diagram is commutative:\mathscr{A}(U)\cross H_{\vartheta}^{p,n}(V, V\backslash U)arrow H_{\vartheta} ^{pn}(V, V\backslash U)|||1\mathscr{A}(U)\cross H_{U}^{n}(V, \mathscr{O}^{(p)})arrow H_{U}^{n}(V, \mathscr {O}^{(p)}).

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Partial derivatives:

We define the partial derivative\frac{\partil}{\partilx_{i} H_{\vartheta}^{0n}(V, V\backslash U)arrow H_{\frac{0}{\vartheta'} ^{n}(V, V\backslash U):

as follows. Let (\xi_{1}, \xi_{01}) represent a hyperfunction on U . We write \xi_{1}=f dzı \wedge\cdot\cdot\cdot \wedge d\overline{z}_{n}and \xi_{01}=\sum_{j=1}^{n}g_{j}d\overline{z}_{1}\wedge\cdots\wedge\hat{d\overline {z}_{j} \wedge\cdots\wedge d\overline{z}_{n} . Then \frac{\partial}{\partial x_{\iota} [\xi] is represented by the cocycle(\frac{\partialf}{\partialz_{i}d\overline{z}_{1}\wedge\cdots\wedge d\overline{z}_{n\backsla h}\sum_{j=1}^{n}\frac{\partialg_{j} \partialz_{i}d \overline{z}_{1}\wedge\cdots\wedge\hat{d\overline{z}_{j}\wedge\cdots\wedge d\overline{z}_{n}).

With this the following diagram is commutative:

H_{\frac{0}{\vartheta'} ^{n}(V, V\backslash U)ar ow^{\frac{} \partial x_{l} \partial}H_{\frac{0}{\vartheta} ^{n}(V, V\backslash U)|1|\}

H_{U}^{n}(V, \mathcal{O})ar ow^{\frac{}{}\partial z_{l}\partial}H_{U}^{n}(V, \mathcal{O})
Thus for a differential operator
is well‐defined.

Differential:F.

(x, D)P(x, D) H_{\vartheta}^{0,n}(V, V\backslash U)ar ow H_{\frac{0}{\vartheta} ^{n}(V, V\backslash U)

Wc define the differential (cf. [16], here we denote \partial by d)d:H_{\vartheta}^{p,n}(V, V\backslash U)arrow H_{\frac{p}{\vartheta} ^{+1n} (V. V\backslash U)(4.1)

by assigning to the class of ( \xi_{1} , \xi 0ı) the class of (-1)^{n}(\partial\xi 1, − \partial\xi 0ı ) . Then the following
diagram is commutative:

H_{\vartheta}^{p,n}(V_{\backslash }V\backslash U)ar ow^{d}H_{\frac{p}{t?} ^{+1, n}(V_{\backslash }. V\backslash U)|1|2

H_{U}^{n}(V, \mathscr{O}^{(p)})arrow^{d}H_{U}^{n}(V, \mathscr{O}^{(p+1)}).

We will see that this leads to the de Rham complex for hyperforms (cf. Subsection 5.3).
Integration of hyperforms:

We take orientations of \mathbb{R}^{n} and \mathbb{C}^{n} as in Subsection 3.2.

K be a compact set in U .

We define the space of P ‐hyperforms on U with support

Letin

K by the exact sequence

0arrow \mathscr{B}_{K}^{(p)}(U)arrow \mathscr{B}^{(p)}(U)arrow \mathscr{B}^{(p) }(U\backslash K)arrow 0.
Then we have:

Proposition 4.2 For any open set V in X containing
phism\mathscr{B}_{K}^{(p)}(U)\simeq H_{\frac{p}{\vartheta} ^{n}(V, V\backslash K)

K,.

there is a canonical isomor‐

125
Let V be a complex neighborhood of U and consider the coverings \mathcal{V}_{K}=\{V_{0}, V_{1}\} and\mathcal{V}_{K}^{l}=\{V_{0}\} , with V_{0}=V\backslash K and V_{1} a neighborhood of K in V . Then we have a canonical\mathscr{B}_{K}^{(p)}(U)=H_{\vartheta}^{p,n}(\mathcal{V}_{K}, \mathcal{V}_{K}')identification. Let R_{1} be a real 2n ‐dimensional su\dagger )manifold of
V_{1} with C^{\infty} boundary \partial R_{1} and set R_{01}=-\partial R_{1} . We define the integration\int_{U}:\mathscr{B}_{K}^{(n)}(U)ar ow \mathb {C}
as follows. Noting that u\in \mathscr{B}_{K}^{(n)}(U)=H_{\overline{\vartheta} (\mathcal{V}_{K}, \mathcal{V}_{K}') is represented by\xi=(\xi_{1}, \xi_{01})\in \mathscr{E}^{(n,n)}(v_{K}, \mathcal{V}_{K}')= \mathscr{E}^{(n,n)}(V_{1})\oplus \mathscr{E}^{(n,n-1)}(V_{01}),

we define\int_{U}u=\int_{R_{1} \xi_{1}+\int_{R_{01}}\xi_{01}.

It is not difficult to see that the definition does not depend on the choice of \xi.

Local duality pairing:

Let K, V and V_{1} be as above. We have a pairing

H_{\frac{p}{\vartheta'} ^{q}(V_{\backslash }V\backslash K)\cros H_{\partial} ^{n-p,n-q}(V_{1})ar ow^{\smile}H_{\overline{\vartheta} ^{n,n}(V_{\backslash } V\backslash K)ar ow^{\int}\mathb {C}_{\backslash }where the first arrow denotes the cup product.
((\xi_{1}\xi_{01}).\eta)\mapsto(\xi_{1}\wedge\eta, \xi_{01}\wedge\eta) . If we set

On the cocycle level, it is given by

H_{\partial_{ar ow} ^{p,q}[K]=1\dot{ \imath} mH_{\partial}^{p,q}(V_{1})V_{1} \supset K,
the above pairing induces a morphism\overline{A}:H_{\vartheta}^{pq}(V, V\backslash K)ar ow H_{\frac{n}{\partial} ^{ -pn-q}[K]^{*}=1\dot{ \imath} mH_{\partial}^{n-p,n-q}(V_{1})^{*}V_{1}\supset Kar ow(4.3)

which we call the \overline{\partial} ‐Alexander morphism. In the above we considered the algebraic duals,
however in order to have the duality. we need to take topological duals.

A theorem of Martineau:

The following theorem of A. Martineau [10] (also [5],[9])may naturally be interpreted in our framework as one of the cases where the \overline{\partial} ‐ Alexandermorphism is an isomorphism with topological duals so that the duality pairing is givenby the cup product followed by integration as described above,

Theorem 4.4 Let K be a compact set in \mathbb{C}^{n} such that H^{q}(K, \mathscr{O}^{(p)})=0 for q\geq 1 . Thenfor any open set V in \mathbb{C}^{n} containiri gK , H_{\frac{p}{\vartheta} ^{q}(V, V\backslash K) and H_{\partial}^{n-pn-q}[K] admits natural
structures of FS and DFS spaces, respectively, and we have:\overline{A}:H_{\vartheta}^{p,q}(V, V\backslash K)ar ow^{\sim}H_{\partial}^{n- p,n-q}[K]'=\{\begin{ar ay}{l } 0 q\neq n \mathscr{O}^{(n-p)}[K]' q=n, \end{ar ay}
where ’ denotes the strong dual.

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The theorem is originally stated in terms of local cohomology for p=0 . In ourframework the duality (in the case q=n ) is described as follows. Let V_{0}=V\backslash K and V_{1}a neighborhood of K in V and consider the coverings \mathcal{V}_{K}=\{V_{0}, V_{1}\} and \mathcal{V}_{K}'=\{V_{0}\} of
V and V\backslash K . Letting R_{1} and R0ı be as before, the duality pairing is given, for a cocycle( \xi_{1} , \xi 0ı) in \mathscr{E}^{(p_{)}n)}(\mathcal{V}_{K}, \mathcal{V}_{K}') and a holomorphic (n-p) ‐form \eta near K , by

\int_{R_{1} \xi_{1}\wedge\eta+\int_{R_{01} \xi_{01}\wedge\eta .

(4.5)

Note that the hypothesis H^{q}(K, \mathscr{O}^{(p)})=0 , for q\geq 1 , is fulfilled if K is a subset of \mathbb{R}^{n}by the theorem of Grauert.

Suppose K\subset \mathbb{R}^{n} and denote by \mathscr{A}^{(p)} the sheaf of real analytic p‐forms on \mathbb{R}^{n} . Then
we have0^{(p)}[K]=1\dot{ \imath} m\mathscr{O}^{(p)}(V_{1})\simeq 1\dot{ \imath} m\mathscr{A}^{(p)}V_{1}\supset KU_{ \imath} \supset Kar owar ow (U{\imath})= \mathscr{A}^{(p)}[K],
where V_{1} runs through neighborhoods of K in \mathbb{C}^{n} and U_{1}=V_{{\imath}}\cap \mathbb{R}^{n}.

Corollary 4.6 For any open set U\subset \mathbb{R}^{n} contaming K , the pa\uparrow ring\mathscr{B}_{K}^{(p)}(U)\cross \mathscr{A}^{(n-p)}[K]ar ow H_{\vartheta}^{n.n} (V_{:}V\backslash K)ar ow^{j}\mathbb{C}is topologically non‐degenerate so that\mathscr{B}_{K}^{(p)}(U)\simeq \mathscr{A}^{(n-p)}[K]'.
\delta ‐function:

We consider the case\Phi(z)=dz_{1}\wedge\cdots\wedge dz_{n}and

K=\{0\}\subset \mathbb{R}^{n}. W^{\tau}é set\Phi_{i}(z)=(-1)^{i-1}z_{i}dz_{{\imath}}\wedge\cdots\wedge\hat{dz_{i}}\wedge\wedge dz_{n}.

The 0‐Bochner‐Martinelli form on \mathbb{C}^{n}\backslash \{0\} is defined asso that\beta_{n}^{0}=C_{n}'\frac{\sum_{i=1}^{n}\overline{\Phi_{i}(z)} {|z|^{2n} \backslash C_{n}'=(-1)^{\frac{rt(rt-1)}{2} \frac{(n-1)!}{(2\pi\sqrt{-1})^{n}\beta_{n}=\beta_{n}^{0}\wedge\Phi(z)is the Bochner‐Martinelli form on \mathbb{C}^{n}\backslash \{0\}.

Definition 4.7 The \delta ‐function is the element in\mathscr{B}_{\{0\}}(\mathbb{R}^{n})=H_{\frac{0}{\vartheta'} ^{n}(\mathbb{C}^{n} , \mathbb{C}^{n}\backslash \{0\})which is represented by(0, -(-1),\mathcal{B}_{n}^{0})\underline{(n+1)}.

Recall the isomorphism in Corollary 4.6 in this case:\mathscr{B}_{\{0\}}(\mathbb{R}^{n})\simeq(\mathscr{A}_{0}^{(n)})_{:}'
where\mathscr{A}_{0}^{(n)} denotes the stalk of \mathscr{A}^{(n)} at 0.

Proposition 4.8 The \delta ‐function is the hyperfunction that assigns the value h(0) to arepresentative \omega=h(x)\Phi(x) of a germ in \mathscr{A}_{0}^{(n)}127\delta ‐form:

We again consider the case K=\{0\}\subset \mathbb{R}^{n}.

Definition 4.9 The \delta ‐form is the element in\mathscr{B}_{\{0\} ^{(n)}(\mathb {R}^{n})=H_{\vartheta}^{n,n}(\mathb {C}^{n}, \mathb {C}^{n}\backslash \{0\})which is represented by(0, -(-1)^{\frac{n(n+1)}{2}\beta_{n})}.

Recall the isomorphism in Corollary 4.6 in this case:\mathscr{B}_{\{0\} ^{(n)}(\mathb {R}^{n})\simeq(\mathscr{A}_{0})^{l}.
Proposition 4.10 The \delta ‐form is the n ‐hyperfom that assigns the value h(0) to a repre‐
sentative h(x) of a germ in \mathscr{A}_{0}.

Remark 4.11 If we orient \mathbb{C}^{n} so that the usual coordinate system (x_{1}, y_{1}, \ldots, x_{n}, y_{n})is positive, the delta function \delta(x) is represented by (0, -\beta_{n}^{0}) . Also: the delta form isrepresented by (0, -\beta_{n}) . Incidentally, it has the same expression as the Thom class ofthe trivial complex vector bundle of rank n (cf. [13, Ch.III, Remark 4.6]).

5

Embedding of real analytic functions

Let M be a real analytic manifold and X its complexification. The embedding of the sheaf\mathscr{A} of real analytic functions into the sheaf \mathscr{B} of hyperfunctions on \Lambda I comes from thenatural identification of 1 as a hyperfunction. Namely, from the canonical identification\mathbb{Z}_{M}=or_{M/X}\otimes or_{M/X} and the canonical morphism or_{11I/X}=\mathscr{H}_{M}^{n}(\mathbb{Z}_{X})arrow \mathscr{H}_{M}^{n} (\mathscr{O}_{X}) , wehave a canonical morphism\mathbb{Z}_{M}=or_{\lambda I}/xXor_{I\iota l/X}arrow \mathscr{B}_{\lambda I}= \mathscr{H}_{M}^{n}(\mathscr{O}_{X})\otimes or_{I1I/X}.
In fact it is injective and the image of 1 is the corresponding hyperfunction. In the sequel
we try to find it explicitly in our framework. For this we consider the complexification
or_{f_{1}I/X}^{c}=\mathscr{H}_{M}^{n}(\mathbb{C}_{X}) of or_{M/X} . Then the above morphism is extended to\mathbb{C}_{M}=or_{\Lambda I/X}^{c}\otimes or_{M/x}arrow \mathscr{B}_{M} .

(5.1)

We analyze the morphism \mathscr{H}_{M}^{n}(\mathbb{C}_{X})ar ow \mathscr{H}_{lI}^{n}-(\mathscr{O}_{X}) by making use of relative de Rhamand relative Dolbeault cohomologies,

5.1

Relative de Rham cohomology

We refer to [1] and [13] for details on Čech‐de Rham cohomology. For relative de Rham
cohomology and the Thom cldss in this context, see [13].

Let X be a C^{\infty} manifold of dimension m . We denote by \mathscr{E}_{X}^{(q)} the sheaf of C^{\infty}q ‐forms
on X . Recall that the de Rham complex (\mathscr{E}^{(\cdot)} , d) gives a fine resolution of the constantsheaf \mathbb{C}=\mathbb{C}_{X} :0arrow \mathbb{C}arrow \mathscr{E}^{(0)}arrow^{d}\mathscr{E}^{(1)}arrow^{d}

.

.

.

arrow^{d}\mathscr{E}^{(m)}arrow 0.

128
The q‐th de Rham cohomology H_{d}^{q}(X) is the q‐th cohomology of (\mathscr{E}^{(\cdot)}(X), d) . The
de Rham theorem says that there is an isomorphism

H_{d}^{q}(X)\simeq H^{q}(X;\mathbb{C}_{X}).

Note that among the isomorphisms, ther (^{\lrcorner} is a canonical olle (cf. [17]).

The Čech‐de Rham cohomology is defined as in the case of Čech‐Dolbeault cohomol‐

ogy, replacing the Dolbeault complex by the de Rham complex. The differential \overline{\varthea} is now
denoted by D . Likewise we may define the relative de Rham cohomology. Thus let S bea closed set in X . Letting V_{0}=X\backslash S and V_{1} a neighborhood of S in X , we consider thecoverings \mathcal{V}=\{V_{0}, V_{1}\} and \mathcal{V}'=\{V_{0}\} of X and X\backslash S , as before. We set\mathscr{E}^{(q)}(\mathcal{V}.\mathcal{V}')=\mathscr{E}^{(q)}(V_{1})\oplus \mathscr{E}^{(q-1)}(V_{01})and define

D:\mathscr{E}^{(q)}(\mathcal{V}, \mathcal{V}')ar ow \mathscr{E}^{(q+1)} (\mathcal{V}_{\backslash }. \mathcal{V}')by

D(\sigma], \sigma_{01})=(d\sigma_{1}, \sigma_{1}-d\sigma_{01}) .

Definition 5.2 The q‐th relative de Rham cohomology H_{D}^{q}(\mathcal{V}, \mathcal{V}') is the q‐th cohomologyof the complex (\mathscr{E}^{(\cdot)}(\mathcal{V}, \mathcal{V}'), D) .

We may again show that it does not depend on the choice of V_{1} and we denote itby H_{D}^{q}(X, X\backslash S) . We have the relative de Rham theorem which says that there is a

canonical isomorphism (cf. [14], [17]):

H_{D}^{q}(X, X\backslash S)\simeq H^{q}(X, X\backslash S;\mathbb{C}_{X}) .

(5.3)

Remark 5.4 The sheaf cohomology H^{q}(X;\mathbb{Z}_{X}) is canonically isomorphic with the sin‐gular cohomology H^{q}(X;\mathbb{Z}) of X with \mathb {Z} ‐coefficients on finite chains and H^{q}(X, X\backslash S;\mathbb{Z}_{X})is isomorphic with the relative singular cohomology H^{q}(X, X\backslash S;\mathbb{Z}) .
Thom class:Let \pi : Earrow f1\prime I be an oriented C^{\infty} real vector bundle of rank l on a C^{\infty}

manifold M. We identify M with the image of the zero section. Then we havc the Thotn
isomorphismT :H^{q-l}(M;\mathbb{Z})arrow^{\sim}H^{q}(E, E\backslash M;\mathbb{Z}).

The Thom class \Psi_{E}\in H^{l}(E, E\backslash M;\mathbb{Z}) of E is the image of [1]\in H^{0}(\lrcorner\eta[;\mathbb{Z} ) by T.

The Thom isomorphism with \mathb {C} ‐coefficients is expressed in terms of de Rham andrelative de Rham cohomologies:
T :H_{d}^{q-l}(M)arrow^{\sim}H_{D}^{q}(E, E\backslash M).

Its inverse in given by the integration along the fibers of \pi (cf. [13, Ch.II, Theorem 5.3]).
Let W_{0}=E\backslash M and TV_{1}=E and consider the coverings \mathcal{W}=\{W_{0}, W_{1}\} and \mathcal{W}'=\{W_{0}\}of E and E\backslash I_{1}I . Then, H_{D}^{q}(E. E\backslash \Lambda I)=H_{D}^{q}(\mathcal{W}, \mathcal{W}') and we have:
Proposition 5.5 For the trivial bundle E=\mathbb{R}^{l}\cross \mathbb{J}I, \Psi_{E} is represented by the cocycle(0, -\eta)l)in\mathscr{E}^{(l)}(\mathcal{W}, \mathcal{W}') .

129
In the above \psi_{\iota} is the angular form on \mathb {R}^{l} , which is given by\psi_{l}=C_{l}\frac{\sum_{i=1}^{l}\Phi_{i}(x)}{\Vert x\Vert^{l} , \Phi_{i}(x)= (-1)^{i-1}x_{i}dx_{1}\wedge\cdots\wedge\hat{dx}_{i}\wedge\cdots\wedge dx_{l}

.

(5.6)

The constant c_{\iota} is given by \frac{(k-1)^{i} {2\pi^{k} if l=2k and by \frac{(2k)!}{lkk:} if l=2k+1 . The important
fact is that it is closed and \int_{S^{l-{\imath} }\psi_{l}=1 for a usually oriented (l-1) ‐sphere in \mathbb{R}^{l}\backslash \{0\}.
Let

X be a C^{\infty} manifold of dimension
l=m-n .

mand

M\subset X a closed submanifold ofdimension . Set
tubular neighborhood theorem and excision, we have a canonical isomorphismn

If we denote by T_{M}X the normal bundle of M in X , by theH^{q}(X_{\backslash }X\backslash M;\mathbb{Z})\simeq H^{q}(T_{M}X, T_{M} X\backslash 11I;\mathbb{Z}).

Suppose X and M are oriented. Then T_{M}X is orientable as a bundle. We orient it sothat its orientation followed by that of M gives the orientation of X . In this case theThom class \Psi_{I1I}\in H^{l} (X,\cdot X\backslash \Lambda I;\mathbb{Z}) of M in X is defined to be the class corresponding to
the Thom class of T_{M}X under the above isomorphism for q=l . We also have the Thom
isomorphismT :H^{q-l}(M_{(}\backslash \mathbb{Z})arrow^{\sim}H^{q}(X, X\backslash M; \mathbb{Z}) .

(5.7)

M is purely l ‐codimensional in X with respect to\mathbb{Z}_{X} and that
From this we see that
the Thom class \Psi_{J1I} may be thought of as the global section of or_{M/X}\simeq \mathbb{Z}_{f\vee I} that gives
the canonical generator at each point of \lrcorner\mathfrak{h}I . Also for the complexification of the reıativeorientation sheaf or_{\Lambda I/X}^{c} and any open set U in M , we have by (5.3),

or_{M/X}^{c}(U)\simeq H_{U}^{l}(V;\mathbb{C}_{X})\simeq H_{D}^{l}(V_{:} V\backslash U)_{:}(5.8)

where V is an open set in X containing U as a closed set.

5.2

Relative de Rham and relative Dolbeault cohomologies

Let X be a complex manifold of dimension n . We define \rho^{q} : \mathscr{E}^{(q)}arrow \mathscr{E}^{(0.q)} by assigning toa q ‐form \omega its (0, q) ‐component \omega^{(0q)} . Then \rho^{q+1}(d\omega)=\overline{\partial}(\rho^{q}\omega) and we have a morphismof complexes

0arrow \mathbb{C}arrow \mathscr{E}^{(0)}arrow^{d}\mathscr{E}^{(1)}arrow^{d}

.

.

.

arrow^{d}\mathscr{E}^{(q)}arrow^{d}

.

.

.

\downarrow L \downarrow\rho^{0} \downarrow\rho^{1} \downarrow\rho^{q}0arrow \mathscr{O}arrow \mathscr{E}^{(0,0)}arrow^{\partial}\mathscr{E}^{(0,1)} arrow^{\partial}

.

.

.

arrow^{\partial}\mathscr{E}^{(0,q)}arrow^{\partial}

.

.

.

Thus, for any open set X' in X , there is a morphism \rho^{q}which makes the following diagram commutative:

H_{D}^{q}(X, X')arrow H_{\vartheta}^{0,q}(X, X') ,H_{D}^{q}(X, X')arrow^{\rho^{q} H_{\frac{0}{\vartheta} ^{q}(X, X')

(5.9)

|| |1
H^{q}(X, X'\cdot \mathbb{C})arrow^{\iota}H^{q}(X_{\backslash ,\prime}X'; \mathscr{O})

.

130
5.3

1 as a hyperfunction

Let M and X be as in the beginning of this section. We now try to find the image of
1 by the morphism (5.1). For simplicity we assume that M is orientable. Let U be acoordinate neighborhood in M and V a complex neighborhood of U in X . Wc orient X
and M so that the orientations give the ones for V and U as in Subsection 3.2. Then we
have canonical isomorphisms\mathbb{C}_{M}(U)\simeq \mathscr{H}_{l_{1}T}^{n}(\mathbb{C}_{X})(U)\simeq H_{U} ^{n}(V;\mathbb{C})\mathscr{B}_{M}(U)=\mathscr{H}_{M}^{n}(\mathscr{O}_{X})(U)\simeq H_{U}^{n}(V; \mathscr{O}) .

and

Note that the first isomorphism above is the Thom isomorphism (5.7) with \mathb {C} ‐coefficientsfor the pair (V. U) . Suppose U is connected and let \mathcal{V} and \mathcal{V}' be coverings as in Subsec‐
tion 3.2. Then we have the commutative diagram (cf. (5.8), (5.9)):\mathbb{C}=H^{0}(U;\mathbb{C})arrow^{T\sim}H_{U}^{n}(V;\mathbb{C})arrow^{\iota} H_{U}^{n}(V;\mathscr{O})
|

|1

?

H_{D}^{n}(\mathcal{V}, \mathcal{V}')ar ow^{\rho^{n} H_{\vartheta}^{0,n} (\mathcal{V}, \mathcal{V}').

The image of ı by T is the Thom class \Psi_{U} which is represented by \tau=(0, -\psi_{n}(y)) in\mathscr{E}^{(n)}(\mathcal{V}, \mathcal{V}') with \psi_{n}(y) the angular form on \mathb {R}_{y}^{n} (cf. (56)). Since \rho^{n}(\tau)=(0, -\psi_{n}^{(0,n-1)}) ,we have:

Theorem 5.10 As a hyperfunction, 1 is locally represented by the cocycle (0, -\psi_{n}^{(0,n-1)})
in \mathscr{E}^{(0,n)}(\mathcal{V}.\mathcal{V}') , where \psi_{n}^{(0,n-1)} is the (0, n-1) ‐component of \psi_{n}(y) .
Using y_{i}=1/(2\sqrt{-1})(z_{i}-\overline{z}_{i}) ) we see that

\psi_{n}^{(0,n-1)}=(\sqrt{-1})^{n}C_{n}\frac{\sum_{\dot{i}=1}^{n}(-1)^{i} (z_{i}-\overline{z}_{\dot{i} )d\overline{z}_{1}\wedge\cdots\wedge\hat{d\overline {z}_{x} \wedge\cdots\wedged\overline{z}_{n} {\Vertz-\overline{z}|^{n} .
In particular, if n=1,

\psi_{1}^{(0,0)}=\frac{1}{2}\frac{y}{|y}.
Embedding of real analytic forms into hyperforms:

Let

U and V be as above.

We define\mathscr{A}^{(p)}(U)arrow \mathscr{B}^{(p)}(U)=H_{\frac{p}{\vartheta} ^{n}(V, V \backslash U)
by assigning to \omega(x) in \mathscr{A}^{(p)}(U) the class [(0, -\psi_{n}^{(0.n-1)}\wedge\omega(z))] , where \psi_{n}^{(0,n-1)} is as above

and \omega(z) denotes the complexification of \omega(x) . Then it induces a sheaf monomorphism\iota^{(p)} : \mathscr{A}^{(p)}arrow \mathscr{B}^{(p)} , which is compatible with the differentials d of \mathscr{A}^{(\cdot)} and \mathscr{B}^{(\cdot)}.

de Rham complex for hyperforms:
the analytic de Rham complex

Let

X be a coml) lex manifold.

0arrow \mathbb{C}arrow^{\iota}\mathscr{O}arrow^{d}\mathscr{O}^{(1)}arrow^{d}

.

.

.

arrow^{d}\mathscr{O}^{(n)}arrow 0

Then we have

131
131
and the diagram (5.9) is extended to an isomorphism of complexes

0arrow H_{D}^{q}(X_{\backslash ,\varrho}X')arrow^{\rho^{q} H_{\vartheta}^{0,q} (X, X')arrow^{d}H_{\vartheta}^{1,q}(X, X')arrow^{d}
|1

|1

.

.

.

|1

0arrow H^{q}(X, X';\mathbb{C})arrow^{\iota}H^{q}(X, X';\mathscr{O})arrow^{d} H^{q}(X, X', \mathscr{O}^{(1)})arrow^{d}

.

.

.

arrow^{d}H_{\vartheta}^{n.q}(X, X')arrow 0|larrow^{d}H^{q}(X, X';\mathscr{O}^{(n)})arrow 0.

Let M and X be as above. Then from the fact that M is purely n ‐codimensional in\mathscr{O}^{(p)} , we see that the above complex for X'=X\backslash M leads

X with respect to to \mathbb{C}_{X} andto the following exact sequence of sheaves on

M :0arrow \mathbb{C}arrow \mathscr{B}arrow^{d}\mathscr{B}^{(1)}arrow^{d}

.

.

.

arrow^{d}\mathscr{B}^{(n)}arrow 0.

References[1] R. Bott and L. Tu, Differential Forms in Algebraic Topology\backslash Graduate Texts in
Mathematics 82, Springer, 1982.

[2] R. Godement, Topologie Algébrique et Théorie des Faisceaux, Hermann, Paris, 1958.
[3] H. Grauert, On Levi’s problem and the imbedding of real analytic manifolds, Ann.
Math. 68 (1958), 460‐472.
[4] R. Hartshorne, Local Cohomology, A seminar given by A. Grothendieck, Harvard
University, Fall, 1961, Lecture Notes in Math. 41, Springer, 1967.

[5] R. Harvey, Hyperfunctions and partial differential equations, Thesis, Stanford Univ.,
1966.

[6] N. Honda, Hyperfunctions and Čech‐Dolbeault cohomology in the microlocal point ofview, in this volume.

[7] N. Honda, T. Izawa and T. Suwa: Sato hyperfunctions via relative Dolbeault coho‐
mology, in preparation,[8] M. Kashiwara and P. Schapira, Sheaves on Manífolds, Grundlehren der hIath. 292,
Springer , 1990.

[9] H. Komatsu, Hyperfunctions of Sato and Linear Partial Differential Equations with
Constant Coefficients, Seminar Notes 22, Univ. Tokyo, 1968 (in Japanese).

[10] A. Martineau, Les hyperfonctions de M. Sato, Sém. N. Bourbaki, 1960‐1961, n^{o} 214,127‐139.

[11] M. Sato, Theory of hyperfunctions I, II, J. Fac. Sci. Univ. Tokyo, 8 (1959), 139‐193,
387‐436.

[12] M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo‐differential equa‐tions, Hyperfunctions and Pseudo‐Differential Equations, Proceedings Katata 1971

(H. Komatsu, ed,), Lecture Notes in Math. 287, Spiinger, 1973, 265‐529.

132[13] T. Suwa, Indices of Vector Fields and Residues of Singular Holomorphic Foliations,
Actualités hIathématiques, Hermann, Paris, 1998.

[14] T. Suwa, Residue Theoretical Approach to Intersection Theory, Proceedings of the
9‐th International Workshop on Real and Complex Singularities, São Carlos, Brazil
2006, Contemp. \perp\backslah Iath. 459, Amer. Math. Soc., 207‐261, 2008.

[15] T. Suwa, Čech‐Dolbeault cohomology and the \overline{\partial} ‐Thom class, Singularities—Niigata‐
Toyama 2007, Adv. Studies in Pure Math. 56, Math. Soc. Japan, 321‐340, 2009.

[16] T. Suwa, Relative Dolbeault cohomology, in preparation.
[17] T. Suwa, Relative cohomology for the sections of a complex of fine sheaves, in prepa‐
ration, a summary is to appear in the proceedings of the Kinosaki Algebraic Geom‐etry Symposium 2017,[18] K. Umeta, Laplace hyperfunctions from the viewpoint of Čech‐Dolbeault cohomology,
in this volume.

Department of Mathematics
Hokkaido University
Sapporo 060‐0810
Japan
E‐mail: tsuwa@sci.hokudai. ac..lp
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