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Multidimensional local residues and holonomic D-modules(Singularities and Complex Analytic Geometry)

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Multidimensional local residues and holonomic

D-modules

新潟大学

田島慎–

(Shinichi

Tajima)

横浜市大

大阿久俊則

(Toshinori Oaku)

お茶の水女子大

中村弥生

(Yayoi

Nakamura)

Multidimensional

$1\mathit{0}$

cal residues are

fundamental

objects

in complex analysis and

geom-etry. However,

if the polar divisors of

a meromorphic

differential form are not in general

position, the

actual calculation of

local residues is difficult in

many

cases. In this

paper

we

study

Grothendieck

local residue from the viewpoint of

$D$

-modules. We mainly consider

the

case

where

the

polar

divisors

are

not

in

general position.

We

propose

a

new

$\mathrm{a}\mathrm{p}\mathrm{p}\dot{\mathrm{r}}\mathrm{o}\mathrm{a}\mathrm{C}\mathrm{h}$

for

calculating

multidimensional

local

residues.

In the appendix we consider the

zero-dimensional

transversal complete

intersection

case.

We

present

a simple method

$\mathrm{f}\mathrm{o}\dot{\mathrm{r}}$

computing

residues for this case.

We use a

computer

algebra

system

Kan for

Gr\"obner

basis

computation

in Weyl algebra

and

a computer

algebra

system

$\mathrm{R}\mathrm{i}\mathrm{s}\mathrm{a}/\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{r}$

for

Gr\"obner

basis

computation,

and

primary

decomposition in

polynomial

rings.

1. Algebraic local cohomologies

Let us

recall

some basic

facts about algebraic local cohomology and holonomic

D-modules.

Let

$X$

be a complex

manifold

$\mathcal{O}_{X}$

the sheaf

on

$X$

of

holomorphic

functions.

Let

$\mathrm{Y}$

a

subvariety in

$X$

.

Let

$J_{\mathrm{Y}}$

be the

sheaf

of

ideal of

$\mathrm{Y}$

in

$X$

.

The k-th algebraic

local

cohomology group

supported

in

$\mathrm{Y}$

is

defined

as

the

inductive limit of extension

groups

$\mathcal{H}_{[\mathrm{Y}]}^{kk}(\mathcal{O}x)=\lim_{\mathit{1}arrow\infty}\mathcal{E}xt_{o_{X}}(\mathcal{O}x/J_{\mathrm{Y}}^{\mathit{1}}; \mathcal{O}\mathrm{x})$

.

Note

that for a hypersurface

case,

we have

$\mathcal{H}_{[\mathrm{Y}]}^{1}(\mathcal{O}\mathrm{x})\simeq \mathcal{O}_{X}[*\mathrm{Y}]/\mathcal{O}_{X}$

, where

$\mathcal{O}_{X}[*\mathrm{Y}]$

stands

for

$\mathrm{t}\mathrm{h}\overline{\mathrm{e}}$

sheaf

of meromorphic

functions

on

$X$

with

poles

along Y.

Let

$D_{X}$

be the sheaf of

rings

on

$X$

of linear

partial

differential

operators

with

holomorphic

coefficients.

Then

$D_{X}$

is coherent

as a

sheaf

of

rings.

It

is

easy to

see that the algebraic

local

cohomology group

$\mathcal{H}_{[\mathrm{Y}]}^{k}(\mathcal{O}_{X})$

is

naturally

endowed

with a structure of left

$D_{X}$

-module.

(2)

Theorem

$(\mathrm{K}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{W}\mathrm{a}\mathrm{r}\mathrm{a}[12], \mathrm{M}\mathrm{e}\mathrm{b}\mathrm{k}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t}[14])$

(1)

$\mathcal{H}_{[\mathrm{Y}]}^{k}(\mathcal{O}x)$

is

coherent over

$D_{X}$

.

(2)

$\mathcal{H}_{[Y]}^{k}(\mathcal{O}_{X})$

is

a holonomic system.

We refer to

[11], [22]

for the notion of a holonomic

system.

Recently,

one of

the

authors

(T.

Oaku

([19], [20]))

constructed an algorithm

to calculating

Gr\"obner

basis of an

algebraic

local cohomology

group.

His

algorithm

has been

implemented

in the computer

algebra

system Kan ([24]), developed by N. Takayama of

Kobe

University.

The following

computation

was

carried out

by

using

Kan.

Example Let

$f(x,y)=(x^{2}+y^{2})^{3}-4x^{2}y^{2},$

$D=\{(x, y)|f(x, y)=0\}$

.

Put

$m=$

(

$\frac{1}{(x^{2}+y)^{3}2-4xy22}$

mod

$\mathcal{O}\mathrm{x}$

)

$\in \mathcal{H}_{[D}1](\mathcal{O}_{X})$

.

The generator

$m$

of the

module

$\mathcal{H}_{[D]}^{1}(\mathcal{O}_{X})$

satisfies

the following

holonomic

system.

$|(-15y^{22}XD_{x}+3yD.+xy_{6}3x_{2}^{3}D15yyy,243+4(_{X^{3}}Dx_{3_{XD3}}-2y^{2_{X}}Dx_{7}+2yx_{1}^{2}D_{y}-y_{2}^{3}D+(-27yx+x_{2}^{4}Dyyx_{D_{y}-}-2215y^{4}D18yx_{2D_{yx}}^{2}(-972yD2^{+}16x_{965}^{4}D-yyx-222^{+0y_{X}0’}275^{+4}6X^{-}D_{y,8}^{-}y^{4}D_{y}^{2}-8748yx(-108^{-}y_{4}^{5}D_{x_{4^{-2}}}6^{+D2}0XD_{y,3}19(-x^{6}3yx-243yx_{X^{2}-3}yx^{2}y^{2}2D24+14x^{2}D_{x_{2^{-}}}129^{+1}6yx_{312+}^{2}1468y^{3}D^{+}+y4x62-y6+92+8y_{5}xD^{+}x_{2^{-y}}+4yD_{y}+2-18y_{42}x+4X)y=yX^{+X}6yxD5D_{y}x4y)m_{15}=0_{D}208y^{2}’ D-xy386y3m_{3}0,9^{+}2y)m=0_{1}45XD-76XDx+5044yX)m=Dy.y+y4xy^{2})m=09yy^{2}xX^{2}D33yD+416’ 16^{-}y_{X^{3}}D\mathrm{o}_{y_{2}}3x\mathrm{o}_{1}x156)m=02y^{2}D_{y}2$

.

Moreover,

these operators

form

a

Gr\"obner

basis of the

annihilator

ideal of the generator

$m$

.

Example

$(\mathrm{c}\mathrm{f}. [25])$

Let

$f(x, y)=x^{6}-x^{2}y^{3}-y^{5},$

$g(x, y)=y$

.

Let

$m$

be the cohomology

class

associated

to the meromorphic

function

$\frac{1}{fg}$

:

$m=[ \frac{1}{fg}]\in \mathcal{H}_{[0,0]}^{2}(\mathcal{O}_{X})$

.

We have

$\{$

$x^{6}m=0$

,

$ym=0$

$(_{XD_{x}+}6)m=0$

.

However,

the

$D_{X}$

-module structure of the algebraic local cohomology group supported

on

(3)

[1]

$|/$

.

sml

sml

$\mathrm{K}\mathrm{a}\mathrm{n}/\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{M}\mathrm{a}\mathrm{c}\dot{\mathrm{h}}$

inel

1991 April

$—$

1996.

Release

2.970417

(c)

N.

Takayama

This

software

may

be

freely distributed

as

is with

no

warranty expressed.

Please

address

bug

reports and

advices

to

$\mathrm{k}\mathrm{a}\mathrm{n}\emptyset \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{s}$

.

kobe-u.

$\mathrm{a}\mathrm{c}$

.

jp

Ready

$\mathrm{s}\mathrm{m}\mathrm{l}>\mathrm{d}\mathrm{r}.\mathrm{s}\mathrm{m}\mathrm{l}$

:

9/26,1995

$—$

Version

4/17,

1997.

$\mathrm{s}\mathrm{m}\mathrm{l}>\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{l}.\mathrm{s}\mathrm{m}\mathrm{l}$

,

1994

$\mathrm{s}\mathrm{m}\mathrm{l}>(\mathrm{b}\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}.\mathrm{S}\mathrm{m}\mathrm{l})$

run

;

bfrest. sml

.

. .

$\mathrm{K}\mathrm{a}\mathrm{n}/\mathrm{s}\mathrm{m}\mathrm{l}$

programs

for

D-modules

Version

970623

by

T. Oaku and

N. Takayama

See

usages

by

(indicial)

usage

:

(restO)

usage

:

(rest-l)

usage

;

$\mathrm{s}\mathrm{m}\mathrm{l}>(\mathrm{b}\mathrm{S}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}.\mathrm{S}\mathrm{m}\mathrm{l})$

run

$*$ $\mathrm{s}\mathrm{m}\mathrm{l}>(\mathrm{t}\mathrm{o}\mathrm{a}\mathrm{S}\mathrm{i}\mathrm{r}.\mathrm{s}\mathrm{m}\mathrm{l})$

run

:

$\mathrm{s}\mathrm{m}1>(\mathrm{X}^{arrow 6}-\mathrm{x}2\wedge \mathrm{y}*3\wedge-\mathrm{y}^{\wedge}5)[(\mathrm{x})(\mathrm{y})]00$

alcl ;

$(\mathrm{x}^{arrow 6^{-}\mathrm{X}2}arrow*\mathrm{y}^{-}3-\mathrm{y}5arrow)[(\mathrm{x})(\mathrm{y})]00$

alcl

:

Computing

an

$\mathrm{F}\mathrm{W}$

-Groebner basis.

Completed.

$\mathrm{s}\mathrm{m}\mathrm{l}>:$

:

$[\-75*\mathrm{y}*_{\mathrm{X}}2arrow \mathrm{D}\mathrm{X}-6**_{\mathrm{X}^{-}}3*\mathrm{D}\mathrm{y}-90*\mathrm{y}^{\sim}2*\mathrm{X}*\mathrm{D}\mathrm{y}+9*\mathrm{x}^{\wedge}2*_{\mathrm{D}}\mathrm{x}-_{3\mathrm{y}2*}*\mathrm{D}\mathrm{x}\star 12*\mathrm{y}*\mathrm{x}*_{\mathrm{D}}\mathrm{y}-450*\mathrm{y}*\mathrm{X}\star 54*_{\mathrm{X}}\sim$

,

$\-9*\mathrm{x}^{\wedge}3*\mathrm{D}\mathrm{x}-15*\mathrm{y}^{\sim}2*_{\mathrm{X}*}\mathrm{D}\mathrm{x}-12*\mathrm{y}*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{y}-18*\mathrm{y}^{-3*}\mathrm{D}\mathrm{y}-54*\mathrm{X}^{arrow 2-}90*\mathrm{y}^{\wedge}2$

,

$375*\mathrm{y}^{\wedge}3*\mathrm{x}*\mathrm{D}\mathrm{X}-\perp 8*\mathrm{x}^{\wedge}4*\mathrm{D}\mathrm{y}\star 30*\mathrm{y}^{\sim}2*\mathrm{X}2\sim*\mathrm{D}\mathrm{y}arrow 450*\mathrm{y}^{\wedge}4*\mathrm{D}\mathrm{y}-54*\mathrm{y}^{\sim}2*_{\mathrm{X}}*\mathrm{D}_{\mathrm{X}^{-}543\mathrm{D}\mathrm{y}}*\mathrm{y}arrow*$

$+2250*\mathrm{y}^{-3^{-}}270*\mathrm{y}arrow 2$

,

$\-21093750*\mathrm{y}2*\mathrm{X}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}-168750\wedge \mathrm{o}*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}-2^{-2}5312500*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}^{\wedge}2\star 421875*\mathrm{x}2*\mathrm{D}_{\mathrm{X}^{-}}2arrow 2\sim$ $+703125*\mathrm{y}^{-}2*\mathrm{D}\mathrm{x}^{\wedge}2+3093750*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}+3375000*\mathrm{y}2\wedge*\mathrm{D}\mathrm{y}^{arrow}2^{-1}01953125*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{x}$ $-6468750*\mathrm{X}^{arrow}2*\mathrm{D}\mathrm{y}-274218750*\mathrm{y}-2*_{\mathrm{D}\mathrm{y}.\mathrm{y}7}+13500\mathrm{o}\mathrm{o}0*\mathrm{x}*\mathrm{D}\mathrm{x}\star 32625\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{D}-61171850*\mathrm{y}$

\star 63281250$ ,

$\-\mathrm{x}-6+\mathrm{y}-3*\mathrm{x}^{-}2+\mathrm{y}^{\wedge}5$

,

$18*\mathrm{x}^{\wedge}5*_{\mathrm{D}+9*}\mathrm{y}\mathrm{y}^{\wedge}2*\mathrm{X}2-\mathrm{D}*\mathrm{x}+15*\mathrm{y}^{\wedge}4*\mathrm{D}\mathrm{x}-6*\mathrm{y}arrow 3*\mathrm{X}^{*}\mathrm{D}\mathrm{y}$

,

$\-3796875\mathrm{o}\mathrm{o}*\mathrm{x}^{arrow}4*\mathrm{D}\mathrm{y}2+263671875*\mathrm{y}-3arrow*\mathrm{D}\mathrm{x}2\sim+28687500*\mathrm{y}*\mathrm{X}^{\wedge}2*\mathrm{D}\mathrm{y}arrow 2$ $\star 32906- 2500*\mathrm{y}3arrow*\mathrm{D}\mathrm{y}2-1\wedge 4501953125*\mathrm{y}^{\wedge}2*_{\mathrm{X}}*\mathrm{D}\mathrm{x}+\perp 6031250*\mathrm{x}^{\wedge}.2*\mathrm{D}\mathrm{x}2arrow$

$-11250000*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{X}2^{-}\wedge 1160156250*\mathrm{y}*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{y}^{-}17402343750*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}\star 3656250*\mathrm{y}*_{\mathrm{X}}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}$ $-236250\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{y}2}-+2274609375*\mathrm{y}*\mathrm{x}*_{\mathrm{D}\mathrm{x}+1}00125000*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{y}+50540625\mathrm{o}\mathrm{o}*\mathrm{y}2\wedge*\mathrm{D}\mathrm{y}$

$-87011718750*\mathrm{y}2\wedge+573750\mathrm{o}\mathrm{o}*\mathrm{X}*\mathrm{D}\mathrm{x}-_{203}062500*\mathrm{y}*\mathrm{D}\mathrm{y}+15925781250*\mathrm{y}-329062500$

,

(4)

$+837\mathrm{o}\mathrm{o}00*\mathrm{X}-3*\mathrm{D}\mathrm{y}3\sim 243\mathrm{o}\mathrm{o}\mathrm{o}00*\mathrm{y}\star-2*\mathrm{x}*_{\mathrm{D}\mathrm{y}^{arrow 3+}\mathrm{y}^{arrow 2*_{\mathrm{X}*}}}14501953125*\mathrm{D}\mathrm{X}2\wedge$ $-16031250*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{X}^{\wedge}3+11250\mathrm{o}\mathrm{o}0*\mathrm{y}-2*\mathrm{D}\mathrm{x}^{-}3+17402343750*\mathrm{y}^{\sim}3*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}$ $-3656250*\mathrm{y}*\mathrm{X}*_{\mathrm{D}\mathrm{D}21}\mathrm{y}*\mathrm{x}^{\sim}-763437500*_{\mathrm{X}3}*\mathrm{D}\wedge \mathrm{y}^{\wedge}2-_{1}3921875\mathrm{o}\mathrm{o}*\mathrm{y}2arrow**\mathrm{x}\mathrm{D}\mathrm{y}^{\sim}2$ $-12555000*_{\mathrm{X}^{arrow}}2*\mathrm{D}\mathrm{y}2\sim \mathrm{D}\mathrm{X}\star 27810*000*\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{y}\mathrm{y}*}arrow 2*\mathrm{D}\mathrm{x}-16740000*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}3$ $-2274609375*\mathrm{y}*\mathrm{X}*_{\mathrm{D}}\mathrm{x}^{\wedge}2+96468750*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}^{-}4948593750*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}$ $+397575000*\mathrm{y}*_{\mathrm{X}}*\mathrm{D}\mathrm{y}2^{-_{1}}160156250*\mathrm{x}2arrow*_{\mathrm{D}_{\mathrm{X}+}101}513671875*\wedge \mathrm{y}2*\mathrm{D}\mathrm{X}-89437500*\mathrm{x}*\mathrm{D}\mathrm{X}^{-}2-$ $-7425\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}+216146250*\mathrm{y}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}-10881\mathrm{o}\mathrm{o}\mathrm{o}0*\mathrm{x}*\mathrm{D}\mathrm{y}^{\sim}2-18141328125*\mathrm{y}*\mathrm{D}\mathrm{x}$ $+12135375\mathrm{o}\mathrm{o}*\mathrm{x}*\mathrm{D}\mathrm{y}-6960937500*\mathrm{X}+28\mathrm{o}\mathrm{o}57500*_{\mathrm{D}\}\mathrm{x}$

,

$\-205031250000*\mathrm{y}^{\wedge}2*\mathrm{x}2*_{\mathrm{D}\mathrm{y}3}4+9887695125*\mathrm{y}3*\mathrm{D}\mathrm{x}4+1044140\sim\wedge 62500*\mathrm{y}^{arrow}3*_{\mathrm{D}2\mathrm{D}}\mathrm{y}^{\wedge}*\mathrm{x}^{\wedge}\simarrow 2$ $+674325\mathrm{o}\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}4\wedge 729\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}000*\mathrm{y}^{arrow}+3*_{\mathrm{D}\mathrm{y}4-}-5438232421875*\mathrm{y}-2*\mathrm{x}*\mathrm{D}\mathrm{x}3arrow$

$+6011718750*\mathrm{x}2arrow \mathrm{D}*\mathrm{X}^{arrow}4-4218750000*\mathrm{y}^{arrow}2*_{\mathrm{D}\mathrm{x}4^{-}}6525878906250*\sim \mathrm{y}^{arrow 3\mathrm{D}}*\mathrm{y}*_{\mathrm{D}\mathrm{X}^{-}}2$

$+1371093750*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{arrow}3+32332500\mathrm{o}\mathrm{o}*\mathrm{x}2\wedge*_{\mathrm{D}\mathrm{y}00}-2*\mathrm{D}\mathrm{X}^{\wedge}2-115200\mathrm{o}\mathrm{o}0*\mathrm{y}^{\sim}2*\mathrm{D}\mathrm{y}^{arrow 2*_{\mathrm{D}\mathrm{x}^{\wedge}}}2$ $-19629843750\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{x}^{\sim}2*_{\mathrm{D}\mathrm{y}264}arrow 3^{-}684375\mathrm{o}\mathrm{o}0*\mathrm{y}^{\sim}3*\mathrm{D}\mathrm{y}3-\wedge 437625000*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}3\sim*\mathrm{D}\mathrm{X}$ $-6331500\mathrm{o}\mathrm{o}0*\mathrm{y}^{arrow}2*\mathrm{D}\mathrm{y}^{arrow}4+852978515625*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{X}^{\wedge}3^{-}20250000000*\mathrm{X}2\wedge*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{-}2$

$+182531250\mathrm{o}\mathrm{o}00*\mathrm{y}2arrow*\mathrm{D}\mathrm{y}*_{\mathrm{D}}\mathrm{x}^{-}2+67668750\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}2*_{\mathrm{D}}\mathrm{x}+37248750\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{X}^{-}2*\mathrm{D}\mathrm{y}^{\wedge}3$

$+285795000000*\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{y}*\mathrm{y}2}arrow 3+435058593750*_{\mathrm{X}^{\wedge}}2*_{\mathrm{D}_{\mathrm{X}}2435}\sim-05859375000\wedge*_{\mathrm{D}\mathrm{x}2}-$

$+45562500000*\mathrm{x}*\mathrm{D}_{\mathrm{X}^{\wedge}}3^{-}1560058593750*\mathrm{y}*\mathrm{X}*_{\mathrm{D}\mathrm{D}\mathrm{x}8}\mathrm{y}*-4048750000*\mathrm{y}*\mathrm{D}\mathrm{y}*_{\mathrm{D}\mathrm{X}^{\wedge}}2-38$ $349843750\mathrm{o}\mathrm{o}*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{y}^{\wedge}2-9076640625000*\mathrm{y}2*_{\mathrm{D}\mathrm{y}0}\simarrow 2+599625\mathrm{o}\mathrm{o}0*\mathrm{x}*\mathrm{D}\mathrm{y}2*\sim \mathrm{D}\mathrm{X}$

$-81164250000*\mathrm{y}*\mathrm{D}\mathrm{y}-3+7652109375000*\mathrm{y}*_{\mathrm{D}}\mathrm{X}^{arrow}2\star 244251562500*\mathrm{X}^{*\mathrm{D}}\mathrm{y}*\mathrm{D}\mathrm{X}$

$+22572225\mathrm{o}\mathrm{o}000*\mathrm{y}*\mathrm{D}\mathrm{y}2\wedge 1\star 80\iota 75781250*\mathrm{X}*\mathrm{D}\mathrm{X}-73665000\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{D}\mathrm{X}^{arrow}2^{-}32455898437500*\mathrm{y}*_{\mathrm{D}\mathrm{y}}$ $-218907\mathrm{o}\mathrm{o}\mathrm{o}000*_{\mathrm{D}\mathrm{y}^{\wedge}241}+75150625\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{D}\mathrm{y}-1\tau 191406250\mathrm{o}\mathrm{o}0$

,

$\-2460375\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}0*\mathrm{y}3arrow \mathrm{y}*_{\mathrm{X}^{*_{\mathrm{D}}}}5^{-98}876953125*\mathrm{y}^{\wedge}3arrow \mathrm{D}*\mathrm{x}^{-5-}949218750\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}^{arrow}2*_{\mathrm{D}\mathrm{X}^{\wedge}}3$ $-4556250\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}^{-4}*\mathrm{D}\mathrm{X}^{-}1730160000*_{\mathrm{X}^{\wedge}}3*_{\mathrm{D}\mathrm{y}*}arrow 5+10497600000\mathrm{y}^{\wedge}2*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}5$

$+5438232421875*\mathrm{y}^{\wedge}2*\mathrm{x}*\mathrm{D}\mathrm{X}^{-4^{-_{601}}}1718750*\mathrm{x}2arrow*\mathrm{D}\mathrm{x}5\sim 4+2187500\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{x}5\wedge$

$+6525878906250*\mathrm{y}3\wedge*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{\wedge}3-1371093750*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}4^{-}\sim 2656125000*\mathrm{x}^{\wedge}2*_{\mathrm{D}}\mathrm{y}2\wedge \mathrm{D}*\mathrm{X}3\wedge$ $+11115000000*\mathrm{y}2*\mathrm{D}arrow \mathrm{y}^{arrow 2\mathrm{D}}*\mathrm{X}^{arrow}3+6264843750\mathrm{o}\mathrm{o}*\mathrm{y}^{-}3*\mathrm{D}\mathrm{y}3*\mathrm{D}\mathrm{x}+569250\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}arrow**\mathrm{X}\mathrm{D}\mathrm{y}3-*_{\mathrm{D}\mathrm{x}}2\wedge$

$+15187500\mathrm{o}\mathrm{o}*\mathrm{x}^{\wedge}3*\mathrm{D}\mathrm{y}^{arrow 4}arrow 415985625\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}2*_{\mathrm{X}^{*_{\mathrm{D}\mathrm{y}^{arrow}*\mathrm{D}4}}}4+259524\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*_{\mathrm{X}2}arrow \mathrm{y}^{\wedge}*\mathrm{D}\mathrm{X}$

$+4615920\mathrm{o}\mathrm{o}0*\mathrm{y}^{arrow}2*\mathrm{D}\mathrm{y}4*\mathrm{D}_{\mathrm{X}+}3460320000*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}5-85297851562arrow 5*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{x}^{arrow}4$

$\star 9808593750*\mathrm{x}^{\wedge}2*_{\mathrm{D}*}\mathrm{y}\mathrm{D}\mathrm{X}-3-1785761718750*\mathrm{y}\wedge 2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{-}3-62353125\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{x}2}\wedge$ $+29991375000*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}^{-3}*_{\mathrm{D}\mathrm{x}-}238561875000*\mathrm{y}^{\sim}2*\mathrm{D}\mathrm{y}^{\sim}3*_{\mathrm{D}\mathrm{x}}+158460300\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}*_{\mathrm{X}*\mathrm{D}}\mathrm{y}^{\wedge}4$

$-435058593750*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{X}^{\wedge}3+48944091796875*\mathrm{y}2*\mathrm{D}\mathrm{x}^{-}3-575859375\mathrm{o}\mathrm{o}*\mathrm{x}*\mathrm{D}\mathrm{x}^{\wedge}4-$

$\star 3039257812500*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}2\wedge\star 81057656250*\mathrm{y}*_{\mathrm{D}\mathrm{D}}\mathrm{y}*\mathrm{x}^{\wedge}3+524742187500*\mathrm{X}^{\wedge}2*_{\mathrm{D}\mathrm{y}*}\sim 2\mathrm{D}\mathrm{X}$ $+10851679687500*\mathrm{y}-2*\mathrm{D}\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{x}-8979750000*\mathrm{X}*\mathrm{D}\mathrm{y}^{\sim}2*_{\mathrm{D}}\mathrm{x}^{\sim}2-190329750\mathrm{o}\mathrm{o}0\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}3$ $+6410061\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{D}\mathrm{y}^{arrow 3\mathrm{D}\mathrm{x}}*+29412720000*\mathrm{x}*\mathrm{D}\mathrm{y}4-8482939453125*\mathrm{y}arrow*\mathrm{D}\mathrm{x}3\wedge$

$-4715578125\mathrm{o}\mathrm{o}*\mathrm{x}*\mathrm{D}\mathrm{y}*_{\mathrm{D}}\mathrm{x}^{\wedge}2-232742250\mathrm{o}\mathrm{o}00*\mathrm{y}*\mathrm{D}\mathrm{y}^{\sim}2*\mathrm{D}\mathrm{x}+587146950000*\mathrm{X}*\mathrm{D}\mathrm{y}^{\wedge}3$ $+428906250\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{X}*\mathrm{D}\mathrm{x}2\sim+_{272925}\mathrm{o}\mathrm{o}000*\mathrm{D}\mathrm{x}^{\wedge}3+47920429687500*\mathrm{y}*\mathrm{D}\mathrm{y}*_{\mathrm{D}_{\mathrm{X}}}$

$-20893415625000*\mathrm{X}^{*}\mathrm{D}\mathrm{y}^{arrow 2}+176690790000*\mathrm{D}\mathrm{y}*25612812500*_{\mathrm{D}*_{\mathrm{D}_{\mathrm{X}}}}\mathrm{y}arrow 2\mathrm{D}_{\mathrm{X}^{-}}58$

+27365625000000*Dx$

,

$36905625000000*\mathrm{y}4*_{\mathrm{D}\mathrm{y}9}-619140625*\mathrm{y}^{-}3*\mathrm{D}\mathrm{x}\wedgearrow 61235\sim 6^{-}10678710937500*\mathrm{y}-3*_{\mathrm{D}\mathrm{y}^{\wedge}}2*_{\mathrm{D}\mathrm{X}4}\sim$

$-341718750000*\mathrm{y}3*\mathrm{D}arrow \mathrm{y}^{arrow 4*\mathrm{D}\mathrm{x}^{\wedge}}2+478224000000*\mathrm{y}*\mathrm{X}^{-}2*\mathrm{D}\mathrm{y}^{-}6-188082\mathrm{o}\mathrm{o}0\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}3\wedge \mathrm{D}*\mathrm{y}^{\wedge}6$

$+679779052734375*\mathrm{y}^{\wedge}2*\mathrm{x}*\mathrm{D}\mathrm{X}5arrow-751464843750*\mathrm{x}2arrow*\mathrm{D}\mathrm{x}6-52734375000\dagger \mathrm{o}*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{x}^{arrow}6$ $+815734863281250*\mathrm{y}^{-}3*_{\mathrm{D}}\mathrm{y}*_{\mathrm{D}\mathrm{x}41713}-86718\sim 750*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}5\wedge$

$-25987\dot{5}0\mathrm{o}\mathrm{o}\mathrm{o}00*\mathrm{x}^{\sim}2*_{\mathrm{D}}\mathrm{y}^{\sim}2*_{\mathrm{D}\mathrm{X}^{-}}4+_{133875}0000000*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{y}-2*\mathrm{D}\mathrm{x}-4$ $+78310546875000*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}^{\wedge}3*_{\mathrm{D}\mathrm{X}^{\wedge}}2+87609375\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}-3*\mathrm{D}\mathrm{x}-3$ $+245754\mathrm{o}\mathrm{o}0\mathrm{o}\mathrm{o}0*\mathrm{X}^{\sim}2*_{\mathrm{D}\mathrm{y}0}arrow 4*\mathrm{D}\mathrm{X}^{-}2\star 39062250000*\mathrm{y}2\wedge*\mathrm{D}\mathrm{y}^{\wedge}4*\mathrm{D}\mathrm{x}arrow 2$

(5)

$-33260625\mathrm{o}\mathrm{o}000*\mathrm{y}*\mathrm{X}^{\wedge}2*_{\mathrm{D}}\mathrm{y}5+96228000\mathrm{o}\mathrm{o}0\mathrm{o}\mathrm{o}\mathrm{o}arrow \mathrm{o}*\mathrm{y}^{\sim}3*\mathrm{D}\mathrm{y}5+88560\mathrm{o}\mathrm{o}00\sim \mathrm{o}*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}5*\mathrm{D}\mathrm{X}$ $-425088\mathrm{o}\mathrm{o}\mathrm{o}000*\mathrm{y}^{arrow 2\mathrm{D}}*\mathrm{y}^{\sim}6-106622314453125*\mathrm{y}*_{\mathrm{X}}*\mathrm{D}\mathrm{X}^{arrow}5-79101562500*_{\mathrm{X}^{\wedge}}2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{arrow}4$ $-218276367187500*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{\sim}4-7129687500000*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}2\wedge*\mathrm{D}\mathrm{x}^{\wedge}3$ $+3969140625000*\mathrm{x}^{\sim}2*_{\mathrm{D}3*_{\mathrm{D}\mathrm{X}}}\mathrm{y}2-\sim\wedge 23622890625\mathrm{o}\mathrm{o}0*\mathrm{y}2arrow*\mathrm{D}\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{x}^{-}2$ $+610065\mathrm{o}\mathrm{o}00000*\mathrm{y}*_{\mathrm{X}}*\mathrm{D}\mathrm{y}^{\wedge}4*_{\mathrm{D}+359110}\mathrm{X}8000000*\mathrm{x}^{\sim}2*\mathrm{D}\mathrm{y}5\sim-35450055000000*\mathrm{y}^{\sim}2*\mathrm{D}\mathrm{y}^{\wedge}5$ $-54382324218750*\mathrm{x}^{\wedge}2*_{\mathrm{D}4}\mathrm{x}\sim+6797790527343750*\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{X}^{\wedge}}4^{-_{87}}01171875000*\mathrm{X}*\mathrm{D}\mathrm{x}^{arrow}5$ $+564807128906250*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}^{arrow}3+9758320312500*\mathrm{y}*\mathrm{D}\mathrm{y}*_{\mathrm{D}\mathrm{x}^{\wedge}}4$

$+803847656250\mathrm{o}\mathrm{o}*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}^{-}2*_{\mathrm{D}}\mathrm{X}^{arrow 2+15}78339843750000*\mathrm{y}2\wedge 2*\mathrm{D}*\mathrm{D}\mathrm{y}^{arrow}\mathrm{X}^{arrow 2}$ $-120684375\mathrm{o}\mathrm{o}00*_{\mathrm{X}}*\mathrm{D}\mathrm{y}2*_{\mathrm{D}\mathrm{x}^{arrow 3^{-_{68}}}}491406250000*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}-3*_{\mathrm{D}\mathrm{x}}arrow$

$+6137167500000*\mathrm{y}*\mathrm{D}\mathrm{y}^{\sim}3*_{\mathrm{D}2^{-}2}\mathrm{x}^{\sim}4440906250000*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}^{arrow 4+}786858046875\mathrm{o}\mathrm{o}00*\mathrm{y}2*\mathrm{D}\sim \mathrm{y}4\wedge$

$+726192\mathrm{o}\mathrm{o}0000*_{\mathrm{X}}*\mathrm{D}\mathrm{y}-4*\mathrm{D}\mathrm{x}-7111368\mathrm{o}\mathrm{o}\mathrm{o}000*\mathrm{y}*\mathrm{D}\mathrm{y}5-11642arrow 21191406250*\mathrm{y}*_{\mathrm{D}\mathrm{x}^{\wedge}}4$ $-925787109375\mathrm{o}\mathrm{o}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}^{arrow}3^{-}292514062500000*\mathrm{y}*\mathrm{D}\mathrm{y}^{-}2*\mathrm{D}\mathrm{x}2\wedge\star 53763975\mathrm{o}\mathrm{o}0000*_{\mathrm{X}}*\mathrm{D}\mathrm{y}^{\sim}3*\mathrm{D}\mathrm{X}$ $-177632527500\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{D}\mathrm{y}-4\star 129748535156250*\mathrm{x}*\mathrm{D}_{\mathrm{X}3}arrow-_{3}887929687500*\mathrm{D}_{\mathrm{X}^{-}}4$ $+8292919921875\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{\sim}2+1160648437500\mathrm{o}\mathrm{o}*\mathrm{X}*\mathrm{D}\mathrm{y}^{arrow}2*\mathrm{D}\mathrm{x}\star 1720872000\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*_{\mathrm{D}2}\mathrm{y}^{\wedge}*_{\mathrm{D}\mathrm{x}^{\wedge}}2$ $+22847796562500000*\mathrm{y}*\mathrm{D}\mathrm{y}^{-3}-261207720\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}0*_{\mathrm{D}}\mathrm{y}^{\sim}4-989171718750000*_{\mathrm{D}*_{\mathrm{D}\mathrm{X}^{\wedge}}}\mathrm{y}2$

$-203966628750000*_{\mathrm{D}\mathrm{y}^{arrow 3}9}+4953515625000000*_{\mathrm{D}2}\mathrm{x}^{\wedge}+1873168593750\mathrm{o}\mathrm{o}0*\mathrm{D}\mathrm{y}2\wedge$

$]$ $\mathrm{s}\mathrm{m}\mathrm{l}>\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}$

;

2.

Residue and residual duality

Let

$X$

be a

domain

in

$\mathrm{C}^{n}$

.

Let

$f_{1},$ $f_{2},$

$\ldots.,$$f_{n}$

be a regular sequence of holomorphic

functions

on

$X$

.

Let

$\mathcal{I}$

be

the ideal

generated

by

$f_{1},$ $f_{2},$

$\ldots.,$

$f_{n}$

over

$\mathcal{O}_{X}$

.

Let

us

denote

by

$\in \mathcal{E}_{Xt_{\mathcal{O}}^{n}}(X\mathcal{O}\mathrm{x}/\mathcal{I}, \mathcal{O}x)$

the

Grothendieck.

residue

symbol

associated to the

meromorphic

function

$\frac{1}{f_{1}f_{2}\cdots f_{n}}$

.

Let

$i$

be the canonical

map

$\mathcal{E}xt_{O}^{n_{X}}(\mathcal{O}x/I, \mathcal{O}_{X})arrow \mathcal{H}_{[A]}^{n}(\mathcal{O}x)$

,

where

$A=\{z\in X|f_{j}(z)=0, j=1,2, \ldots,n\}$

.

Set:

$m=i()\in H_{[A]}^{n}(\mathcal{O}x)$

.

We

assume

that the

common

locus

$A$

consists

of

finite

number of points

$A_{k},$

$k=1,2,$

$\ldots,N$

.

Corresponding

to

the

decomposition of the algebraic local

cohomology

group

$\mathcal{H}_{[]}^{n_{A}}(\mathcal{O}x)$

(6)

we have

$m=m_{1}+m_{2}+\cdots+m_{N}$

with

$m_{k}\in H_{[A]}^{n_{k}}(\mathcal{O}x)$

.

Let

$\Omega_{X}$

be

the sheaf on

$X$

of holomorphic

differential

$\mathrm{n}$

-forms. The

canonical

pairing

$\Omega_{X}\mathrm{x}H_{[A}^{n_{k}}](\mathcal{O}_{X})arrow H_{[A]}^{n_{k}}(\Omega_{X})$

composed with

$H_{[A]}^{n_{k}}(\Omega_{X})arrow \mathrm{C}$

defines the residue

pairing at the point

$A_{k}$

.

Put

${\rm Res}_{A_{k}} \langle\phi(z), m\rangle=\frac{1}{(2\pi i)^{n}}\oint_{A_{k}}\phi(_{Z})mdz$

.

We regard

${\rm Res}_{A_{k}}$$\langle \cdot, m\rangle$

as a linear map

$\Omega_{X}\ni\phi(z)dZarrow{\rm Res}_{A_{k}}\langle\phi(_{Z}), m\rangle\in \mathrm{C}$

.

Note There exists

$m_{k}\in \mathrm{N}$

and complex constants

$c_{\beta,k}(0\leq|\beta|\leq m_{k})$

such that for every

$\phi dz\in\Omega x$

,

${\rm Res}_{A_{k}} \langle\phi(_{Z}), m\rangle=\sum_{0\leq|\beta|\leq m_{k}}C_{\beta,k}\cdot((\frac{\partial}{\partial z})^{\beta}\phi)(A_{k})$

.

3.

Main Theorems

Let

$X$

be

a

domain in

$\mathrm{c}^{n}$

:

Let

$f_{1},$ $f_{2},$ $\ldots.,$

$f_{n}$

be

a regular sequence of holomorphic

functions

on

$X$

.

Let

$A=\{z\in X|f_{j}(z)=0, j=1,2, \ldots, n\}$

.

Let

us denote by

$m$

the

residue class

$m=i()\in H_{[}^{n_{A}}(\mathrm{J}\mathcal{O}_{X}).$

.

We assume that

the

common

locus

$A$

consists of finite number of points

$A_{k},$

$k=1,2,$

$\ldots,$

$N$

.

We

have

$m=m_{1}+m_{2}+\cdots+m_{N}$

with

$m_{k}\in H_{[A]}^{n_{k}}(\mathcal{O}x)$

.

The following theorem asserts that the

cohomology

class

$m$

can

be

characterized as a

solution

of linear

partial differential

equations up to

constant factor.

Theorem

A

Let

$J=\{P\in D_{X}|Pm=0\}$

be the annihilator ideal

of

$m$

.

Then at

each

point

$A_{k}$

,

we

have

$\{u| Pu=0, u\in H_{[A]}^{n_{k}}(\mathcal{O}_{X}), P\in J\}=\{cm_{k}|c\in \mathrm{C}\}$

.

Proof

Put

$\mathcal{M}_{k}=\mathcal{H}_{[]}^{n_{A_{k}}}(\mathcal{O}_{X})$

.

We

have

$m_{k}\in \mathcal{M}_{k}$

. Since

the

algebraic local cohomology group

(7)

Hence

we have

$\mathcal{H}om_{\mathcal{D}_{X}}(Dx/J, \mathcal{M}_{k})$

$=\mathcal{H}om_{\mathcal{D}_{X}}(Dxm,\mathcal{M}_{k})$

$=\mathcal{H}omv_{X}(Dxm_{k}, \mathcal{M}_{k})$

$=\mathcal{H}om_{\mathcal{D}_{X}}(\mathcal{M}k, \mathcal{M}_{k})$

.

The claim

follows from

th.e

fact that

$\mathcal{H}om_{\mathcal{D}_{X}}(\mathcal{M}_{k}, \mathcal{M}_{k})=\mathrm{c}|_{A_{k}}$

.

q.e.d.

Let

us recall the fact that

$\Omega_{X}$

is

naturally

endowed with

a structure of a right

$D_{X}$

-module.

The

right action of

$P= \sum a_{\alpha}(z)(\frac{\partial}{\partial z})^{\alpha}$

is described explicitly as follows.

$(\phi(z)dZ)P=(P^{*}\phi)(Z)d_{Z}$

,

where

$P^{*}= \sum(-\frac{\partial}{\partial z})^{\alpha}a_{\alpha}(z)$

is the formal

adjoint

of

$P\in D_{X}$

.

If

$P\in J$

,

then we have

${\rm Res}_{A_{k}}\langle(P^{*}\psi(z))d_{Z,m\rangle}={\rm Res}_{A_{k}}\langle(\psi(z)dz,Pm\rangle=0$

.

Furthermore,

we

$\mathrm{h}\mathrm{a}.\mathrm{v}\mathrm{e}$

the

$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{i}_{0}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$

Theorem,

which is a

direct

consequence of a result

of

Kashiwara

[11].

Theorem

$\mathrm{B}$

Let

$J$

be

the annihilator ideal

of

$m$

.

Then, we

have

$\{\phi(z)d_{Z}\in\Omega_{X}|{\rm Res}_{A_{k}}\langle\phi(z)dz,m\rangle=0,\forall k=1,2, \ldots, N\}$

$=\{(P*\psi)(z)dZ|P\in J, \psi(\mathcal{Z})dz\in\Omega \mathrm{x}\}$

.

Note

that,

for

the

case

of

one

variable,

Theorem

$\mathrm{B}$

provides

a new theoretical foundation

of the

Horowitz-Ostrogradski algorithm

([10])

for

$\dot{\mathrm{t}}$

he

integration of rational functions.

4.

Examples

Example Let

$X=\{(x, y)|x,y\in \mathrm{C}\},$

$f(\dot{x}, y)=y^{2},$

$g(x,y)=y-x^{2}$

.

The

$\tilde{\mathrm{m}}$

ultiplicity

of

intersection

of these two curves at the

origin

is equal to 4. The

coho-mology class

(8)

satisfies

the

following

system

of

linear

partial

differential equations.

$\{$

$y^{2}m=0$

,

$(y-x^{2})m=0$

,

$(xD_{x}+2yD_{y}+6)m=0$

.

It

is

easy to see

that the

annihilator ideal

$J$

of

$m$

is

generated

by

these three

operators:

$J=\langle y^{2}, y-x^{2}, xD_{x}+2yD_{y}+6\rangle$

.

Put

$m= \sum a_{\alpha,\beta}[\frac{1}{x^{\alpha}y^{\beta}}]$

.

Since

$(xD_{x}+2yD_{y}+6) \frac{1}{x^{\alpha}y^{\beta}}=(-\alpha-2\beta+6)\frac{1}{x^{\alpha}y^{\beta}}$

,

we

have

$m=[ \frac{a_{2,2}}{x^{2}y^{2}}+\frac{a_{4,1}}{x^{4}y}]$

.

The second equation

$(y-x^{2})m=0$

implies

$m= \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}[\frac{1}{x^{2}y^{2}}+\frac{1}{x^{4}y}]$

.

Let

$\mathcal{I}=\langle y^{2}, y-X^{2}\rangle$

be the

ideal

generated by

$y^{2}$

and

$y-x^{2}$

over

the ring

$\mathcal{O}_{X}$

.

Then the

quotient

space

$\Omega_{X}/\Omega_{X}\mathcal{I}$

is

a

4-.dimensional

vector

space.

Put

$K=$

{

$\phi(x,$

$y)dx$

A

$dy|{\rm Res}_{[]}0,0(\phi(x,$

$y)dX$

A

$dy,$

$m)=0$

}.

Obviously

we have

$\Omega \mathcal{I}\subset K$

.

Since

$P^{*}=-xD_{x}-2yD_{y}+3$

,

we have

$P^{*}1=3,$

$P^{*}x=2x,P^{*}x^{2}=x^{2},$

$P^{*}x^{3}=0$

.

Therefore,

the differential

forms

$dx$

A

$dy,$

$xd_{X}$

A

$dy$

and

$x^{2}dx$

A

$dy$

belong

to

$K$

and the

differential

form

$x^{3}dx$

A

$dy$

gives a

representative of a

non-trivial

element of

$\Omega_{X}/IC$

.

Example

Take

$f(x, y)=(x^{2}+y^{2})^{2}+3x^{2}y-y^{3},$

$g(x, y)=y-x^{2}$

.

Let

$A=\{(x, y)|f(x, y)=g(x, y)=0\}$

.

Then

$A=\{(0,0)\}\cup\{(x, y)|y-x^{2}=0, y^{2}+y+4=0\}$

.

Put

$m=[ \frac{1}{fg}]\in \mathcal{H}_{[A]}2(\mathcal{O}X)$

.

Let

$J\subset D_{X}$

be the

annihilating

ideal

of

$m$

.

Then

$\{Q_{1}, Q_{2}, P_{1}, P_{2}, P_{3}\}$

is an

involutory

base of

the ideal

$J$

,

where

$Q_{1}$

$=$

$-x^{2}+y$

,

$Q_{2}$

$=$

$-y^{4}-y^{\mathrm{s}_{-4y}2}$

,

$P_{1}$

$=x(y^{2}+y+4)D_{x}+2y(y^{2}+y+4)D_{y}+10y^{2}+8y+24$

,

$P_{2}$

$=$

$y(y^{2}+y+4)D_{x}+2xy(y^{2}+y+4)D_{y}+2x(4y^{2}+3y+8)$

,

(9)

Since

$P_{1}$

$=$

$(xD_{x}+2yD_{y}+6)(y^{2}+y+4)$

,

$P_{2}$

$=$

$(yD_{x}+2xyD_{y}+4X)(y^{2}+y+4)$

,

$P_{3}$

$=$

$(-yD_{x}^{2}+6yD_{y}+12)(y^{2}+y+4)$

,

hold,

the annihilator ideal

$J$

of

the cohomology

class

$m$

is

generated

by

$y-x^{2},$

$y^{2}+y+4$

over

$D_{X}$

at

$\{(x, y)|y-x^{2}$

.

$=0, y^{2}+y+4=0\}$

.

5.

Appendix

Let

$f_{1},$ $f_{2},$

$\ldots,$

$f_{n}\in \mathrm{C}[z_{1}, Z_{2}, \ldots, z_{n}]$

be

a regular

sequence

of

polynomials.

Let

$A=\{z\in$

$\mathrm{C}^{n}|f_{1}(z)=f_{2}(z)=\cdots=f_{n}(z)=0\}$

be the

common locus of

$f_{1},f_{2},$

$\ldots,$

$f_{n}$

.

We assume

that

$f_{1},$$f_{2},$$\ldots,f_{n}$

are

in general position,

i.e.,

the

Jacobian determinant

$Jac= \frac{\partial(f_{1},f_{2},\ldots,fn)}{\partial(z_{1},z2,\ldots,Zn)}$

does

not

vanish

at any point

$A_{k}\in A$

.

Then

we

have

${\rm Res}_{A_{k}} \langle\phi(_{Z),m\rangle}=\frac{\phi(A_{j})}{Jac(A_{k})}$

.

By

rewriting

the above

relation:

we

get

$Jac(A_{k})\cdot ResAk\langle\phi(z), m\rangle-\phi(Aj)$

.

Let us introduce a new indeterminant

$t$

.

We see that the residues of

$\frac{\phi(z)d_{Z}}{f_{1}f_{2}\cdots f_{n}}$

should

satisfy

$\{$

$JaC(\mathcal{Z})t-\phi(z)=0$

,

$f_{1}(z)=f2(Z)=\cdots=f_{n}(\mathcal{Z})=0$

.

We arrive

at

the

following method for computing the residues of

$[ \frac{\phi(z).d_{Z}}{f_{1}f_{2}\cdot\cdot f_{n}}]$

.

$\bullet$

Set

$I=\langle f_{1}(z), f2(_{Z)}, \ldots, f_{n}(z), JaC(_{\mathcal{Z})-}t\phi(Z)\rangle\subset \mathrm{C}[z_{1,2,n}z\ldots, Z,t]$

.

$\bullet$

Compute

the

Gr\"obner

basis of the

$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\dot{\mathrm{l}}$

$I$

with respect to pure.lexicographic order

$z\succ t$

and then perform the primary decomposition of the

polynomial ideal

$I$

.

Note that

the above method

is a natural

generalization

of

$\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{r}- \mathrm{L}\mathrm{a}\mathrm{Z}\mathrm{a}\mathrm{r}\mathrm{d}- \mathrm{R}\mathrm{i}\mathrm{o}\mathrm{b}_{0}\mathrm{o}$

and

(10)

Example Let $f(x, y)=y-x2,$

$g(x, y)=y-X-2,$

$\phi(x, y)=1$

.

Put

$h(x, y, t)=Jac(x, y)\cdot t-1$

,

where

$Jac(X, y)=-2x+1$

is the jacobian

determinant

of

$f,g$

.

Let

$I=\langle f,g, h\rangle\subset IC[x, y, t]$

.

Then the

Gr\"obner

base

of

$I$

with respect to the pure

lexicographic

$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}x\succ y\succ t$

is

$\{-9t^{2}+1,2y+9t-5,2X+9t+1\}$

.

The

primary decomposition

of this

ideal is given

by

$\langle 3t+1, y-4,x-2\rangle,$

$\langle 3t-1,y-1, x+1\rangle$

.

We thus get

${\rm Res}_{(2,4)}\langle 1,$ $[ \frac{1}{fg}])=-\frac{1}{3}$

,

${\rm Res}_{\mathrm{t}^{-1,1})} \langle 1, [\frac{1}{fg}]\rangle=\frac{1}{3}$

.

Example Let

$f(x, y)=(x^{2}+y^{2})^{2}+3x^{2}y-y^{3},$

$g(x, y)=3x^{2}+3y^{2}-1,$

$\phi(x, y)=1$

.

Put

$h(x, y, t)=JaC(x, y)\cdot t-1$

,

where

$Jac(x.’ y)$

is

the

jacobian determinant of

$f,g$

.

Let

$I=\langle f,g, h\rangle\subset K[x, y,t]$

.

Then the

Gr\"obner

base of

$I$

with

respect

to

the

pure

lexicographic ordering

$x\succ y\succ t$

is

(11)

References

[1]

A. Altman and

S.

Kleiman, Introduction to

Grothendieck

duality theory, Lecture Notes in

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