INDEX THEOREMS AND MICROSUPPORT

INDEX THEOREMS AND MICROSUPPORT

L. Boutet de Monvel

(Univ.ParisVI)

パリ 6 大学

Vn thesenotes

we propose

to describe again theAtiyah-Singer indextheorem for systems of differentialoperators, and related extensions such

as

theindex theorem for Toeplitzoperators

or

the relative indextheorem proved by B. Malgrange and the author. We make

a

special emphasis

on

the microlocal contribution produced by the sheaf in which the solutions of the differentialequations

are

computed,

as

described by P.Schapira and

J.P.Schneiders, believing thatthispoint ofview, although

as

yetincomplete, sheds

a

new

and unifying light.

We describe this presentation oftheindex theorem in \S 1. Vt

was

notpossible in these short notesto give completeand detailed proofs,and

we

havelimited ourselves to give in

3

appendixes short, buthopefullyuseful, descriptions of the main ingredients: K-theory, $\mathscr{D}-$

modules, and theidea ofthe proof.

Description of the index theorem

Let

us

recall thatcomplex ofvectorspaces is a sequence

(1) $p$: $...arrow E_{k}^{\underline{p}}SE_{k+1}arrow\ldots$ $(k\in Z)$

of complex vector

spaces

$E_{k}$ and linearmaps$p_{k}\in L(E_{k};E_{k+1})$ such that$p_{k+1^{o}}p_{k}=0$ (wemay

identifyalinear

map

$p:E_{o}arrow E_{1}$toacomplex of length 2, concentrated in degrees$0$and 1). The cohomology $H^{*}(p)$ is the gradedvector

space

$H^{*}(p)=kerp/V_{l}np(H^{k}(p)=kerp_{k}/{\rm Im} p_{k- 1})$.

a

is

a

Fredholm complexif$H^{*}(p)$is finitedimensional, ie. the $H^{k}(p)$

are

finite dimensional

and vanishexceptfor

a

finitenumberof indices$k$. Then the index (Euler characteristic) of

a

is defined

as

the alternating

sum:

(2) $Vndp=\sum(- 1)^{k}di_{I}nH^{k}(p)$

The index theorem is$concel\cdot ned$ with the index of complexes inwhich the $p_{k}$

are

differential operators on a manifoldX and the $E_{k}$

are

suitablespaces of distributions

on

X

or

partsof X. Vt has long been known that undersuitable ellipticity conditions the indexexists,

and that itis quite stableunder smallperturbations

or

$defo\iota lnations$_,

so

one

expectsthatit

can

be computed in terms ofsimpler topological $inval\cdot iants$of the data. We firstrecall what

these formulas look like.

The modelfor all indexformulas isHirzcbruch‘s$f^{\backslash }o11nu1\cdot nion$ of the Riemann-Roch

are

the Chem character ofthe sheafand theToddclassof the

space.

LaterA.Grothendieck

gave a

relativeversion ofthe Riemann-Roch theorem, in which the topological ingredientis

the behaviourunder directimage ofa K-theoretical elementassociated to the sheaf. Let

us

recall the formulation ofBaum, Fulton and Mac Pherson ofGrothendieck’s theorem: let X be a projective analytic

space

and $Z\subset X$

a

subspace. TheGrothendieck

group

$K_{Z}^{an}(X)$ is the

group

generated by isomorphy classes ofcoherent $\mathcal{O}_{X}$-modules with supportin $Z$, and the

relations $[M]=[M’]+[M^{\dagger\prime}]$ for eachexact

sequence

$0arrow M^{\dagger}arrow Marrow M’arrow 0$. There isacanonical

homomorphism$K_{Z}^{an}(X)arrow K_{Z}^{top}(X)$,where $K_{Z}^{top}(X)$ is the Atiyah

group

of“virtual vector

bundles with support in $Z’$, whichdescribes theadditive and deformation invariant

propertiesof complexes ofvectorbundles which

are

exactoutside of Z. Accordingto Baum,

Fulton,Mac Pherson, therelative Riemann-Roch theorem statesthat thishomomorphism

commutes to

proper

directimages (italsocommutes to inverseimages). Thisshould be

complemented bythe description of theK-theoretical image, whichis constructedby

means

ofthe Bott periodicitytheorem. Onemay furthertranslatethis in teims ofcohomology, using the Chemcharacter:

(3) ch : $K_{Z}^{top}(X)\otimes \mathbb{Q}arrow\sim H_{Z}^{p\tau ir}(X,\mathbb{Q})$

The Hirzebruch-Riemann-Rochtheorem is the cohomologicaltranslation when then goal manifoldis

a

point (recall that$Earrow dimE$ defines thecanonical isomorphism $K(point)\simeq Z$). In

this analysisthe Todd class

appears

when

one

interpretsthe K-theoretical directimage

(whosecohomological intelpretation isnotintegration alongfibers).

The Atiyah-Singer index formuladeals withelliptic complexes of differential operators

on

a realmanifold. Since holomorphic functions

are

solutionsof the elliptic system of Cauchy-Riemann equations, this formula contains the complex Riemann-Roch formula from whichit

was

inspired. Althoughthe first proof of this formula

was

cohomological and closeto thatofHirzebruch, the published proof of

1968

is

more

inspired by thatof Grothendieckandconvincingly shows thatK-theory is

a

naturaltoolin this context.

A similar

case

where

one

hasan index theorem isthefollowing: ifXis

a

complex

manifold,$U\subset X$ arelatively compactopen subset with smooth boundary $\partial U$,and

$p$ a complex ofholomorphic differentialoperators

on

X,

_one

defines the

non

characteristicity of

$\partial U$for

$p$ ; thiscondition is closely relatedto theellipticity condition inthe realcase, and

when it istrue,the complex $p$ acting

on

germsof$ho1_{omo1])}hic$ sections

near

$Uu\partial U$is

Fredholm and its indexis essentially given by the

same

Riemann-Roch formula(the

same

In thesenotes

we

restrictourselvesto operators withanalyticcoefficients

on

analytic manifolds. Thisis notan important$resl\cdot iction$ for the topological aspect oftheindex formula

because this deals with homotopyclasses ofcontinuous functions, _{which usually contain}

real-analytic functions. It does make

a

difference forthe analysisand geometry of the differential operators involved:

some

pathologies

are

avoided , butmostly in the analytic

setting

we

disposeof

a

good algebraic and geometricformalism similar tothose of algebraic

or

analytic geometry, whichwould notexist othelwise,

eg.

there is

a

good notion ofsupports andcharacteristic sets, and direct

or

inverseimages.

Inthissetting thedatafor the index theorem is the following: first

we

have

a

complex manifoldX;

a

realmanifold willalways beconsidered

as

the gernofa subset$X_{0}\subset X$ (the

set of realpoints in acomplexmanifold). OnX wehavea differential system withanalytic

coefficients,_bestdescribed

as a

coherent$\mathscr{D}$-module $\mathscr{M}(\mathscr{D}$ denotes thesheafofanalytic

differential operators). Finally

we

haveasheaf of coefficients,9‘ _{in which}

_we

_{compute the}

solutions. We will represent the differential system by

a

sheaf$\mathscr{M}$ ofright$\mathscr{D}$-modules(or

more

generally by

an

$ob\backslash |ect$ withcoherentcohomology of the derivedcategoryofthese); $\mathscr{F}’$

is

a

sheaf of left $\mathscr{D}$-modules (or

more

generallyan

$ob\dot{\mathfrak{s}}ect\backslash$of the$del\cdot ived$ category); thesheaf

of solutions is thecomplete tensorproduct $\Lambda l^{L}\otimes_{\emptyset}\mathscr{T}$ (an object of the derivedcategoryof

sheavesofvectorspaces), and the index (ifitexists) isthealtemating

sum

ofthe Betti numbers ofits globalsections:

(4) Index$( \Lambda\ell,\mathcal{T})=\chi(\mathscr{M}^{L}\otimes_{\emptyset}\mathscr{T})=\sum(- 1)^{1}(\otimes_{\emptyset^{ff)}}$

(One

can

equivalentlyrepresent the differentialsystem by

a

left $\mathscr{D}$-module,the sheaf of

solutions being RHom$(\mathscr{M},\Psi))$.

Let

us

point outtypical

cases

forthesheaf$\mathscr{F}’$ ofcoefficients:

a

first exampleisthe

case

where$\mathscr{F}’$ is the sheaf$\mathcal{O}$ of holomorphic functions; in this

case

(Xcomplexcompact),

we

maytake $\mathscr{M}=m\otimes_{\mathcal{O}}\mathscr{D}$ with$m$

a

coherent9-module, and the index theorem willgive back

the Riemann-Roch theorem (in fact

one

gets alittle

more

since coherent left $\mathscr{D}$-modules

are

not all oftheform $m\otimes_{\mathcal{O}}\mathscr{D}$ ). More generally

one can

choose$\mathscr{T}$

a

coherentleft $\mathscr{D}$-module;

theellipticity conditionis then that

ssne

and SS.9“ meetalonga compactset (inthe

zero

section of$T^{*}X$). The index formulaisthen the formulabelow.

Asecond exampleis thefollowing: let$Y$ beaclosed subsetofX and $\mathscr{T}=\mathcal{O}_{Y}$thesheaf

ofgelms ofholomorphic functions along $Y(F=i_{*}i^{-1}\mathcal{O}$if $i$ isthe canonical inclusion $Yarrow X$,

so

that for thestalks

we

have$\mathscr{T}_{x}=\mathcal{O}_{x}$if$x\in Y$, and$()$ otherwise).The

case

ofToeplitz

ofX,with smooth boundary $\partial U$. The

case

ofoperators

on a

real manifold corresponds tothe

case

where$Y=X_{0}$isthe set ofreal points ofX

as

above. More generally Schapira and

Schneiders haveexamined the

case

$T=\mathcal{O}\otimes$

’

where$f$ is

a

$real- const\iota uctible$ sheaf.

Wemay

now

describe the structure ofthe indexformula: to

ne

is associated its microsupport SS$\mathscr{M}$, anda K-theoreticalelement

$[\prime M]\in K_{SS\mathscr{M}}(T^{*}X)$ (when $\mathscr{M}$ is given

as a

complexofdifferential operators withsymbolexactoutsideof$Z=SS\mathscr{M},$$[\ovalbox{\tt\small REJECT}]$ is the element

of$K_{SS\mathscr{M}}(T^{*}X)$definedby the symbol; in general cf. Boutet deMonvel-Malgrange). In the

examples describedabove we

can

also define themicrosupport$SS\mathscr{T}$, andin the best

cases

a

K-theoretical element $[\mathscr{T}]\in K_{SS\mathscr{T}}(T^{*}X)$. The ellipticity condition is that$SS\mathscr{M}\cap SS\mathscr{T}$ is

compact.Then theindex formula then reads

as

follows: Index theorem: (5) Index$(\mathscr{M},\mathscr{T})=\chi^{top}([\prime M][\mathscr{T}])$

where$\chi^{top}$is the canonical K-theoretical character_{$K_{Z}(T^{*}X)arrow Z$} arising from the complex

stlucture of$T^{*}X$ (cf. appendixl), and $[ffi][ff]$ is the K-theoreticalproduct (ithascompact

support $SS\ovalbox{\tt\small REJECT}\cap SS\mathscr{T}$). In the

cases we

will describe below$\mathscr{T}$ is associated to asimple set $U$,

and $SS\mathscr{T}=SSU$has acomplextubularneighborhoodwhoseBottelementis precisely $[\mathscr{T}]$,

so

the indexformula

can

be$rewl\cdot itten$

(5)$bis$ Index$(\Lambda l,\mathscr{T})=\chi^{top}([\mathscr{M}]|SS\mathscr{T})$

If,9‘ _is $\mathscr{D}$-coherent, $SS\mathscr{F}’$ is its charateristic setas mentionned; if$\mathscr{T}=\mathcal{O}\otimes$

’

with$r$

a

real-constructiblesheaf, SSSi isthe microsupport of$r$

as

defined by Kashiwara and

Schapira. The index formula inthe

case

$T=\mathcal{O}\otimes f$ with$f$ a$\iota\cdot eal$-constluctible sheaf

was

described by Schapiraand Schneiders,in terms of the microlocalEulerclasses

_of’

and

ne.

Here

we

havedescribed the index formula in terms ofK-theory; this

seems more

natural in view of Grothendieck’s.and $Atiyah- Singer^{t}s$ work,and also forformulaswith parameters. At

this stage thisdescription is notcomplete, although it containsallprevious

cases

ofthe index theorem:

we

need toassociate to the sheaf ofcoefficients$\mathscr{T}$

a

K-theoretical element

$[\mathscr{T}]\in K_{SST}(T^{*}X)$. This

was

done by M.Ohana only inthe simpler

cases

$\mathscr{T}=\mathcal{O}_{Y}$ if$Y$is

a

real-analytic submanifoldwith

comers

of$Y$; itremains tobe done in

more

general

cases.

To conclude letus pointoutthat the preceeding description also applies to the relative case, asdescribed inBoutet deMonvel-Malgrange: let$f:Xarrow Y$ bea submersion of complex analytic manifolds. Then

one

defines the transfer module $\mathscr{D}_{Xarrow Y}$(the sheafonX of

“differentialoperators’t oftype $\mathcal{O}_{Y}arrow \mathcal{O}_{X}$: $\varphiarrow P((Q\varphi)_{0}f)$with$P$,

resp.

$Q$,

a

differential

$H\simeq T^{*}Y\cross YX$of horizontal covectors. Let$U\subset X$ be an open setwith smoothboundary $\partial U$:

then SSO$U$

was

defined above (itis thezerosection above$U$ andthe set ofoutgoing normal

covectors on $\partial U$). We set

$\mathscr{D}_{1Iarrow Y}=\mathscr{D}_{Xarrow Y}\otimes_{\mathcal{O}\chi}\mathcal{O}_{1I}$ ; then, although the microsupportis not

generally defined in thiscontext,

a

reasonable definitionis $SS\mathscr{D}_{Uarrow Y}=SS\mathscr{D}_{Xarrow Y}+SS\mathcal{O}_{U}$(at leastwhen the

map

SS$\mathscr{D}_{Xarrow Y^{\cross}X}SS\mathcal{O}_{U}arrow T^{*}X$isproper). We mayextend the projection $F$:

$SS\mathscr{D}_{Xarrow Y}=T^{*}Y\cross YXarrow T^{*}YtoamapF_{e}:SS\emptyset_{1Iarrow Y}$ by requiring thatit isconstantalong real

half-lines parallel tothe normal outgoing real half-lines along$\partial U$.

When $\mathscr{M}$ is acoherent 9-module

on

X, westudied in Boutetde Monvel-Malgrange

the directimage of the $geI^{\cdot}m$ of

ne

along $Uu\partial U$:

(6) $f_{U^{+}}(\mathscr{M})=Rf_{*}(\mathscr{M}\otimes_{\emptyset x^{\mathscr{D}_{Uarrow Y)}}}$

(whichdescribesintuitively thedifferential relations

on

$Y$between

germs

of solutions of$\mathscr{M}$

along the fibersof $f$ in $Uu\partial U$). Thegeometric ellipticity condition is that $SS\mathscr{M}\cap SS\mathscr{D}_{Uarrow Y}$

iscontained in $H$; thegeometric finiteness condition is thattherestriction Fe:

$SS\mathscr{M}\cap SS\mathscr{D}_{Uarrow Y}arrow T^{*}Y$is

proper.

Therelativeellipticity condition (with respectto U) is

that any section of$\mathscr{M}$ is killed by

some

veItical operatorwhich is non-characteristic along

$\partial U$ anditcannotbe read onthe principal symbol alone. It impliesgeometric ellipticity, and

together with the strongercompactness conditionabove, _{as was}shown by Houzel-Schapira, it impliesrelative finiteness, ie. that $f_{U^{+}}(\ovalbox{\tt\small REJECT})$ hascoherent cohomology. In this

case we

have

$SSf_{U^{+}}(A\ell)\subset Z=F_{e}(SS\mathscr{M}\cap SS\mathscr{D}_{1Iarrow Y})$and the relative index formula of Boutet de

Monvel-Malgrange

can

bewritten:

Relativeindex formula: (7) $[f_{lI^{+}}(\swarrow u)]_{Z}=F_{e*}[\mathscr{M}]_{SS\mathscr{M}}|$ SS$\mathscr{D}\iota\intarrow Y$

where$F_{e*}$ is the K-theoretical image (the relation to theformula above isthat the product

$[\mathscr{M}]_{SS\mathscr{M}}[ss\mathscr{D}_{tIarrow Y]conesponds}$to $[\mathscr{M}]_{c}\backslash |$ SS$\mathscr{D}_{tIarrow Y}$ by the Bottisomorphism from

Appendix 1. $K$ theory

$a$

.

Definitions Let X be aparacompacttopological space.We recall that the Atiyah group

$K(X)$isthe

group

generated byisomorphyclassesofvectorbundlesand the groupoid$law\oplus$.

If$Z\subset X$

a

closed subset,_{$K_{Z}(X)$} is the

group

ofequivalence classes $[a]$ of bounded complexes

a ofC-vectorbundles onX, exactoutside of$Z$_, wherethe equivalencerelationis generated

by the relations:

(All) (i) $[a]+[b]=[a\oplus b]$

(ii) $[a]=()$if thereexistsa deformation ofato anexact complex, exactoutof Z.

Infact

any

elementof$K_{Z}(X)$ is theclass oflength 2: if

(A12) a: $...arrow E_{k}^{\underline{a}}5E_{k+1}arrow\ldots$ $(k\in Z)$

is

a

bounded complex

on

Xexact out of$Z$,

we

have $[a]=6(a)\in K_{Z}(X)$,where$\delta(a)$ is the

operator ($=complex$oflength 2) $a+a^{*}$ : $\sum E^{2k}arrow\sum E^{2k+1}$.

K-theory is equipped with aproduct, colTesponding to the tensorproduct of complexes ofvectorbundles: if$u\in K_{Z}(X)$ and$v\in K_{Z^{\dagger}}(X)$then$uv\in K_{Z\cap Z’}(X)$;in particular$K(X)$ in an algebraand $K_{Z}(X)$ ida K(X)-module.

Let$H$ be aHilbertspaceand let Fred$(H)\subset L(H)$ be theset ofFredholm operators. If

$Z\subset X$

as

above

we

denote _{$F_{Z}(X)$} thegroup of homotopyclasses ofcontinuous functions $A$

:

$Xarrow Fred(H)$invertible outside of Z. SinceGL(H) is contractible(by N.Kuiper’s theorem)

thisis identical the groupof homotopyclasses of Fredholm Hilbert bundleswhich

are

exact

outside of$Z$ (acomplex ofHilbert bundles isFredholm ifits cohomologyisfinite

dimensional ateach point). Thereis

an

obvious map $K_{Z}(X)\in F_{Z}(X)$ because

a

finite-dimensional complex is

a

particular

case

of Fredholm Hilbertcomplex. $J\ddot{a}nisch^{t}s$ theory shows thatthis is

an

isomorphism if$Z$iscompact,

or

if theCech dimension ofXis finite.

Theinverse

map

is the index

map

and is denoted$Ind_{Z}$.

$b$

.

Inverseimage If$f$is

a

continuous

map

$Xarrow Y$,theinverse image forvectorbundles

induces

an

inverse image$f^{-1}$ for K-theory:

$K_{Z}’(Y)arrow K_{Z}(X)$ ,if$Z\supset f^{-1}Z’$. In $pal\ddagger icular$ if$U$ is

an open

setof X and $Z\subset U$ isclosedin X, there is

a

restriction

map

_{$K_{Z}(X)arrow K_{Z}(U)$}. IfXis

finite dimensional this is always

an

isomorphism (excision) (aFredholm

map

a on

$U$

invertibleoutside of$Z$ can be deformed into 1 outsideof

some

smallneighborhood of$Z$by

$Kuiper^{t}s$theorem,

so

itshomotopy class

can

beextended).

$c$

.

Products K-theoryis multiplicative: if$\xi=[a]\in K_{Z}(X)$ and $\eta=[b]\in K_{Z’}[X]$ where$a$ ,

$Z$resp. $Z^{t}$,then _{$\xi\eta=[a\otimes b]\in K_{Z\cap Z’}(X)$} , where$a\otimes b$ isthetensorproduct of complexes (itis

exactwherever

one

ofthefactorsis). Slightly

more

generallyif

we

have$Z\subset Y\subset X$, and

$\xi=[a]\in K_{Z}(Y),$ $\eta=[b]\in K_{Y}(X)$,

we

definetheproduct$\xi\eta\in K_{Z}(X)$ : it isthe class $[\sim a\otimes b]$ where

5

is anyextension ofato X (ifa isa Fredholmfamily,

we

may takeanyextension,

Fredholm

or

not, of

a:

theproduct $\sim a\otimes b$ willstillbeexact(thus Fredholm) outsideof$Y$

because$b$ is).

$d$

.

Bott isomorphism If$N$ isacomplexvector bundle

on

X,

we

defineits Koszul

complex, whichis

a

complex ofvectorbundles

on

$N$ considered

as a

topological

space:

(A13) $k_{N}$

...

$\Lambda^{-k}(p^{-1}N^{*})arrow\Lambda^{-k+1}(p^{-1}N^{*})arrow\ldotsarrow(p^{-1}N^{*})arrow Carrow 0$

where $pNarrow X$is theprojection, $N^{*}$ the dual bundleof$N$_, and thediferentialat

a

point$n\in N$

is theinteliorproduct $\omegaarrow n_{L}\omega$.

$k_{N}$isexactoutside of the

zero

section (which

we

identifywithX) anddefines an

element$[k_{N}]\in K_{X}(N)$. The Bottmap$\beta_{NX}$isthe K-theoretical multiplication by $[k_{N}]$

(A14) $\beta_{NX}$ : _{$K_{Z}(X)arrow K_{Z}(N)$}

$Atiyah|sfo\iota mulation$ofthe Bottperiodicity theorem is that(ifX is finite dimensional)

this

map

is always an isomolphism (cfBoutetdeMonvel-Malgrangefor the

case

wherethe support $Z$is$\neq X$). Theproof in Boutet de Monvcl-Malgrange describes theinverse

map

as

the index of families of Toeplitz operators. Thus the Bott periodicitytheorem

appears

as

the firstand fundamental

case

of all index theorems, and essentiallyall proofs oftheindex theorem consist in reducing tothis formula.

$e$

.

K-theoretical image Weend thissection by thedescription of the K-theoretical push-forward. Let

us

notice that the Bott element $[k_{N}]$ is

as

well defined by the bundle

map

$\delta(k_{N})=k_{N}+k_{N}^{*}$ : $A_{N}^{+}arrow A_{\overline{N}}$

where $A_{N}^{+}=\Lambda^{even}N^{*}$ and $A_{\overline{N}}=\Lambda^{odd}N^{*}$(wehave chosen

some

hermitian metric to define

the adjoint$k_{N}^{*}$). Thusthe Bott element andthe Bottmap

are

still well definedif$N$is only

a

real vectorbundle equippedwith

a spinc

structure.1

Spinc

$structUl\cdot es$also give riseto

a

$\wedge I^{1}$

Aspincstructureon areal vectorbundle$N$consistsof aeuclideanmetricon$N$_,andasimple graded

Cliffordbundle(ie _avectorbundle$A_{N}=/6_{N}^{+}+A_{\overline{N}}$endowed withastructure of$C_{N}$-module, where$C_{N}$isthe

Cliffordalgebra of N-generatedby$NaI\iota d$the relationsn.n$=-||_{I\iota||2}$ for each_{$n\in N$,} whichissimple ateach

Grothendieck

group

$K^{S}P^{in^{C}}(X)(A_{N\oplus N’}=A_{N}\otimes^{gr}A_{N’})$of virtual spinc structures. This hasthe

following property(which doesnot hold forcomplex structures): if$E$is areal vector bundle,

and $\xi\in K^{S}P^{in^{C}}(X)$

a

virtual

spinc

stmcturewith underlyingrealvirtual bundle [E], then there

is aunique compatible

spinc

stlucture

on

$E^{2}$

If X and$Y$

are

twosmooth manifolds and$f$

a

differentiable

map

$Xarrow Y$,

a

spinc structure

on

$f$is a spinc_{$stluctuI^{\cdot}e$}

on

the nonnal bundle$N(f)=f^{-1}[TY]-[TX]\in K(X)$. Amap

between complex (orsymplectic) manifoldscarries acanonical spincstructure.

For$f$spinc and

proper

on $Z\subset X$ ,the K-theoretical push-forward _{$K_{Z}(X)arrow K_{f(Z)}(Y)is$} defined bythefollowingaxioms:

- itis covariant: $(fg)_{*}=f_{*}g*$

-if$f:Xarrow N$is the

zero

section of

a

spincbundle _(equipped with theobvious spinc

structure),then$f_{*}$ is the Bott isomorphism.

- it is compatible with

a

change ofbasis, ie. if

we

have

a

diagram

X $-Garrow X$ $\downarrow F$ $\downarrow f$

$Y-garrow Y$

where X,$Y,$ $X’,$$Y^{\dagger}$ aremanifolds _$f,$

$g,$ $F,$ $G$ differentiablemaps,$f$and _$g$

are

transversal and

$Y^{1}=Y\cross_{x}X’,$ $f$and $F$

are

equipped with compatiblespincstructures. Iffurther

we

have

supports $Z\subset X,$$Z^{t}\subset X’,$$T\subset Y,$ $T^{t}\subset Y’$ with $Z^{1}\supset G^{-1}(Z),$_{$T’\supset g^{-1}(T),$ $T\supset f(Z),$ $T’\supset F(Z’)$} then

$G^{-1}f*=F*g^{-1}$: $Kz(Y)arrow K_{T}’(X^{\dagger})$.

Thus $f_{*}$ is defined if$f$is

a spinc

immersion (as thecomposition oftheBott

isomorphism of

a

tubular neighborhood, and theexcision map). Itis also definedif$f$is the

projection$X=Y\cross C^{n}arrow Y$(with thecanonical

spinc

structure),

as

the composition of the

restriction

map:

$K_{Z}(X)arrow K_{B(X)}$ where$B\supset Z$is aball bundlewith basis_$f(Z)$ and of radius

some

suitably large continuousfunction, and oftheinverseofthe Bott isomorphism

$K_{B}(X)arrow K_{f(}z)(Y)$(this iswell defined because the pair$(X,f(Z))$

can

be deformed in thepair

(X,B)). In the general casef isthe compositionoftwosuch

maps.

In particularif X has

a spinc

structure (eg. a complex

or

asymplectic structure),and

$Z\subset X$ iscompact, thecharacter _{$\chi^{top}:K_{Z}(X)arrow Z=K(point)$} is the_{$push- f_{01}ward$} mapby the

spinc map$Xarrow point$.

2 This followsfrom tbefact that if$NaI|dN^{t}\subset N$areequipped withspincstrutures, andweset$N”=N^{\prime\perp}$

,then

$A_{N’’}=Hom^{gr_{C_{N}’}},(A_{N’},A_{N})$is(upto$i_{L}\backslash omo_{c}\backslash 1]1hi\backslash m$)theuniquespincstructure such that

Appendix2.

9-Modules

In the introduction

we

mentionned that the natural framework to describe linear differentialsystemsis the theory of$\mathscr{D}$-modules. Thissectiongives

a

r\’esum\’e ofthe basic

definitionsandconstructions concerning 9-modules. Forfurther details

see

thebookof

Sato-Kawai-Kashiwa,

or

Boreletal seminar (in the algebro-geometricsetting), andthe

books of Kashiwara5, Schapira, and Bjork,

or

the Grenobleseminar(Boutet de Monvel-Lejeune-Malgrange, 1965). Manyoperations

on

$\mathscr{D}$-modules

are a

superposition ofseveral

manipulationsand

are

suitably described in terms of$de\iota$ived categories; forthis

we

referto

the same, andVerdier,Borel etal 2, Kashiwara-Schapira2.

$a$

.

Introduction LetXbe

an

analytic manifold. We denote by $\mathcal{O}$, thesheafof

complex valued analytic functions, $\Omega$thesheaf ofdifferential folms of maximum degree (densities) and

9

the sheaf of analytic differential operators: $\mathcal{O}$ is

a

left$\mathscr{D}$-moduleand$\Omega$

a

right$\mathscr{D}$-module. $\mathscr{D}$ has

a

canonical filtration (

$\mathscr{D}_{m}$is thesub-sheafof operators of degree$\leq m$)

and gr$\mathscr{D}$ identifieswith the sheaf ofsections ofthesymmetric algebra STX

(equivalently-the gradedsheafof polynomial functions on$T^{*}X$_, with coefficients in $\mathcal{O}_{X}$). $\mathscr{D}$ is coherent

(because$gr\mathscr{D}$ is). A system of differential equations onX isoften described

as a

complex of

differential operators

(A21) $p$

:

... $E^{k}arrow p_{k}$ $E^{k+1}arrow\cdots$

where the $E^{k}$

are

analytic vectorbundles

on

X (here

we

denote by the

same

letter the sheaf

ofanalytic sectionsofE), and wherewe areinterested in the solutionsheaves,_ie. the cohomologysheaves of$p^{3}$ One is alsointerested bythe solutionswith $C^{\infty}$

or

distribution

coefficients,

_{or more}

_{generally with coefficients} in a sheaf$N$of left $\mathscr{D}$-modules.

A

more

systematic description consists in defining

a

differential system$P$

as a

complex of right$\mathscr{D}$-modules (orrather

an

$ob\backslash \dot{\mathfrak{s}}ect$ ofthe derived category ofthese)

4;

the

sheafof solutionswith coefficients in $N$is the completetensorproduct

(A22) $Sol(P, N)=P\otimes_{\emptyset}^{\llcorner}N$

3_{Forexample if}$p:E^{0}arrow E^{1}$ isof length 2, thehomolo\’oysheaf$H^{0}$ is$tl$₎$e$sheaf of solutions of theequation

$pf=0$,and$H^{1}$ is the sheafofobstructionstosolving the equation_$pf=g$.

4 _{the sheaf of right9-modules associated to}

$p$(A-3) is

$P^{d}$ : $arrow 9^{k}arrow\delta^{k+1}arrow\cdots$ withS_{$k=Diff(O,E^{k})$}

thesheafofdifferentialoperatorsoftype$0arrow E^{k}$_{, anddifferential$Qarrow p_{0}Q$,} so_that$P^{d}\otimes O$ istheinitial

complex$p$. Simnilarilyoneassociatesto$p$thecomplex$pg$ ofleft$\emptyset$-modules Diff$(E,O)$,for which Sol(P)$=Hom_{\emptyset}(P^{g},0)$.

which isan object of the derived categoryofsheaves. Familiar systems correspondto

coherent$\mathscr{D}$-modules (orcomplexes withcoherentcohomology). Infact

we

will be mostly

dealing with $\mathscr{D}$-modules which

possess

good

filtrations.5

$b$

.

Characteristic set

In general, if$M$is a left (orlight) $\mathscr{D}_{X}$-module, thecharacteristic set,

or

analytic

microsupport SS$M$is theconic set$supp(\delta\otimes_{p^{-1}9}p^{-1}M\rangle=T^{*}X$, where$p$is theprojection

$T^{*}Xarrow X$and

8

is the sheaf

on

$T^{*}X$ofanalytic pseudo-differential operators. This definition

extends tocomplexes

or

objects of the derivedcategory. If$M$iscoherent (orhas coherent

cohomology), SS $M$is also thesupportof

gr

$M$for any good filtration, and itis

an

analytic

involutive subsetof$T^{*}X^{6}$ (cfthe bookofSato, Kawai,Kashiwara).

$c$

.

Directandinverseimage

Let X, $Y$ be analytic manifolds and _{$f:Xarrow Y$} an analytic map.We define the

transfer module$\mathscr{D}_{Xarrow Y}$

as

the sheaf

on

X off-differentialoperatorsoftype $\mathcal{O}_{Y^{arrow}}\mathcal{O}_{X}$ (local

sections

are

operatorsofthe form $u\in 0yarrow P_{X}(Qyu_{o}f)$ with $P_{X}\in \mathscr{D}_{X},$ $Q_{Y}\in \mathscr{D}_{Y}$). $\mathscr{D}_{Xarrow Y}$is

a

left$\mathscr{D}_{X}$-module and

a

right$f^{-1}\mathscr{D}_{Y}$-module. Similalily

one

defines the transfermodule$\mathscr{D}_{Yarrow X}$

as

the sheafonX: itis thesheaf off-differential operatorsoftype$\Omega_{Y}arrow\Omega_{X}$,

a

$(f^{-1}\mathscr{D}_{Y},\mathscr{D}_{X})-$

bimodule.

Examplel-If $f$ is submersive

we

have the relative DeRham complex _{$d_{X/Y}$}:

(A23) $d_{X/Y}$ : $0arrow \mathcal{O}_{Y}arrow T^{*}X/Yarrow\cdots\Lambda^{k}T^{*}X/Yarrow\Lambda^{k+1}T^{*}X/Yarrow\cdots$

As

a

left $\mathscr{D}_{X}$-module, $\mathscr{D}_{Xarrow Y}$ is generated bythepullback operator$\epsilon(uarrow uoD$; it is

a

flat

$f^{-1}\mathscr{D}_{Y}$-module (as$\mathcal{O}_{X}$ is flat

on

$f^{-1}\mathcal{O}_{Y);}$ and thecomplex $DR_{X}^{g_{!Y}}$ ofleft$\mathscr{D}$-modules

associated to$d_{X/Y}$ is

a

locally free resolution of$\mathscr{D}_{Xarrow Y}$ (the augmentation is $P\in \mathscr{D}xarrow P_{O}\epsilon$).

If$f:Xarrow Y$is

an

immersion

we

have$\mathscr{D}_{Xarrow Y}\simeq \mathcal{O}_{X}\otimes_{f^{-1_{\mathcal{O}_{Y}}}}f^{1}\mathscr{D}_{Y}$. Itis

a

locally$free\mathscr{D}_{X^{-}}$

module; locally$f$is isomorphic with the

zero

section of

a

vectorbundle, whose Koszul

complex defines (extendingcoefficientsto $\mathscr{D}_{Y}$) alocallyfree resolution of$\mathscr{D}_{Xarrow Y}$

on

$\mathscr{D}_{Y}$.

5 _{Agoodfiltrationon a9-module}$M$isafiltration$M=\cup M_{k}$,with$M_{k}O_{X}$coh\’erent,$M_{k}=0$if$k<<0,$$M_{k}9_{p}\subset$

$M_{k+p}$,with equality if$k\gg O$.AfD-moduleiscoherent iffitpossesseslocally goodfiltrations. Algebraic

coherent9-modulesandholonomic9-modulespossessglobal goodfiltrationsbutingeneralexistenceofa

globallydefinedgood filtrationon acoferent9-moduleisnot known.The canonicalflitrationof9 isagood

filtration.

6_ie.if$f$and$g$vanishonchar,$\mathscr{M}$ thensodoes their Poisson bracket {f,g}$= \sum\partial f/\partial\xi_{j}\partial g/\partial x_{j}-\partial f/\partial x_{j}\partial g/\partial\xi_{j}$ (inany

If$.\parallel l$ is aright

$\mathscr{D}_{X}$-module (resp. aleft $\mathscr{D}_{Y}$-module) thedirect image _{$f_{+}M$} is the object

ofthe derivedcategory ofright $\mathscr{D}_{Y}$-modulesdefinedby

(A24) $f_{+}M=Rf_{*}(M^{\llcorner}\otimes 9_{X}\mathscr{D}_{Xarrow Y})\in obD^{b}(\mathscr{D}_{Y})$

(resp. the inverse image$f^{+}M$is

(A25) $f^{+}M=F^{1}(M)\otimes\emptyset_{Y})\mathscr{D}_{Yarrow X}\llcorner$[-d] _{$\in obD^{b}(\mathscr{D}_{Y)}(d=dimX- dimY)$}

one

defines similarilythe directimage ofaleft$\mathscr{D}_{X}$-module

or a

right $\mathscr{D}_{Y}$-moduleusingthe

transfer module $\mathscr{D}_{Yarrow X)}$.

Example2-If$f:Xarrow Y$is aclosedimmersion, the directimage$f_{+}$ is defined for$\mathscr{D}$-modules:

thefunctor$Marrow f_{*}M\otimes \mathscr{D}_{Xarrow Y}\emptyset_{X}$ isexact; itis

a

categoryequivalence (Kashiwara

equivalence)between $\mathscr{D}_{X}$-modules and$\mathscr{D}_{Y}$-modules “algebraically” supportedby X(ie.

any

of whosesectionsiskilled by

some

powerof the ideal ofX).

Duallyif$f:Xarrow Y$is asubmersion withcohomologically$tl\cdot ivial$ fiber-eg.X is the

germ of

some

manifold alonga continuoussection off-itis shown in Boutetde Monvel-Malgrangethat theinverse image$f^{+}\backslash$ realizes a

categoryequivalencebetween coherent$\mathscr{D}_{Y^{-}}$

modules andthe categoryof coherent $\mathscr{D}_{X}$-modules which

are

$regulal\cdot ily$characteristic along

horizontalcovectors of the submersion (thecharacteristicset of$d_{X/Y}$). This is particularily

useful in thefollowing

case:

$Y$ is

a

complex manifold,_{$X=Y_{R}$} the real sublying

manifold-germ of$Y\cross\overline{Y}$

along thediagonal, and $f:Xarrow Y$ is thecanonical projection.

$d$

.

Symbol

,

K-theoretical element associated toa

9-module

LetX be

a

complex manifold, and$M$ a $\mathscr{D}$-modulepossessing agood filtration. If

$Z\subset T^{*}X$ is

a

closedconic set containing SSM

we

define:

(A26) $[M]_{Z}^{an}\in K_{Z}^{an}(T^{*}X)$

the elementof the Grothendieck

group

of homogeneous sheaves

on

$T^{*}X$. This element(or

ratherits restrictionto any subset with compact basisin$T^{*}X$) doesnot depend

on

the choice

of

a

goodfiltration

on

M.

We also definethe topological symbol (A27) $[M]_{Z}^{top}\in K_{Z}(T^{*}X)$

of right$\mathscr{D}$-modulespossessinggood filtrations(orsimilatily for

a

left$\mathscr{D}$-module) by the

(i)itis additiveforexact

sequences

ie. $[M]_{Z}^{top}=[M’]_{Z}^{top}+[M^{\dagger\prime}]_{Z}^{top}$if thereexistsanexact

sequence $0arrow M’arrow Marrow M^{\dagger\dagger}arrow 0$

(ii) if$M$corresponds (as above) to

a

complex$P$ of differential operators whose symbol $\sigma(P)$ isexactoutside of$Z$,then $[M]_{Z}^{top}=[\sigma(P)]_{Z}$(theelement of $Kz(T^{*}X)$ defined by$\sigma(P)$).

(iii) itis compatible with submersiveinverse images. More$P^{lecisely}$ let$f:Xarrow Y$be

a

submersion.If$M$is

a

coherent 9-module (possessing

a

good filtration)

we

have$f^{+}M=$

$f^{-1}M\otimes_{r-\iota_{9}}\mathscr{D}_{Yarrow X}[d](d=dimX1Y)$. Denote $F:f^{-1}T^{*}Y=X\cross_{Y}T^{*}Y\simeq’ T^{*}X$thecotangent

map: itsimageis $H=car\mathscr{D}_{Yarrow X}\subset T^{*}Y$, theset of$ho\iota$izontalcovectors; denote $\overline{f}$

:

$F^{1}T^{*}Yarrow T^{*}Y$ the projection. Then

we

_havecar$(t(M))=F\overline{F}^{1}$_(car$M$) $\subset H$, and

(A28) $[fM]_{\Gamma Z}^{to_{\vdash}p}=\overline{f}^{-1}[M]_{Z}^{top}$ . $[\mathscr{D}xarrow Y[d]]_{Z}^{to,p}=F_{*}f^{-1}[M]_{Z}^{top}-(d=dimX/Y)$

the K-theoreticalimage. These axiomsdefine $[M]_{Z}^{top}$ if Xis the germof

a

complex manifold along

a

compactset,

or

if X is

a

$pro_{\backslash }|ective$manifold, since $M$then possessesa ftgood“ locally free resolution (correspondingto a locally free $1^{\cdot}esolution$ or_$grM$), whose symbol

defines $[M]_{Z}^{top}$. In thegeneral

case

the real sublying manifold$X_{R}$ is Stein,

so

[$M_{R]_{Z_{R}}^{top}}$is well

defined, so as $[M]_{Z}^{top}$since in thiscasethe K-theoretical map $F_{*}\Gamma^{1}$: _{$K_{Z}(T^{*}X)arrow K_{fZ}(T^{*}X_{R})$}

is

one

to

one

(itis theBott isomorphism). Reducing similarilyto the casewhere Xis real

one

shows that the symbol $[M]_{Z^{()}}^{tp}$iswell behaved underclosed immersions:

(iv) If$f:Xarrow Y$ is aclosedimmersion, $M$ a$\mathscr{D}_{X}$-module witha good filtration and

car

$M=$

Z. Then $f_{+}M$

possesses

agood filtration,

we

have$carf_{+}M=\overline{f}F^{-1}(Z)$_, and

(A29) $[f_{+}M]_{Z}^{to}l^{1}=\overline{f}_{*}F^{-1}[M\}_{Z}^{op}$ (K-theoretical image($with\overline{f}:F^{1}(T^{*}Y)arrow T^{*}Y$ the

projection, and $F:f^{-1}(T^{*}Y)arrow T^{*}X$ the cotangent_map)

Remark-if$m$ is

a

coherent$\mathcal{O}_{X}$-modulewith $support\subset Z$,

we

setnote $[m]_{Z}^{top}\in K_{Z}(X)$

whoseinverseimage

on

$T^{*}X$ is $[m\otimes_{\mathcal{O}_{X}}\mathscr{D}_{X}]_{p^{-1}}^{top_{Z}}\in K_{p^{-1}Z}(T^{*}X)$. Thisdefinitioncommutes

with submersionsandclosedimmersions. 7 _{This definition of}

_course

_{coincides withthatof}

Baum, Fulton, Mac-Pherson when itis defined (eg. if X is

a

Stein

or

projective manifold

so

$m$ has finite locally free resolutions). In their workBaum, Fulton, Mac-Pherson

prove

the

result forimmersions by “deformation to the normal cone“.

7 _{The canonicalmap}$K_{Z}^{an}(X)arrow K_{Z}^{top}(X)$wasdefinedbyBaum-Fulton-Mac PhersonwhenX isaprojective

space,usingadeformationtothenolmalcone.Ourdefinitionusesthereal sublyingmanifold$X_{R}$soX doesnot

needstobeprojective. Notethatgoing from Xto$X_{R}$,oneloosesnothing at thelevel of$\emptyset$-modules,butone

Appendix

3.

Sketch ofthe proof oftheindexformula

The data fortheindex theorem consistsinacomplex manifoldX, asystem of differentialequationsonX described byacoherentright$\mathscr{D}$-module$M$ (possessing agood

filtration),andasheafofcoefficients$N$ (aleft$\mathscr{D}$-module).

In whatfollows $N$will alwaysbe of the form _{$N=0_{U}$}where $U$is

a

realanalytic

submanifold with boundary (orcorners).8 The microsupport SS $N=SSU$is then thesetof all ”outgoing“ nolmalcovectors,ie complex covector $\zeta$ suchthat${\rm Re}\zeta$is negative

on

the 1st

order jet of$U$(thismakes

sense

unambiguously if$M$is

a

real analytic submanifold with

corners).

example-if$U$ isatotally realsubmanifold in X (real case), SSU is the setof

pure

imaginarycovectors atpoints of U.

If$U$is asubmanifold ofrealdimension $2dimX$_, withboundary$\partial U$a real-analytic

hypersurface, $SSU\subset T^{*}X$is theunion ofthe

zero

section of$U$ and theoutgoingconormal

bundle of$\partial U$(setofall $\zeta$atpoints.

As mentionned above the$\mathscr{D}$-module _$M$ iselliptic with respectto _$U$ifSS$M\cap SSU$ is

containedin the

zero

section: the stronger $f_{1}^{\vee}niteness$condition is thatSS$M\cap SSU$ is

compact. Wethendispose of thefollowing $ob|ects:\backslash$

-the characteristic set $Z=SSM$ , _andthe symbol$[M]_{Z}^{top}\in K_{Z}(T^{*}X)^{9}$

-the microsupport SSN: in the

case

considered here $(N=O_{U})$ thisalways has

a

tubular

neighborhoodwith

a

canonical spincstructure, and by definition [N] isthe Bott element corresponding tothis. 10

Thusthe termsin theindex folmula (5)

are

well defined. The main idea of the proofis

to embed everything inanumeric space where thefolmula is known. Howeverellipticity is notpreserved byclosedembeddings (asystem whose solutions

are

carried by

a proper

submanifold cannotbeelliptic),

so

it isuseful to slightly enlarge the definition. In the

cases

we

aredealing with($U$a realanalytic manifold withcorners) it iseasyto

see

that$U$

can

be

8 _thecasewhere$N$isacoherentleft$\emptyset$-modulecanform$\prime Uly$be reducedtothecase$M=O$.‘Riere clearly should bea moregeneral case, making the symmetry between$M$and$N$moreapparent, buttheK-theoreticalaspectin

moregeneralcasesremainstobedevelopped.Asmentionned above$Ule$casewhere$N$is associatedtoa

constructible sheafonXhas been exalniIled bySchapiraandSchneiders.

9_{this isreally only definedabovecompact}$sub\backslash \backslash ets$ofX,butthisisenou\’oh for the$i_{11}dex$ formulawhereitonly

needstobedefineddearthe compactsetSS$M\cap SS$U.

10 _{for example if}$U$isanopensubset with analytic boundtary.SS$U$

can

bedefoimedinto thezerosection;for thezerosection the tubularneighborhoodis$T^{*}X$_,whichisa_$c(lnplex$vectorbundle.TheK-theoretical element

deformed and thickened in the following

sense:

thereexistes

a

continuous

one

parameter family $U_{\epsilon}$of neighborhoods of$U$,such thateach $U_{\epsilon}$is amanifold with boundary, $U_{\epsilon}\subset U_{\epsilon’}$if

$\epsilon<\epsilon’$,and SS _{$U_{\epsilon}arrow SSU$} if$\epsilonarrow 0$. We willsay that such afamilyofneighborhoods is adapted

to $U$,

a use

thefollowing generalization ofellipticity:

D\’efinition A3- $M$is almostelliptic with respectto$U$if thereexists

an

adapted family $U_{\epsilon}$

of neighborhoods suchthat$M$ is adapted with respectto $U_{\epsilon}$for small$\epsilon$.

examples:

an

elliptic module is almost elliptic. Products ofalmostelliptic modules

are

almostelliptic. If$M$ is almost ellipticwith respectto $U$, and$f$is

an

analytic embedding, _{$f_{+}M$}

is almostelliptic withrespect to $f(U)$. Finally aholonomic module is alwaysalmostelliptic,

with respectto anyU.

The index foimula extends naturally to almostelliptic systems (replacingthe product $[M]_{carM}^{top}[\mathcal{O}_{U}]_{SSU}$(orthe restriction$[M]_{carM}^{top}|SSU$) by the limit of the deformations

$[M]_{carM}^{top}[\mathcal{O}_{11_{8}}]_{SSU_{\epsilon}})$,and

we

proveit inthis framework,which allows embeddings.

Theindex theorem may then be proved as follows:

1.We firstreplace the manifold X by thesublying real manifold $X_{R}$, and $M$by $M_{R}$

2. The choose

a

closed immersion $f$ to embed everything in

a

numelic

space

$R^{n}$_, and

possiblythickenin $C^{n}$toreduce to the

case

where $U$is asmall ellipsoYd neighborhood ofthe

real unitball; in this

cas

theformula is alreadyestablished: it reduces to the index foimula

for Toeplitzoperators

on a

ball and is

a

particular

case

ofthe Bottperiodicity theorem. Thetopological character$\chi^{top}$

or

the K-theoretical push-forward

were

precisely

constmctedtofollow in theseoperations.

As

was

shown by Atiyah-Segal, the absolute index $fo\iota mula$ has

a

natural

generalization tosystemsto systems depending

on

parameters.i1 In

our

analytic framework these

are

described

as

follows:

we

first have

an

analytic map $f:Xarrow Y$ (ofreal

or

complex

manifolds), acomplex$M$ ofright$\mathscr{D}_{X/Y}$-modules (withgood filtrations), representing

an

analytic family of differential systems

on

thefibers,and

a

sheaf ofcoefficients$N=0_{U}$,

associated

as

aboveto

a

real analyticsubmanifold with

comers

$U\subset X$. Therelative

characteristic variety$Z=carM/Y\subset T^{*}X/Y$ is _{the support}of$grM$ (forany goodfiltration). To

$M$

we

associatethe symbol

(A31) $[M]_{Z}^{top}\in K_{Z}(T^{*}X/Y)$

11_{alsoto G-equivariantsystems,}$G$acompactgroup(theindexisthen avirtual representation ofG),towhich case ourproofadaptseasily-using equivariallt embeddings.

defined

as

above (itis only defined abovecompact subsetsofX). Thedirect image of$MIU$

(describingsolutions along fibers ofU) is$f_{+}M=Rf_{*}(M\otimes_{9}N)L$_{. To} $N$

we

associateitsrelative

microsupport$SSN\subset T^{*}X/Y$_, whosefiberabove$y\in Y$ is $SSU_{y}$. Thefiniteness (ellipticity)

condition is that theprojection $SSM\cap SSNarrow Y$be

proper,

ie. that$M_{y}$be elliptic along$U_{y}$ for all$y\in Y$ and this

ensures

that$f_{+}M$ hascoherentcohomology. In that

case

thesupportof

$f_{+}M$ iscontained in theprojection $Z^{t}$ of$Z=carM/Y\cap SSU/Y$ , and thefollowing formulais

the natural generalizationof the indexformula:

(A32) $[f_{+}M]_{Z}^{to}P=\overline{f}*([M]_{Z}^{top}[N_{U}])\in K_{Z’}(Y)$ (K-theoreticalimage)

where$\overline{f}$

isthe projection$T^{*}X/Yarrow Y$. Theproof is animmediate adaptationofthe proof

sketched above.

Relative index theorem

We endthese notes byabrief$desc\iota\cdot iption$ oftherelative

case.

In the relative index

folmula we

are

given ananalytic map$f:Xarrow Y$ between analytic manifolds, a$\mathscr{D}$-module $M$

on

X, and

a

subset$U$definingthe sheafofcoefficients. We

are

interested in the directimage

$f_{+}(MlU)$: a relative ellipticitycondition will

ensure

thatthis is coherent,andthe relative

indexformula will then describe its symbolbundle $[f_{+}(MlU)]$ interms of[M].

Let

us

$descl\cdot ibe$ this

more

precisely. First therelativeindex formulabelow,

as

all

folmulas above,iscompatible with closed immersions: replacingX by $X\cross Y,$$M$ and $U$by

theirdirect image by the graphmapIdxf,and$f$by the projection $X\cross Yarrow Y$,

we

are

reduced

to the

case

where $f$issubmersive (aprojection), which

we

willalways

suppose

from

now

on.

As above

we

denote

(A34) $H=X\cross YT^{*}Y$

7:

$Harrow T^{*}Y$ the second $pro_{\backslash }|ection$ $F:Harrow T^{*}X$ the cotangent

map

to

our

set$U$ defining thecoefficient sheaf

we

associate the transfermodule$\mathscr{D}_{Uarrow Y}=$

$C_{U}\otimes \mathscr{D}_{Xarrow Y}$. We definethemicrosupport SS$\mathscr{D}_{1Iarrow Y}=SSC_{U}+SS\mathscr{D}_{Xarrow Y}$(the setofall

covectorofthe form $\xi+\eta$ with $\xi\in SSU$ and$\eta\in H=SS\mathscr{D}_{Xarrow Y}$atpointsofU). We denote

further

(A35) $U_{e}=SS\mathscr{D}_{1Iarrow Y}=SSU\cross x^{H}$

$F_{e}$ : $U_{e}arrow T^{*}X$ the

map

which extends $F$ by _{$F_{e}(\eta,h)=\eta+h$}

Therelative ellipticity ellipticity condition is the following:

we

have

seen

above the definition of ellipticitywith respectto $U$for

a

vertical$\mathscr{D}_{X/Y}$-module, whichisthe

straightforward generalization of theellipticity condition inthe absolute

case.

The$M$is

relatively elliptic iflocally (nearany point of$\partial U$) itis

a

quotientof

a

$\mathscr{D}$-module of theform

$N\otimes_{9_{X/Y}^{\mathscr{D}}X}$for

some

coherent$\mathscr{D}_{X/Y}$-module$N$, elliptic withrespect toU. 12Relative

ellipticity implies the followinggeometric condition:

(A36) SS $M$ andSS$\mathscr{D}_{Uarrow Y}$ meetalong $H$ ( $the$

zero

section of$T_{H\Pi^{*}Y}^{*}$).

Thefiniteness condition isthis relative ellipticitycondition, _{plus the condition that the}

projection: SS $M\cap SS\mathscr{D}_{Uarrow Y}arrow T^{*}Y$is

proper.

Thereis alsoa notion of almost ellipticity.

Howeverthe relative ellipticity condition is

more

complicated than in the absolute case;it

cannotbe read

on

theprincipal symbol ofoperators alone and is harderto manipulate(and

less stable).

Under thisrelative ellipticity and finitenesscondition, it

was

shownby Houzel-Schapira thatthe directimage $f_{+}M_{U}=Rf_{*}(M^{L}\otimes_{\emptyset}\mathscr{D}_{Uarrow Y})$

is coherent and its characteristicset

iscontained inthe set$Z^{(}=\overline{f}(SSM\cap SS\mathscr{D}_{Uarrow Y})$. Therelative indexformula of Boutet de Monvel-Malgrangeis in this

case

the straightforward generalization of(5)$bis$

:

$[f_{+}M_{U}]_{Z}^{to_{t}p}=f_{e*}F_{e^{-1}}[M]_{Z}^{top}=$

We referto thepaperof Boutetde Monvel-Malgrange for theproof ofthe relative index formulaandgive hereonlythebriefest indication. Asabove

one

maysimplifythe situation replacing $M$by $M_{R}$ andembedding in

a

numeric

space;

we

maythusreduceto the

case

$U=Y\cross Q_{8}$ where$Q_{\epsilon}$ isafixedcomplexellipsoid, neighborhood of

a

realball. Inany

case,it is practicalto

use a

resolution of$M$ by ’vertical“ D-modules ofthe type$N\otimes_{\emptyset_{X!Y}^{\mathscr{D}_{X}}}$,

andmake

use

of vertical filtrations, ie. double filtrations of the form $M_{pq}=M_{p}\mathscr{D}_{X/Y}^{q}$. The graded objectassociated to this is avertical $\mathscr{D}_{H\Gamma\Gamma^{*}Y}$ -module towhich

we

may apply the

theoryabove (withparameters): its direct image iscoherent,and it is the firsttermof

a

spectral

sequence

which

converges

to$grf_{+}M_{U}$. There still remains

some

work todo to

compare

the K-theoretical element associated to the “vertical” graded object\dagger ’ $gr^{v}M$,which

lives

on

$T_{H/T^{*}Y^{\sim}Y}^{*}-H\cross T^{*}Y$ and $[M]_{Z}^{top}$which lives

on

$T^{*}X$(in factthey

are

bot compared to

their “cones\dagger t _{which live along the}

_zero

_{section of}$H$in $H\cross T^{*}Y$) andcheck that theygive the

same

elementby theK-theoretical image

12 equivalentlyanysestion$s$of$M$atapointof$\partial lI$is killedbyaverticaloperator_{$P\in 9_{X/Y}$}noncharacteristic

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Atiyah M.F. -K-theory, Benjamin, Amsterdam.

AtiyahM.F., BottR., _{-On theperiodicitytheorem forcomplex} vectorbundles. Acta Math.

112(1964) _229-247.

AtiyahM.F., Bott R., Schapiro A. -Cliffordmodules.Topology 3, suppl\’ement(1964), _3-83.

AtiyahM.F., HirzebruchF. -Vector bundles and homogeneous

spaces,

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