Hermitian vetor bundles and harateristi lasses
Jose I. BurgosGil
Abstrat. Thispaperreviews the realization ofBeilinson's regulator map
usinghermitian dierential geometry. This onstrution isa generalization
ofChern-Weiltheoryofharateristilassesofvetorbundles,tohigherK-
theory.
1. Introdution
Chern lasses of omplex vetor bundles an be onstruted using dierent
tehniques. Forinstane, in algebraitopology, one anuse theEuler lassof an
oriented sphere bundle to dene the top Chern lass and then dene the other
lassesindutively(see [25℄). Or,in dierentialgeometry, weanuseChern-Weil
theory, whih produes Chern lasses in de Rham ohomology by means of the
urvature of a onnetion (see for example [5℄, [34℄ or [16℄). Another aproah,
due to Grothendiek [17℄, starts with the rst Chern lass of a line bundle, and
then usesan expliitformula forthe ohomologyof aprojetivebundle to dene
higherChern lasses. Thisapproahis veryuseful inalgebraigeometry and an
be used to produe Chern lasses in the Chow ring, in etale ohomology or in
Deligne-Beilinsonohomology.
TheGrothendiekmethodhasbeengeneralizedbyGillet[11℄toproduehar-
ateristilassesfromhigherK-theorytoanyarbitraryohomologysatisfyinger-
tainaxioms. InthepartiularasewhentheharateristilassistheChernhar-
ater and the ohomology theory is real Deligne-Beilinson ohomology, the map
obtainedis Beilinson'sregulator map [2℄. This mapis ageneralizationof Borel's
regulatoranditisinvolvedinverydeeponjeturesin ArithmetiGeometry.
Beilinson'sregulatorisstillaverymysteriousmap,in partbeausehigherK-
theoryisarihandomplexworld. Thusitisusefultohaveasmanyapproahesto
Beilinson'sregulatoraspossible. Gilletand Soule[13℄havegiven adesriptionof
Beilinson'sregulatorforK
1
usingBott-Chernforms. Thisdesriptionanbeseen
asageneralizationofChern-Weiltheoryto K
1
. Inthepaper[8℄,S.Wangandthe
authorhaveextendedthisdesriptionofBeilinson'sregulatortohigherK-theory.
Theaimofthispaperistoreviewtheonstrutionofharateristilassesfor
higherK-theory. Thispaperismeanttobeintrodutory,andsothefousisplaed
PartiallysupportedbyGrantPB96-0234.
0000(opyright holder)
onthebasiideasofthetheory. Butreferenesaregivenwherethereaderannd
detailedproofsandgeneralstatements.
Themainobjetofstudywillbealgebraivetorbundlesoversmoothomplex
algebraivarieties. Theseobjetsanbestudiedfromthepointofviewofomplex
geometry or from algebrai geometry. Sine our objetive is to give a omplex
geometri onstrution of an objetdened in the algebrai geometry setting we
willshiftfromonepointofviewtotheotherin dierentsetions.
Theplanofthestudyisasfollows. Insetion2,Grothendiek'sonstrutionof
Chernlassesis presentedin thepartiularaseof sheafohomologywithinteger
oeÆients. Insetion3, wereallthe Chern-Weiltheoryof harateristilasses
of hermitian vetor bundles. Setion 2 is devoted to a simple version of Gillet's
onstrutionof harateristilasses forhigherK-theory. In setion5,wereview
real Deligneohomology. Bott-Chernforms arethe topi of setion6. Insetion
7, we introdue the omplex of exat ubes. By a result of R. MCarthy [24℄
therationalhomologyofthisomplexisisomorphitorationalK-theory. Finally,
setion 8is devoted to thedenition of harateristilassesfor higherK-theory
usingexatubesof hermitianvetorbundles. Forsimpliity, weonlydisussthe
aseofprojetivevarieties. Butnotethatanimportantingredientintheomparison
betweenthisonstrutionandGillet'sonstrution,istheextensionofthistheory
toquasi-projetivevarieties.
2. Chern lasses ofvetorbundles
Therearemanydierentonstrutionsof Chernlassesofvetorbundles(see
for example[25℄, [34℄ or[18℄). In this setion we will review averygeneral one
due to Grothendiek[17℄. Inthis onstrution,the properties of theohomology
theorythatareneededtodeneChernlassesaregivenasaxiomsforaohomology.
Theseaxiomsaresatisedbymanytheories,forinstane, Chowringsofalgebrai
varieties. MoreoverGrothendiek'sonstrutionisthebasisofGillet'sonstrution
ofChernlassesforhigheralgebraiK-theory[11℄.
Inthissetion,wewill speializetheonstrutionofChernlassesto thease
ofsheafohomologyofsmoothomplexvarietieswithintegeroeÆients. Wewill
use the lassial topology. To stress this point we will work with holomorphi
vetorbundles. Insetion4wewilldisusstheaxiomatiapproahinthealgebrai
geometryontext.
Letus introdue therstChernlass ofaholomorphiline bundle. This will
atasanormalizationfortheChernlasses. LetX beaomplexmanifold,andlet
O
X
bethesheafofholomorphifuntions onX. LetO
X
bethesheafofinvertible
holomorphifuntions. Thenthereisanisomorphism
8
<
:
Isomorphismlasses
ofholomorphi
linebundles 9
=
;
!H 1
(X;O
X ):
Theexponentialsequene
0 ! (2i)Z ! O
X exp
! O
X
! 0
givesusalongexatsequenein ohomology
H 1
(X;O
X
) ! H
1
(X;O
X )
Æ
! H 2
(X;Z(1)) ! H 2
(X;O
X );
Definition2.1. The rst Chern lass of a line bundle L, denoted
1 (L) 2
H 2
(X;Z(1)),istheimageofitslassinH 1
(X;O
X
)bytheonnetionmorphismÆ.
OnewehavedenedtherstChernlass,themaintooltodenehigherChern
lasseswillbetheDold-Thomisomorphism. LetE bearanknholomorphivetor
bundle over X. Let P(E) be the assoiated projetive bundle and letus denote
byp:P(E) !X theprojetion. Thenthevetorbundlep
E hasatautologial
subbundle,whose breat eah point istheline determinedbythat point. Letus
denotebyO
P(E)
( 1)thislinebundleandletO
P(E)
(1)betheduallinebundle. Let
bethe rst Chern lass of O
P(E)
(1). Let us denote by Z(p)the onstantsheaf
(2i) p
ZC.
Theorem 2.2(Dold-Thom isomorphism). Foreah pair of integers i; m,the
map
n 1
X
k =0 p
()[ k
: n 1
M
k =0 H
m 2k
(X;Z(i k)) !H m
(P(E);Z(i))
isan isomorphism.
Proof. When X is onepoint, the result is the lassial formula for the o-
homology of the projetive spae. The general ase follows from the fat that
the existene of the global lasses k
, implies the triviality of the Leray spetral
sequeneofthemorphismp:P(E) !X.
ThistheoremallowsustodeneChernlassesinthefollowingway.
Definition2.3. The Chern lasses of the vetor bundle E are the lasses
i
(E)2H 2i
(X;Z(i))determined bytheequation
p
(
n
(E))+p
(
n 1
(E))[++p
(
1
(E))[ n 1
+ n
=0:
(2.1)
ThetotalChernlassisthesum
(E)=1+
1
(E)++
n (E):
TheChernlassesareharaterizedbytherstChernlassandthebehaviour
underinverseimagesandexatsequenes. Thispropertyisveryuseful,forinstane,
whenomparingdierentdenitionsofChernlasses.
Theorem 2.4. Thereexistsauniquewaytoassign,toeahholomorphivetor
bundleE,atotalChernlass (E)satisfying the following properties:
1. Normalization: If L isaline bundle then(L)=1+
1
(L), where
1 (L)is
denedin 2.1.
2. Funtoriality: Forany morphism ofomplex varietiesf :X !Y wehave
(f
E)=f
(E).
3. Whitneysum formula: Forany exat sequeneof vetorbundles
0 !S !E !Q !0 (2.2)
wehave(E)=(S)[(Q).
Skethof proof. Letusstartbyprovingtheuniqueness. Assumethatthere
existsatheoryofChernlassessatisfyingonditions1,2and3. LetE beavetor
bundle over a smooth omplex manifold X. Let us write Q = p
E=O ( 1).
ThenQisarankn 1vetorbundleoverP(E). Thebasiideaistousetheexat
sequene
0 !O( 1) !p
E !Q !0:
First,using indutionand properties1and 3oneeasily provesthat
i
(E)=0for
i>n. Then, sinetherstChernlass ofO
P(E)
( 1) is , properties1,2and 3
implythat
p
(E)=(1 )(Q):
Sine
n
(Q) =0and (1 ) 1
=1++ 2
+ 3
+:::, the Chernlasses should
satisfytheformula(2.1). BytheDold-Thomisomorphismthisequationdetermines
theChernlasses.
To provetheexistene we needto showthat Chernlasses, asdened in 2.3
satisfy theonditions 1, 2and 3. The rst onditionfollowsfrom thedenition.
Theseond oneis easy. Forthethird oneweneedto usethe existeneand some
propertiesoftheGysinmorphism(see[17℄formoredetails).
3. Chern lassesof hermitian vetor bundles
In this setion we will explain the Chern-Weil onstrution of Chern lasses
using omplexdierentialgeometry. Thistheory isexplainedin manyplaes (see
forinstane[5℄,[34℄ or[16℄)andwereferthereadertothem fordetails.
LetB beasubringofR. WewillwriteB(p)=(2i) p
B. Thesymmetrigroup
on nelements, S
n
, ats on B[T
1
;:::;T
n
℄ by permuting the variables. Let IP(n)
bethe subsetof invariant polynomials. Then IP(n)=B[
1
;:::;
n
℄, where
i is
thedegreeisymmetrielementaryfuntion inn variables. An analogousresultis
trueifwereplaepolynomialsbyformalpowerseries. Wewilldenoteby
IP(n)the
set of invariantpowerseries in n variables and by IP(n)
k
the spae of invariant
homogeneouspolynomialsofdegreek.
Let be M
n
thevetor spae of nn omplex matries. Let ': M
n
! C
beamap suh that '(A) is ahomogeneouspolynomialof degreek in theentries
of A, with oeÆients in B. We say that ' is invariant if, for all A 2 M
n and
g2GL
n
(C)wehave
'(A)='(gAg 1
):
LetusdenotebyI
k (M
n
)thespaeofinvarianthomogeneouspolynomialsofdegree
k. There is an isomorphism I
k (M
n
) ! IP(n)
k
whih sends any ' to its value
in the diagonal matrixwith entries T
1
;:::;T
n
. Due to this isomorphismwe will
identifybothspaes.
Let X be a omplex manifold. Let E
denote the sheaf of omplex smooth
dierential forms,E p;q
the sheaf of (p;q)-forms, E
R
the subomplex of real forms
and E
R
(p)=(2i) p
E
R . LetE
(X), E p;q
(X),E
R
(X)and E
R
(p)(X)denote theor-
respondinggroupsofglobalsetions. LetE beaholomorphivetorbundleonX.
ThenwewilldenotebyE
(E)thesheafofE-valuedsmoothdierentialformsand
byE
(X;E)theorrespondingspaeofglobalsetions. E p;q
(X;E)willdenotethe
spaeofformsoftype(p;q)withvaluesin E.
AonnetiononEis aC-linearmap
0 1
satisfyingtheLeibnitz rule
D()=d+D;
(3.1)
forany2E 0
(X)and 2E 0
(X;E).
GivenanyonnetionD weanextendittoobtainoperators
D:E k
(X;E) !E k +1
(X;E)
by imposing the Leibnitz rule. But in general (E
(X;E);D) is not a omplex,
beausetheoperator
K=D 2
:E 0
(X;E) !E 2
(X;E)
maybenon zero. Infat the operatorK is E 0
(X)-linearandthus it anbe seen
as a setion K 2 E 2
(X;End(E)). This operator is alled the urvature of the
onnetion.
Dueto thedeomposition E 1
(X;E)=E 0;1
(X;E)E 1;0
(X;E)weandeom-
poseD=D 0;1
+D 1;0
,with
D 0;1
:E 0
(X;E) !E 0;1
(X;E)
D 1;0
:E 0
(X;E) !E 1;0
(X;E):
Sine E is holomorphi, there is a well dened -operator, whih in a loal
frameisgivenby
(f
1
;:::;f
n )=(f
1
;:::;f
n ):
Let us assume that E is provided with a hermitian metri h. Let us denote
byh ;i theorrespondinginner produt. Then thereis auniqueonnetion that
isompatible withboththeomplexstrutureandthehermitian metri. That is,
thereisauniqueonnetionD=D(h) satisfyingthefollowingonditions:
1. \Compatibilitywiththehermitianmetri": Forany;2E 0
(X;E)
dh ;i=hD;i+h;Di:
2. \Compatibilitywiththeomplexstruture": D 0;1
=.
Letuswrite K=K(h)=D 2
,theurvatureofthemetrih. Letf bealoal
frame,ifwedenotebyh(f)thematrixofthemetrihinthisframe,then(see[34℄
pag. 82).
K=(h 1
(f)h(f)):
(3.2)
Fromthisequationit islearthat K2E 1;1
(X;End(E)). Thus inaloal frameit
isgivenbyamatrixof(1;1)forms.
Let ' 2
IP(n) and let us denote by '
k
its omponent of degree k. The
invarianeof'
k
impliesthatwehaveawelldenedelement'e
k
(E;h)='
k
( K)2
E 2k
(X). To see this, one rst denes '
k
( K) in a loal frame, where K is a
matrixof(1;1)formsandthenoneusestheinvarianetogluetogethertheseloal
denitions(seeforinstane[34℄III.3). SineI= L
k 1 E
k
(X)isanilpotentideal,
wehavealsoawelldened form'(E;e h)= L
e '
k (E;h).
Theorem 3.1. Let E ! X be a holomorphi vetor bundle and let h be a
hermitianmetri. Then
2. The form' (E;e h) satises
e
'(E;h)2 M
p0 E
2p
R
(p)(X):
3. Thelassof' (E;e h)indeRhamohomology isindependentofthemetrih.
Sineanyholomorphivetorbundleadmitsahermitianmetritheabovethe-
oremallowsustousehermitianmetristodeneharateristilasses.
Definition 3.2. LetXbeaomplexmanifold. LetEbearanknholomorphi
vetorbundleandlet'2R[[ T
1
;:::;T
n
℄℄beaninvariantpowerseries. Letushoose
anyhermitianmetrihonE. Thentheohomologylass of'(E;e h),denoted
' dR
(E)2 M
p H
2p
dR
(X;R(p))
willbealled thedeRhamChernlassofEassoiatedtothepowerseries'. The
dierentialform' (E;e h) willbealled theChernform.
Remark 3.3. This distintion between Chern lasses and de Rham Chern
lasses is provisional. We will distinguish between them until we see that they
agree.
Examples3.4.
1. If'=
i
,thei-thelementarysymmetrifuntionthen dR
i
(E)=' dR
(E)is
thei-thdeRhamChernlassofE (seetheremarkabove).
2. If'=1+ P
i then
dR
(E)=' dR
(E) isalled thetotalde RhamChern
lass. Observethat thetotalChernformisgivenby
e
(E;h)=det(1 K):
3. IfB ontainstheeldQ, thentheChernharaterisdened bythepower
series
h(T
1
;:::;T
n )=
n
X
i=1 exp(T
i ):
ThenaturalinlusionZ(p) !E 0
R
(p)induesamorphism :H
(X;Z(p)) !
H
dR
(X;R(p)). InordertoseethatthetwodenitionsofChernlassesareompat-
iblewehavetoompare ((E))with dR
(E). Thekeytoomparebothdenitions
ofChernlassesisthetheorem2.4. Thusweonlyneedto showthatthedeRham
Chernlassesalsosatisfythepropertiesgivenin theorem2.4.
Theorem 3.5.
1. Let f : X ! Y be a morphism of omplex manifolds. Let (E;h) be a
hermitian vetorbundleon Y: Then
f
e
(E;h)=e(fE;f
h):
2. LetLbeaholomorphi linebundle. Then
dR
1
(L)= (
1 (L)):
3. Let
0 !S !E !Q !0
bean exatsequeneof holomorphi vetorbundles onX. Then
Proof. Theproofofproperties1and3arestandardandanbefoundinany
referenefor Chern-Weil theory. Forinstane, see [34℄. Sine property 2is more
speiof theomparison weare making,wewill provideaproof. Theideais to
represent
1
(L) asa
Ceh oyle and dR
1
(L) as adierential form. So we will
usetheomparisonbetween
CehanddeRham ohomologiesto omparethetwo
lasses.
Let us start representing
1
(L) using
Ceh ohomology. Let us assume that
U =fU
i
gis a good overof X (see for instane [6℄). This means that the open
sets U
i
and all their nite intersetions are ontratible. This impliesthat
Ceh
ohomologyfortheonstantsheafZ(1)agreeswithsheafohomology. Givenasheaf
ofabeliangroupsF,wewilldenotebyC
(U;F)theomplexof
Cehohainswith
respettotheopenoverU.
Letf(U
i
;s
i
)gbeatrivializationofthelinebundleL. Thuss
i
isanonvanishing
setionof (U
i
;L). Foreah pairi;j letuswrite U
i;j
=U
i
\U
j . Letg
ij
=s
j
=s
i
be thetransition funtions. So g
ij
belongs to (U
i;j
;O
) and fg
ij g 2C
1
(U;O
)
is a1-oyle that representsthelass ofL in H 1
(X;O
). Wehave toapply the
onnetionmorphismto this lass. Sinetheopen set U
i;j
is ontratible,wean
hooseadeterminationofthelogarithmlogg
ij overU
i;j
. OverU
ijk
=U
i
\U
j
\U
k
let us write t
ijk
= logg
jk
logg
ik
+logg
ij
. Then t
ijk
is onstant and it is an
integermultiple of2i. Theset ft
ijk gis a
Ceh 2-oylefor thesheafZ(1)that
represents
1 (L).
Nextwewanttorepresent dR
1
(L)asanexpliitdierentialform. SineLhas
rankone,theurvatureK2E 1;1
(X;End(L))=E 1;1
(X)is adierentialform. By
3.2thisform isgiven,ineahopenU
i ,by
K=(h(s
i )
1
h(s
i
))=log(h(s
i )):
Sine this form does notdepend on thesetion s
i
it is aglobal dierential form.
Thus therst deRham Chern lassis representedby theform e
1
(L;h) = K =
log(h(s
i )).
Toomparethetwolasseswewillfollowtheomparisonbetween
Cehandde
Rham ohomologiesgivenin [6℄. Themain toolisthe doubleomplexC
(U;E
).
It hasnaturalmorphismsfrom E
(X)and from C
(U;C) and is quasi-isomorphi
tobothomplexes. Let
d 0
:C p
(U;E q
) !C p+1
(U;E q
)
d 00
:C p
(U;E q
) !C p
(U;E q+1
)
denotethedierentials,whered 0
isthedierentialof
Cehohainsandd 0 0
is( 1) p
timesthedierentialofforms.
Thenft
ijk g=d
0
flogg
ij g,and
d 00
flogg
ij
g=f d(g
ij
=g
ij
)g=f s
i
=s
j d(s
j
=s
i )g:
therefore the
Ceh ohain fs
i
=s
j d(s
j
=s
i
)g also represents
1
(L). On the other
hand, K= d 00
flog(h(s
i
))g,and
d 0
f log(h(s
i
))g=f log(h(s
j )=h(s
i )) g=
f log(s
j s
j
=s
i s
i ) g=
f s
i
=s
j d(s
j
=s
i )g:
Therefore K representsalsothelass (L).
As aonsequeneofthis theorem weseethat ((E))= dR
(E). Inpartiu-
larthis implies that the Chern lass
i
orresponds to the elementary symmetri
funtion
i
. Thisjustiesthefollowingdenition.
Definition3.6. LetXbeaomplexmanifold. LetEbearanknholomorphi
vetorbundleandlet'2B[[T
1
;:::;T
n
℄℄ beaninvariantpowerseries. Let
'(T
1
;:::;T
n )=(
i
;:::;
n )
betheexpressionof'intermsofsymmetrielementaryfuntions. ThentheChern
lass of E assoiated tothe powerseries'is:
'(E)=(
1
(E);:::;
n (E))2
M
p H
2p
(X;B(p)):
Corollary 3.7. Let X be a omplex manifold and let E be a holomorphi
vetorbundleofrankn. Let'2B[[
1
;:::;
n
℄℄beaninvariantpowerseries. Then
' dR
(E)= ('(E)):
Inviewofthisresultwewilldropthesupersript dR
fromthenotation.
TheChern harater(see example3.4.3)isone of themostinterestingpower
seriesofharateristilasses. ThemainadvantageoftheChernharaterlassis
thatitbehavesverywellunderexatsequenesandtensorproduts. Fortheproof
ofthenextpropositionwereferalsoto [34℄.
Proposition3.8.
1. Let 0 ! S ! E ! Q ! 0 be an exat sequene of vetor bundles.
Then
h(E)=h(S)+h(Q):
2. LetE andF bevetor bunles. Then
h(EF)=h(E)^h(F):
Unlike theChern harater lass, theChern harater form does notneed to
behaveadditivelyforexatsequenes. Let
0 !(S;h 0
) !(E;h) !(Q;h 00
) !0
beanexatsequeneofhermitianvetorbundles. Letuswrite(S;h 0
)(Q;h 00
)for
theorthogonaldiretsum. Then
e
h((S;h 0
)(Q;h 00
))= e
h(S;h 0
)+ e
h(Q;h 00
):
Butingeneral
e
h(E;h)6=
e
h((S;h 0
)(Q;h 00
)):
Atrst glanethis mayseemunfortunate. But, in setion8, we willsee that, as
Shehtmannpointedout([30℄), thelakofadditivityat thelevelofformsanbe
4. Chern lassesfor higherK theory
Inthis setionwewillreview Gillet'sonstrutionofChern lassesforhigher
K-theory[11℄. Wewillfollowthesimpliedversiongivenin [31℄. Wewillassume
thatthereaderhassomefamiliaritywiththelanguageofsimpliialobjets(seefor
instane[23℄ or[14℄).
In this setion we will be in the algebrai geometry ontext. To stress this
fat,insteadof working withomplexnumbers,let usx agroundeld k. Let V
betheategoryofallsmoothquasi-projetiveshemes,equippedwiththeZarisky
topology. LetusdenotebyD(V)(resp. D +
(V))thederivedategoryofomplexes
ofabeliansheavesonV (resp. whih arebounded below). Let usassumethat we
haveaxedgradedomplexofsheaves
F
()= M
i2Z F
(i); withF
(i)2D +
(V):
Thusforeahsmoothquasi-projetiveshemeX2Ob(V)wehavetheohomology
groups H
(X;F
()). We need to assume ertain properties of this ohomology
theoryinorder tomimitheonstrutionofChernlassesof setion2.
Let us denote by O
the sheaf of invertible rational funtions. It denes an
elementofD +
(V)assumingthatitisaomplexonentratedindegreezero. Then
H 1
(X;O
)parameterizesisomorphismlassesofalgebrailinebundlesoverX. The
rstpropertyweneedforourohomologytheoryis:
P 1: There isamorphisminD +
(V)
O
[1℄ !F
(1):
In partiular, for eah X 2 Ob(V), we obtain a morphism
1 : H
1
(X;O
) !
H 2
(X;F
(1))whih allowsusto denetherstChernlassofalinebundle.
Thenextpropertyweneedisamultipliativestruturefortheohomology.
P 2: Foreahn; m2Z,therearehomomorphismsinD +
(V)
[:F
(n) L
Z F
(m) !F
(n+m); ande:Z !F
(0)
whihmakeF
()anassoiativeandgraded ommutative(withrespet to
therstdegree)algebrawithunit.
Thethird propertyweneedisaformula fortheohomologyof theprojetive
spae,inother words,weneedtheDold-Thom isomorphismtobesatised.
P 3: ForX 2Ob(V),letp:P n
X
!X bethen-dimensional projetivespae
overX. Let be therst Chern lass of theline bundle O(1). Then, for
eahpairofintegersi; m,themorphism
n 1
X
k =0 p
()[ k
: n 1
M
k =0 H
m 2k
(X;F
(p k)) !H m
(P n
X
;F
(p))
isanisomorphism.
ByaLerayspetralsequeneargument,from propertyP3itfollowsthatthe
Dold-Thomisomorphismissatisedforanyprojetivebundle.
WithpropertiesP1,P2andP3,weanrepeattheproedureofthesetion
2 and dene, for eah vetorbundle E over X 2 Ob(V), Chern lasses
i (E) 2
H 2i
(X;F
(i)). Butin orderto havetheWhitney sumformulaweneed toassume
P 4: Leti:Y ,!X bealosedimmersioninV ofpureodimension1andlet
[Y℄2H 1
(X;O
)bethelassofthedivisorY:Thenforanyx2H 2i
(X;F
(i))
suhthati
x=0,wehave
x[
1
([Y℄)=0:
ThenextstepistoextendthedenitionofChernlassestosimpliialshemes.
Let bethe ategorywhose objetsarethe orderedsets [n℄ =f0;:::;ng, n0
andwhosemorphismsareorderedmaps. LetSV denotetheategoryof simpliial
objetsin V,thatis, theategoryofontravariantfuntorsbetweenandV.
Definition4.1. Let X
2 ObSV be a simpliial sheme. A rank n vetor
bundle over X
is an objet of SV, E
, together with a morphism of simpliial
shemes
:E
!X
satisfyingthefollowingonditions
1. Foranyk0themorphism
k :E
k
!X
k
isaranknvetorbundle.
2. Foranyj; k0andany2Mor
([j℄;[k℄)theommutativediagram
E
k E()
! E
j
?
?
y k
?
?
y k
X
k X()
! X
j
isarelativemorphismof vetorbundles.
Alternatively,following[12℄,weandeneavetorbundleoverX
asavetor
bundle E
0
over X
0
, together with an isomorphism : Æ
0 E
0
! Æ
1 E
0
of vetor
bundlesoverE
1
,suhthat Æ
2 ÆÆ
0 =Æ
1 .
IfX
isasimpliialshemethenagainH 1
(X
;O
)parameterizesisomorphism
lassesoflinebundles(see[12℄ex1.1). Moreover,thepropertyP3impliesthatthe
Dold-Thomisomorphismissatisedforarbitraryprojetivebundlesoversimpliial
shemes (see [11℄ Lemma 2.4). Therefore Grothendiek's onstrution of Chern
lassesanbeappliedtosimpliialshemes. Inpartiularitanbeapplied inthe
universalase.
LetB
GL
n
=k bethelassifyingshemeofthegroupshemeGL
n
overk. The
simpliial sheme B
GL
n
=k is provided with a universal rank n vetor bundle,
denoted by E
n
. Letus denote by (n)
i
=
i (E
n )2 H
2i
(B
GL
n
=k;F
(i)) the i-th
Chernlassoftheuniversalbundle.
Thenextobjetiveistoexplainhowlassesintheohomologyofthelassifying
shemegiveriseto mapsbetweenK-theoryandohomologyofshemes.
LetS
beasimpliialset andlet X be asheme. Then weanonstrutthe
simpliialshemeS
X suh that
(S
X)
n
= a
p2Sn
fpgX;
and the faes and degeneraies are indued by those of S
. Let A be a nitely
generated k-algebrasuhthat U =SpeA is asmoothsheme. ThenB
GL
n (A)
is asimpliial set. Thus we anonstrut thesimpliial sheme B
GL
n
(A)U.
SineanelementofB
j GL
n
(A)isamorphismbetweenU andB
j GL
n
=kweobtain
atautologialmorphismofsimpliialshemes
:B
GL
n
(A)U !B
GL
n
=k:
Thus,weobtainlasses
( (n)
)2H 2i
(B
GL
n
(A)U;F
(i)).
For asimpliial set S
, letus denoteby ZS
thehomologial omplexwhih,
in degree j is the free abelian group generated by S
j
, and whose dierential is
d= P
( 1) i
Æ
i .
Lemma 4.2. Let S
bea simpliial set andlet X be asmooth quasi-projetive
sheme. Then thereisanatural shortexatsequene
0 ! Y
p+q=n 1 Ext
1
Z (H
p (ZS
);H
q
(X;F
(i)) !H n
(S
X;F
(i))
! Y
p+q=n
Hom(H
p (ZS
);H
q
(X;F
)) !0:
Proof. Thisresult isthedual of theKunnethformula(seefor instane [29℄
Thm. 11.32).
Thus,from eah lass (n)
i 2H
2i
(B
GL
n
=k;F
(i)) weobtainafamilyofmor-
phisms
Æ( (n)
i )
j :H
j (ZB
GL
n
(A)) !H 2i j
(U;F
(i)):
Now wewant to go to the limit when n goes to innity. Form > n, let us
onsidertheinlusioni
n;m GL
n
=k !GL
n+1
=k givenby
A7 !
A 0
0 I
;
whereI istheidentitym nm n-matrix. Sine
i
n;m E
m
=E
n
(atrivialvetorbundle);
wehavethati
n;m
(m)
i
= (n)
i
. Therefore,thefamilyofmorphismsf Æ'(
(n)
i )
j g
n1
denesamorphism
i;j :H
j
(GL(A);Z)=lim
!
n H
j (ZB
GL
n
(A)) !H 2i j
(U;F
(i)):
TheK-theory groupsof A are thehomotopygroupsof the+-onstrutionof
B
GL(A). Sinethe+-onstrutionisayli,wehavethat
H
(B
GL(A)
+
;Z)=H
(B
GL(A);Z)=H
(GL(A);Z):
Definition 4.3. LetAbeanitelygeneratedk-algebrasuhthatU =SpeA
is a smooth sheme. For eah pair of integersi; j, the i-th Chern lass map in
K
j
(A)istheomposition
i;j :K
j
(A)=
j (B
GL(A)
+
)
Hurewiz
! H
j
(GL(A);Z) i;j
!H 2i j
(U;F
(i)):
Thefollowingresultisaneasyonsequeneofthedenition.
Proposition4.4. Let f
℄
: A ! B be a morphism of nitely generated k
algebrassuhthatU =SpeAandV =SpeB aresmoothshemes. Letf :V !
U be theorresponding morphismof k-shemes. Then thefollowing diagram
K
j (A)
i;j
! H 2i j
(U;F
(i))
f
?
?
y
f
?
?
y
K
j (B)
i;j
! H 2i j
(V;F
(i))
InordertoextendtheonstrutionofChernlassestoarbitrarysmoothquasi-
projetiveshemeswewill requireapropertyofinvarianeunder homotopy. This
will allow us to useJouanolou's trik. But stritly speaking this property is not
neessary(see[11℄).
P 5: For any sheme X 2 Ob(V), the natural map A 1
X
! X indues an
isomorphism
H
(X;F
()) !H
(A 1
X
;F
()):
This property implies that, for any bre bundle p : Y ! X, with bre A n
,
(not neessarilyavetorbundle), themap p
H
(X;F
()) !H
(Y;F
()) isan
isomorphism.
ForanyshemeX2V,byJouanolou'slemma([20℄lemma1.5)thereisabre
bundlep:Y !X,withY aÆne,whihisatorsoroveravetorbundle.
Definition4.5. Let X bea smoothsheme overk. Let us hoosean aÆne
torsorp : Y !X. For eah pair of integersi; j, thei-th Chern lass map for
K
j
(X)istheomposition
i;j :K
j (X)
p
=K
j (Y)
i;j
!H 2 i j
(Y;F
(i)) p
=H 2i j
(X;F
(i)):
Thanks to the naturality of the map
i;j
in the aÆne ase (proposition 4.4)
and[20℄proposition1.6,thisdenitiondoesnotdependonthehoieoftheaÆne
torsorY.
SofarwehaveonstrutedChernlassesforhigherK-theory. Nowwewantto
onstrutharateristilassesforarbitrarypowerseries. LetagainBbeasubring
ofR andassumethatF isasheafofB-modules. Observethat,replaingT
n+1 by
0,wehaveamorphismbetweentheringofinvariantpowerseries
B[[T
1
;:::;T
n+1
℄℄
Sn+1
!B[[T
1
;:::;T
n
℄℄
Sn
:
Intermsofelementarysymmetrifuntionstheabovemorphismisthemorphism
B[[
1
;:::;
n+1
℄℄ !B[[
1
;:::;
n
℄℄
thatsends
n+1
to 0. Wewill denoteby
i
n;m :
IP(m) !
IP(n)
theinduedmorphisms. These morphismsmakef
IP(n)g
n
aninversesystem. Ob-
servethatwehaveaommutativediagram
f
IP(m)g
n
! Q
i H
2i
(B
GL
m
=k;F
(i))
i
n;m
?
?
y
i
n;m
?
?
y
f
IP(n)g
n
! Q
i H
2i
(B
GL
n
=k;F
(i))
wherethehorizontalarrowssendtheelementarysymmetrifuntion
i
tothei-th
Chernlass.
Definition4.6. Astable invariantpowerseries isanelement
f' (n)
g2lim
IP(n):
Observethat anystableinvariantpowerseriesisgivenbyanelement
'2B[[
1
;:::;
n
;:::℄℄:
ProeedingaswiththeChern lasses,foranyinvariantstable powerseries',
weobtainharateristilasses
':K
j
(X) ! M
H 2 i j
(X;F(i)):
Example4.7. AssumethatB Q. Letusdenotebyh (n)
thereduedChern
haraterofrankn. Thatis
h (n)
(T
1
;:::;T
n
)=h(T
1
;:::T
n ) n:
Sine h (n+1)
(T
1
;:::;T
n
;0) =h (n)
(T
1
;:::;T
n
), then fh (n)
gis astable invariant
powerseries. ThusitdenesareduedChernharaterlass,denotedh.
Definition 4.8. TheChernharateristhemap
h:K
i () !
M
j H
2 j i
(;F(j));
givenby
h= (
h; ifi>0;
rank+h; ifi=0:
Reallthatthere isaprodutstruture([21℄)
K
i K
j
!K
i+j
thatextendsthemultipliativestrutureinduedinK
0
bythetensorprodut. The
Chernharateris alsoompatiblewiththis produt. Foraproofofthefollowing
resultsee[11℄or[31℄.
Theorem 4.9. Forx2K
i
andy2K
j then
h(xy)=h(x)[h(y):
5. Real Deligneohomology
Let us reall the denition of real Deligne ohomology and some omplexes
thatan beused toomputeit. AgainletX beasmoothproperomplexvariety.
As in setion 3let B beasubring ofR. Letus write B(p) =(2i) p
B C. We
will denote alsobyB(p)the onstant sheaf. Let
X
bethesheaf ofholomorphi
dierentialformsonX.
Definition5.1. The(B-)Deligneohomology ofX,denoted H
D
(X;B(p)),is
thehyperohomologyof theomplexofsheaves
B(p) !O
X
!
1
X
! ! p 1
X :
Thisdenition hasbeenextendedbyBeilinsontotheaseof smoothomplex
algebraivarieties,notneessarilyproper(see[2℄,[1℄,[10℄and[19℄). Thisextension
is known as Deligne-Beilinson ohomology. Beilinson also showed that Deligne-
BeilinsonohomologyanbewrittenassheafohomologyforasheafintheZariski
topology, satisfyingthe properties 1to 5 of last setion. Thus wean apply the
onstrution of the last setion and obtain harateristi lasses from higher K-
ChernharaterinrealDeligne-BeilinsonohomologyisalledBeilinson'sregulator
anditisageneralizationofBorel'sregulator(see[4℄,[2℄,[27℄).
LetF denote theHodgeltration:
F p
X
= M
p 0
p
p 0
X :
Thenreal Deligneohomologyanbedened alsoasthehyperohomologyofthe
simpleomplexassoiatedtothemorphismofomplexes
R(p)F p
X
!
X :
WewanttohaverealDeligneohomologyastheohomologyofanexpliitomplex,
asdeRhamohomologyistheohomologyoftheomplexofdierentialforms. To
this end wewill resolvetheonstant sheafand the sheaves ofholomorphi forms
usingsmoothdierentialforms. Wewilluse thesamenotation asin setion3for
thesheavesandomplexesofdierentialforms.
TheHodgeltrationof theomplexE
isgivenby
F p
E n
= M
p 0
p E
p 0
;n p 0
:
Thesheavesofdierentialformsarene,heneayli. SineE
R
(p)isaresolution
oftheonstantsheafR(p),andE
isaresolutionof
X
,ompatiblewiththeHodge
ltration,weobtainthat realDeligne ohomology ofX isthe ohomologyof the
simpleomplexassoiatedtothemorphismofomplexes
u
p :E
R
(p)(X)F p
E
(X) !E
(X);
givenbyu
p
(r;f)=f r. Letusdenotethesimpleomplexassoiatedtou
p as
s(u
p )=s(E
R
(p)(X)F p
E
(X) !E
(X)):
An elementofs n
(u
p
)isgivenbyatriple
(r;f;!)2E n
R
(p)(X)F p
E n
(X)E n 1
(X);
andthedierentialisgivenbyd(r;f;!)=(dr;df;f r d!).
Following Deligne [9℄ we anuse a simpleromplex to ompute real Deligne
ohomology.
Definition 5.2. LetD
(X;p)denotetheomplexgivenby
D n
(X;p)= 8
>
>
>
>
>
<
>
>
>
>
>
: E
n 1
R
(p 1)(X)\ M
p 0
+q 0
=n 1
p 0
<p;q 0
<p E
p 0
;q 0
(X); forn2p 1
E n
R
(p)(X)\ M
p 0
+q 0
=n
p 0
p;q 0
p E
p 0
;q 0
(X); forn2p:
Thedierentialof thisomplex,denoted byd
D
, isinduedbyd indegreegreater
orequalthan2p, by d indegreelessorequalthan2p 2andis equalto 2
indegree2p 1.
A proof that the ohomology of this omplex is Deligne ohomology an be
1. Themorphismu:E
R
(p)(X)F E (X) !E (X)isinjetiveforn2p 1
andtheokernelis
E n 1
R
(p 1)(X)\ M
p 0
+q 0
=n 1
p 0
<p;q 0
<q E
p 0
;q 0
(X):
2. Theabovemorphismuis surjetiveforn2p 1andthekernelis
E n
R
(p)(X)\ M
p 0
+q 0
=n
p 0
p;q 0
q E
p 0
;q 0
(X):
3. Inpartiular,forn=2p 1themorphismuisanisomorphism. Moreover,
if!2E 2p 2
(X),thendu 1
d!= 2!.
Example5.3. Observethat D 2p
(X;p)=E 2p
R
(p)(X)\E p;p
(X). Thusif(E;h)
isahermitianholomorphivetorbundleofranknand'2R[[T
1
;:::;T
n
℄℄isareal
invariantpowerseries, then
e
' (E;h)2 M
p D
2p
(X;p):
InordertowritedownexpliitmorphismsbetweenD
(X;p)ands(u
p
)weneed
tointroduesomenotations. Givenadierentialforma,wewill writea= P
a p;q
itsdeompositionin formsofpurebidegree. LetF p;p
bethemorphismdenedby
F p;p
a= X
p 0
p
q 0
p a
p 0
;q 0
:
Let
p
bethemorphismgivenby
p
a=(a+( 1) p
a)=2:
Observethat
p
isthe projetionofE
(X)onto E
R
(p)(X),and that, forn <2p,
p
=
p 1 ÆF
n p+1;n p+1
istheprojetionofE n
(X)ontotheokernelofu
p .
Let : s n
(u
p
) ! D n
(X;p) and ' :D n
(X;p) !s n
(u
p
)be themorphisms
givenby
(r;f;!)= (
p
(!); forn2p 1;
F p;p
r+2
p (!
p 1;q+1
); forn2p;
'(x)= (
(x p 1;q
x q;p 1
;2x p 1;q
;x); forn2p 1;
(x;x;0); forn2p;
where q=n p. Then and 'are homotopyequivalenesinverseto eah other
(see[7℄).
RealDeligneohomologyhasaprodut[2℄,[1℄thatanbedesribed,interms
oftheomplexess(u
p
)in thefollowingway. Let01bearealnumber. For
(r;f;!)2s n
(u
p
)and(s;g;)2s m
(u
q
),letuswrite
(r;f;!)[
(s;g;)=(r^s;f^g;
(!^s+( 1) n
f^)+(1 )(!^g+( 1) n
r^)):
This is afamilyof produts, allof them homotopially equivalent. Moreover,for
=1=2thisprodut isgraded ommutative,whereasfor=0;1thisprodutis
Deligneohomology. ThisprodutinduesaprodutintheomplexD
(X;p)(see
[7℄). Thisomplexisonlyassoiativeorgradedommutativeuptohomotopy.
Inordertohaveaomplexwithagradedommutativeandassoiativeprodut
we will use the Thom-Whitney simple, introdued in [26℄. The Thom-Whitney
simple assoiates, to a strit osimpliial dierential omplex, a new dierential
omplex. This new dierential omplex is homotopially equivalent to the total
omplex of the original osimpliial omplex. The main interest of the Thom-
Whitney simpleis that,ifwestart withastrit osimpliialgraded ommutative
assoiativealgebra,theomplexthatweobtainisagainadierentialgradedom-
mutativeassoiativealgebra.
Inourase,weanonsider themorphismu
p
astritosimpliialomplexu
p
writing
u 0
p
=E
R
(p)(X)F p
E
(X);
u 1
p
=E
(X);
u i
p
=0; fori2;
withmorphismsgivenby
Æ 0
(r;f)=f; Æ 1
(r;f)=r;
andtheothermorphismsequaltozero.
LetusdesribetheThom-Whitney simpleinthisase. LetL
1
betheomplex
of algebraiforms in theaÆneline A 1
R
. That is,L 0
1
=R[t℄, and L 1
1
=R[t℄dt. Let
Æ
0
;Æ
1 : L
1
! R be the morphisms given by evaluation at 0 and 1 respetively.
ThatisÆ
0
(f(t)+g(t)dt)=f(0),and Æ
1
(f(t)+g(t)dt)=f(1).
Definition 5.4. The Thom-Whitney simple of u
p
, denoted s
TW (u
p
) is the
subomplexof
E
R
(p)(X)F p
E
(X)L
1 E
(X)
formedbytheelements(r;f;!)suh that
f =(Æ
0
Id)(!);
r=(Æ
1
Id)(!):
Thedierentialand theprodutof theThom-Whitney simplearegivenom-
ponentwise:
d(r;f;!)=(dr;df;d!)
and
(r;f;!)^(s;g;)=(r^s;f^g;!^):
Withthese denitionsofdierentialandprodut,thediretsum L
p s
TW (u
p )isa
dierentialgradedommutativeassoiativealgebra.
Weanonstrutexpliitequivalenes(see[26℄)
s
TW (u
p )
I
!
E s(u
p )
givenby
and
I(r;f;(h(t)+g(t)dt)!)=
r;f;
Z
1
0
g(t)dt!
:
WewillwriteI 0
= ÆI :s
TW (u
p
) !D
(X;p)andE 0
=EÆ':D
(X;p) !
s
TW (u
p
). We will later use that I 0
and E 0
is a pair of homotopy equivalenes,
inverse to eah other, between D
(X;) and a omplex with an assoiative and
gradedommutativeprodut.
6. Bott-Chern forms
For anystableinvariantpowerseries'andanyexatsequene
:0 !(S;h 0
) f
!(E;h) !(Q;h 00
) !0
ofhermitianvetorbundles,theWhitneysumformulaimpliesthattheassoiated
Chernlassessatisfy
'(E)='(SQ):
(6.1)
But in general this equation is no longer true for the Chern forms. That is, the
form
e
' (E;h) '((S;e h 0
)(Q;h 00
))2 M
D 2p
(X;p)
maybe nonzero. Nevertheless, equation6.1 andthe-lemma ([16℄)imply that
thereexistsadierentialform'e
1
suhthat
e
' (E;h) ' ((S;e h 0
)(Q;h 00
))= 2'e
1 : (6.2)
SineD 2p
(X;p)=E 2p
R
(p)(X)\E p;p
(X)and 2 isapurelyimaginaryoperator,
bihomogeneousofbidegree(1;1),weanhoose
e '
1 2
M
p E
2p 2
R
(p 1)(X)\E p 1;p 1
(X)= M
p D
2p 1
(X;p):
In other words, the form '(Ee ;h) '((S;e h 0
)(Q;h 00
)) is exat in the omplex
L
D
(X;p)Theaimofthissetionistosolvetheequation6.2inafuntorialway.
Wewillsaythattheexatsequene issplitif(E;h) istheorthogonaldiret
sum(S;h 0
)and(Q;h 00
).
Theorem 6.1(Gillet andSoule[13℄). Let'beastableinvariantpowerseries.
Toeahexat sequeneof hermitian vetorbundle
:0 !(S;h 0
) f
!(E;h) !(Q;h 00
) !0
we an assign a dierential form 'e
1 () 2
L
p D
2p 1
(X;p), alled the Bott-Chern
form. satisfying thefollowing properties.
1. 2'e
1
='(E;e h) '((S;e h 0
)(Q;h 00
)).
2. If f :X !Y isamorphism of omplexmanifolds then
e '
1 (f
)=f
e '
1 ():
3. If isasplit exatsequeneof hermitianbundles, then'e
1
()=0.
Moreover, these properties haraterize 'e
1
up to Im+Im. That is, up to a
boundaryin the omplex L
D
(X;p).
TheproofthatthesepropertiesharaterizeBott-Chernformsanbefoundin
[3℄. Letus giveaonstrutionofBott-Chernforms. Themethod wewill followis
amodiationofthemethod ofGilletandSoulethat avoidstheneedto hoose a
partition ofunity. In thiswayweobtainwelldened Bott-Chernformssatisfying
theonditionsoftheorem6.1and notonlylassesuptoIm+Im.
Astandardproedure toprovethattwodierentialformsand,denedon
a dierential variety Y, are ohomologousis thefollowing. First, oneonstruts
ageometrihomotopy betweenand. Thatis, adierentialform dened on
Y R suhthat j
Yf1g
=and j
Y0
=. Fromthis homotopyone obtainsa
primitivefor byintegration:
d Z
1
0
= : (6.3)
Letusdenoteby 1
theurrent
1
()= Z
1
0 :
Thentheequation6.3anbeexpressedastheequationinurrents
d 1
=Æ
1 Æ
0
; (6.4)
whereÆ
0 andÆ
1
aretheDiradeltaurrentsenteredat 0and1respetively. We
willadapt theaboveproeduretoDeligneohomology.
Letusassumeforawhilethat thehermitianmetrih 00
onQisthehermitian
metri indued by the metri hof E. The rst step is to onstrut a geometri
homotopybetweenthehermitianvetorbundles(E;h)and(S;h 0
)(Q;h 00
). This
homotopywillbeparametrizedbytheomplexprojetivelineinsteadofbytheunit
interval. Let (x : y)behomogeneous oordinates of P 1
=P 1
C
. Then x and y are
setionsofthebundleO
P
1(1). ThestandardmetriofC 2
induestheFubini-Study
metrionO
P
1(1). Letusdenotebyg thismetri. Then
g(x)= xx
xx+yy
and g(y)= yy
xx+yy :
Let p
1
: XP 1
! X and p
2
:X P 1
! P 1
denote the projetions. Let
us write E(1) = p
1 E p
2
O(1) and S(1) = p
1 Sp
2
O(1). Let us onsider the
morphism
: S ! S(1)E(1)
s 7 ! sy+f(s)x:
ObservethatthevetorbundleS(1)E(1)hasametriinduedbythemetrisof
S,E andO(1).
Definition 6.2. Thetransgressionbundle assoiatedto theexatsequene
isthehermitianvetorbundle
tr
1
()=oker( )
withthehermitianmetriinduedbythemetriofS(1)E(1).
Therestritionsofthetransgressionbundletr
1 ()are
tr
1 ()j
X(0:1)
=(E;h);
(6.5)
tr ()j =(S;h 0
)(Q;h 00
):
(6.6)
Thus tr
1
() is ageometri homotopy between (S;h)(Q;h ) and (E;h). Note
thatto obtain6.5wehadto assumethat h 00
istheinduedmetri.
Theseondstepisto obtainageometrihomotopyofdierentialforms. This
homotopyisgivenbytheform'(tre
1
()). ThenthefuntorialityoftheChernforms
andtheequations6.5and6.6implythat
e ' (tr
1 ())j
X(0:1)
='e((E;h)); (6.7)
e ' (tr
1 ())j
X(1:0)
='e((S;h 0
)(Q;h 00
)): (6.8)
The third step is to integratethe geometri homotopy to obtaina primitive.
Let t = x=y be an absolute oordinate in P 1
. Let [1=2logtt℄ denote the urrent
dened by
[1=2logtt℄()= 1
2i Z
P 1
1
2 logtt:
Theurrent[1=2logtt℄will playtherole,in Deligneohomology,thattheurrent
1
playsin de Rham ohomology. Theanalogueof equation 6.4 isthe Poinare-
Lelongequation(seeforinstane[16℄)
2[1=2logtt℄=Æ
(0:1) Æ
(1:0) : (6.9)
Definition 6.3. TheBott-Chernform assoiatedtotheexatsequene and
thepowerseries 'isthedierentialform
e '
1
()=[1=2logtt℄(tr
1 ()):
(6.10)
Equations6.7,6.8and6.9implytheondition1oftheorem6.1. Moreoverthe
funtoriality(ondition2) islearfromtheonstrutionofBott-Chernforms.
Lemma 6.4. If isasplit exatsequenethen'e
1
()=0.
Proof. Letusonsiderthemorphism:P 1
!P 1
givenby(x:y)=(y:x).
Theline bundle O(1)with theFubini-Studymetriis invariantunder
. Sine
issplit,themap inthedenitionofthetransgressionbundleis
: S ! S(1)S(1)Q(1)
s 7 ! sy+sx+0:
Therefore,bytheinvarianeofO(1),
tr
1
()is theokernelofthemorphism
0
: S ! S(1)S(1)Q(1)
s 7 ! sx+sy+0;
whih is isometri to tr
1
(). Therefore
e '(tr
1
()) = ' (tre
1
()). Thus it is an
even form. On the other hand, the urrent [1=2logtt℄ is odd:
[1=2logtt℄ =
[1=2logtt℄. Hene'e
1
()=[1=2logtt℄(tr
1
())=0.
Letusassumenowthatthemetrih 00
ofQisarbitrary. Leth 000
bethehermitian
metrionQinduedbythemetrih. Thenfromtheexatsequeneweandene
twonewexatsequenes.
1
:0 !(S;h 0
) !(E;h) !(Q;h 000
) !0
2
:0 !(Q;h 00
) !(Q;h 000
) !0 !0:
Definition 6.5. Let
:0 !(S;h 0
) !(E;h) !(Q;h 00
) !0
beanexatsequeneofhermitianvetorbundles. Let'beastableinvariantpower
series. ThentheBott-Chernform assoiatedto'anf is
e '
1 ()='e
1 (
1
)+'e
1 (
2
)
Remark 6.6. Sine,byLemma6.4theBott-Chernoftheexatsequene
0 !(Q;h) !(Q;h) !0 !0
is zero,if isanexatsequene withthe thirdmetriinduedby theseond one
thendenition 6.3givesthesameresultasdenition 6.5.
7. Exatubes
Let X be a smooth quasi-projetive variety over C. Let us x a small full
subategoryE=E(X)oftheategoryofalgebraivetorbundlesoverX,whihis
equivalenttoit.
Let h 1;0;1i be the ategory assoiated to the ordered set f 1;0;1g. Let
h 1;0;1i n
bethen-thartesianpower,andleth 1;0;1i 0
betheategorywithone
elementandonemorphism. FollowingLoday[22℄,wedeneexatubesasfollows:
Definition 7.1. Anexatn-ubeofEisafuntorFfromh 1;0;1i n
toEsuh
that,forallintegers1in,andalln 1-tuples(
1
;:::;
n 1
)2h 1;0;1i n 1
thesequene
F
1;:::;i 1; 1;i;:::;n
1
!F
1;:::;i 1;0;i;:::;n
1
!F
1;:::;i 1;1;i;:::;n
1
isashort exatsequene. Wehavewritten F
1;:::;n
forF(
1
;:::;
n ).
We will denote by C
n
E the ategory of exat n-ubes. It is a small exat
ategory. WewillwriteC
n
E=Ob(C
n E).
Definition 7.2. Given an exat n-ube F and integers i 2 f1;:::;ng, j 2
f 1;0;1g,the fae j
i
F istheexatn 1-ubedenedby
j
i F
1;:::;n
1
=F
1
;:::;
i 1
;j;
i
;:::;
n 1 :
Examples7.3.
1. Anexat0-ubeisanelementofOb(E).
2. Anexat1-ubeisanexatsequeneofobjetsofE.
3. Foreahi2f1;:::;ng,weanseeanexatn-ubeF astheexatsequene
ofexatn 1-ubes
0 ! 1
i
F !
0
i
F !
1
i
F !0
This exatsequenewill bedenoted
i
F. Notethat F is haraterizedby
anyoftheexatsequenes
i F.
Let ZC
n
(E) be the free abelian group generated by C
n
(E). Let us dene a
dierentiald:ZC
n
(E) !ZC
n 1
(E)bytheformula
d= n
X 1
X
( 1) i+j+1
j
i :
It iseasyto see thatd =0;thus wehaveobtainedahomologyomplexdenoted
ZC
(E). Sine weareusingaubitheory,in orderto obtaintherighthomology,
weneedtofatoroutbythedegenerateelements.
Foreahexatn 1-ubeF,andeahintegeri2f1;:::;ngwewilldenoteby
s i
1
F theexatn-ubedenedbytheexatsequene(seeexample7.3.3)
i (s
i
1
F):0 !0 !F Id
!F !0:
Analogouslywedenes i
1
bytheexatsequene
i (s
i
1
F):0 !F Id
!F !0 !0:
The exat ubes in the image of s
1 and s
1
are alled degenerate n-ubes.
Clearly the dierential d sends a degenerate ube to a linear ombination of de-
generateubes. Thereforethesubgroupgeneratedbyalldegeneratedubesforma
subomplexof ZC
(E) denotedbyD
.
Definition7.4. Thereduedubial omplex oftheategoryEis
e
ZC
E=ZC
E/D
:
ThehomologyoftheomplexZC red
EisloselyrelatedtotheK-theoryofX.
Forinstane,letS
EbetheWaldhausenspaeassoiatedwiththeategoryE[32℄.
Then
K
i
(X)=
i+1 jS
Ej:
Now,as in[33℄, [8℄or[24℄,oneanonstrutamorphismofomplexes
:ZS
E[1℄ ! e
ZC
E:
Composing withtheHurewiz morphismoneobtainsanaturalmap
K
i
E !H
i (
e
ZC
E):
Forthepurposeofonstrutingharateristilassesthismapisenough(see[8℄).
But R. MCarthy [24℄ has given a preise desription of the homology of e
ZC
E
thatmakesthisomplexmuhmoreinteresting.
Theorem 7.5. The homology of e
ZC
E is the homology of the algebrai K-
theoryspetrumof the ategory E. In partiular
K
i
(E)Q
= H
i (
e
ZC
E)Q:
Moreover,theuseofubesmakes e
ZC
Everywellbehavedtostudy produts.
Definition 7.6. Let F be an exat n-ube and let G be an exat m-ube.
ThenFGistheexatn+m-ubegivenby
(FG)
1;:::;n+m
=F
1
;:::;
n G
n+1
;:::;
n+m :
This produt makes e
ZC
E an assoiativedierentialalgebra whih is homo-
topiallyommutative. Thereforeitshomologyhasthestrutureof anassoiative
andommutativealgebra.
Theorem 7.7(R. MCarthy[24℄.). The morphism
K
(E) !H
(
e
ZC
E)
Wewanttointroduehermitianmetrisin thevetorbundles.
Definition 7.8. LetE=E(X)betheategorywith
ObE= (
(E;h)
E2ObE
hhermitianmetrionE )
and
Hom
E
((E;h);(F;g))=Hom
E (E;F):
Foreah vetorbundle E2Eletus hooseahermitianmetrih
E
. Thisgives
us a funtor F : E ! E. Let G : E ! E be the funtor forget the metri.
These funtors are equivalenes inverse to eah other. In partiular, this implies
thefollowingresult.
Lemma 7.9. Thefuntors FandGinduemorphisms of omplexes
F: e
ZC
E !
e
ZC
E;
G: e
ZC
E !
e
ZC
E;
whih arehomotopy equivalenes, inverseoneof the other.
Proof. ItislearthatGÆF=Id. InadditionthehomotopyhbetweenFÆG
andtheidentityisgiveninthefollowingway. LetF beanelementof e
ZC
n
E. Then
hF istheexatn+1ubedened bytheexatsequene
n
(hF):0 !FÆG(F) Id
!F !0 !0:
TodenehigherBott-Chernformsforexatubesofhermitianvetorbundles,
extendingthetehniqueof setion6,weneedthe third metriin any short exat
sequenetobeinduedbythemiddlemetri. Tothisendweintroduethefollowing
notation.
Definition 7.10. LetF=f(E
;h
)gbeanexatn-ubeofhermitianvetor
bundles. WesaythatFisanemi-n-ube,if,foreahn-tuple=(
1
;:::;
n ),and
eahiwith
i
=1,themetrih
isinduedbythemetrih
(1;:::;i 1;0;i+1;:::;n) .
LetZC emi
EbethesubomplexofZCEgeneratedbythetheemi-n-ubes,and
let D emi
be the subomplex generated by the degenerate emi-n-ubes. We will
write
e
ZC emi
E=ZC emi
E Æ
D emi
e
ZCE:
Letusseethattheomplexes e
ZCEand e
ZC emi
Earehomotopiallyequivalent.
LetF=f(F
;h
)g2C
n
E. Fori=1;:::;nlet 1
i
F bedened by
1
i F
= (
(F
;h
); if
i
= 1;0;
(F
;h 0
); if
i
=1;
where h 0
is the metriindued by h
(
1
;:::;
i 1
;0;
i+1
;:::;
n )
. Thus theoperator 1
i
hanges the metris of the fae 1
i
F by those induedby the metris of the fae
0
i F.
Let 2
i
F betheexatn-ubedeterminedbytheexatsequene
( 2
F):0 ! 1
F !
1
1
F !0 !0:
Thisn-ubemeasuresthedierenebetweenF and
i F.
Letuswrite
i F=
1
i F+
2
i
F, andletusdenote bythemap
: ZC
n
E ! ZC
n E
F 7 !
(
n :::
1
F; ifn1;
F; ifn=0:
Themap is amorphismofomplexes. Moreover,the imageof liesin the
set of emi-n-ubes and sends degenerate ubesto degenerate ubes. Therefore
themorphisminduesamorphismofomplexes
e
: e
ZCE !
e
ZC emi
E:
Proposition7.11. Themorphism e
isahomotopy equivalene.
Proof. Letusdenoteby
: e
ZC emi
E !
e
ZCE
theinlusion. Observethat,ifF isanemi-n-ubethen
F =F+ degenerateelements.
Therefore e
Æ =Id. LetF and Gbethemorphisms of omplexesof lemma 7.9.
SineFÆGishomotopiallyequivalenttotheidentity,weobtaintheequivalene
FÆGÆÆ e
Æ e
:
But, sine the funtor G forgets the metri, G(ÆF) G(F) onsists only in
degenerateubes. ThereforeFÆGÆÆ e
=FÆG:InonsequeneÆ e
Id .
8. Higher Bott-Chern forms
Let X bea smoothomplex projetivevariety. Theaim of this setion is to
give a morphism between the omplex of emi-ubes and the omplex D
(X;).
This morphismwill realizetheharateristilassesfrom higherK-theoryto real
Deligneohomology.
Observethat,sinewewanttorealizetheharateristilassesasamorphism
ofomplexesofabeliangroups,wewillobtainamorphismofgroups. Inpartiular
theinduedmap
K
0
(X) ! M
p H
2p
D (X;p)
willbeadditive. ThisforesustohoosetheChernharaterasourharateristi
lass. Nevertheless,sinetheChernlassesanbereoveredfromtheomponents
oftheChernharaterform,theformulaeweobtainanbeappliedtoanyhara-
teristilass.
Thereasonwerestritourselvestoprojetivevarietiesistoavoidthetehnial
diÆulties ofthelogarithmisingularities at innity. Butnotethat amain ingre-
dient in the proof that higher Bott-Chern forms give Beilinson's regulator is the
extensiontoquasi-projetivevarieties(see[8℄)
Letus see thatBott-Chernforms are thedegreeonestepofthe morphismof
omplexeswearelookingfor. If
0 f
00
isanexatsequeneofhermitianvetorbundles,then
d=(E;h) (Q;h 00
) (S;h 0
):
Therefore
d
D e
h
1
()= 2
e
h
1 ()
= e
h(E;h) e
h(Q;h 00
) e
h(S;h 0
)
= e
h(d):
Toextendthismorphismto higherdegrees, wewill iterate thedenition ofBott-
Chernforms.
LetF=fF
gbeanemi-n-ube. Therststepistoonstrutageometrin-th
order homotopybetweenthevertexesofF. Thereasonfor alling itahomotopy
will belear inproposition8.3. This homotopywill beahermitian vetorbundle
denedoverX(P 1
) n
andwillbealledthen-thtransgressionbundle. Moreover,
we want the n-th transgression bundle to be an exat funtor. Let us dene it
indutively. Ifn=1,anemi-1-ubeisashortexatsequene ofhermitianvetor
bundles withthethird metriinduedby theseond one. Then, thedenition of
the rst transgression bundle is given in 6.2. It follows from the denition that
tr
1
is an exatfuntor. Assume nowthat wehavedened thefuntor tr
n 1 and
thatitisanexatfuntor. Asin 7.3.3,Theemi-n-ubeF anbeseenasanexat
sequeneofemi-n 1-ubes:
n F:
n
(F)0 ! 1
n
F !
0
n
F !
1
n
F !0:
Applyingthefuntortr
n 1
tothisexatsequeneofemi-n 1-ubes,weobtain
anexatsequeneofhermitianvetorbundlesonX(P 1
) n 1
,denotedtr
n 1 (
n F).
Definition 8.1. LetF beanemi-n-ube. Thenthen-thtransgressionbundle
is
tr
n
(F)=tr
1 (tr
n 1 (
n F)):
Sinetr
1
isanexatfuntor,andbyindutionhypothesiswemayassumethat
tr
n 1
isalsoanexatfuntor,weobtainthattr
n
isalsoanexatfuntor.
Remark 8.2. An emi-n-ubeanbeseenas anexatsequene ofemi-n 1-
ubesin n dierentwaysdependingon whih faeswetake. Thus theaboveon-
strutionmaydepend,in priniple,onthehoieofanorderingofthesubindexes.
Nevertheless, the resultis independent of this order. See for instane [8℄ Deni-
tion3.8 foramoresymmetridenitionor[28℄Proposition 2.1for aproof ofthe
invarianeunderpermutations.
Thebasipropertyofthetransgressionbundleisthefollowing.
Proposition8.3 ([8℄Proposition3.9). LetF be anemi-n-ube. Let(x
i :y
i )
behomogeneousoordinates inthe i-thfator of(P 1
) n
. Then
tr
n (F)j
fxi=0g
= tr
n 1 (
0
i F);
tr
n (F)j
fy
i
=0g
=tr
n 1 (
1
i F)
?
tr
n 1 (
1
i F):
Inviewof thisproposition,then-thtransgression bundleof anemi-n-ubeis
Theseondstepistogofromahomotopyofvetorbundlesto ahomotopyof
dierentialforms. Thisstepissimple;therequiredhomotopyis e
h(tr
n
(F))beause
bythefuntorialityoftheChernharaterformandProposition8.3weobtainthat
e
h(tr
n (F))j
fxi=0g
= e
h(tr
n 1 (
0
i F));
(8.1)
e
h(tr
n (F))j
fyi=0g
= e
h(tr
n 1 (
1
i
F))+ e
h(tr
n 1 (
1
i F)):
(8.2)
Thethird stepisto integratethedierentialform e
h(tr
n
(F))dened onX
(P 1
) n
,toobtainadierentialform, e
h
n
(F),denedonX.
To this end wewill introdue some urrents on (P 1
) n
. Let us introdue the
homologialanalogueof the omplexD
, where these urrents will live. For any
smooth omplex projetive variety Y, let D
(Y) be the omplex of urrents on
Y. That is, D
n
(Y) is the topologial dual of E n
(Y). We will denote by D R
(Y)
thesubomplexof realurrentsand byD
p;q
(Y) theurrentsof typep;q(i.e. the
topologialdualofE p;q
(Y)). WewillwriteD R
(Y)(p)=(2i) p
D R
(Y).
Definition8.4. LetD
(Y;)betheomplexdenedby
D
n
(Y;p)= 8
>
>
>
>
>
<
>
>
>
>
>
: D
R
n
(p)(Y)\ M
p 0
+q 0
=n
p 0
p;q 0
p D
p 0
;q 0
(Y); forn2p:
D R
n+1
(p+1)(X)\ M
p 0
+q 0
=n+1
p 0
>p;q 0
>q D
p 0
;q
0(X); forn2p+1:
The homology of the above omplex is the Deligne homology of Y. If Y is
equidimensionalofdimensionnthen,foranyform!2D j
(Y;p), wewilldenoteby
[!℄2D
2n j
(Y;n p)theurrentdenedby
[!℄()= 1
(2i) n
Z
Y
!^: (8.3)
This morphismrealizesthe Poinareduality. If ! is aloally integrableform,we
willusealsothenotation[!℄todenoteitsassoiatedurrent.
Let us denote by d i
j : (P
1
) n 1
! (P 1
) n
, for i = 1;:::;n and j = 0;1 the
inlusionsgivenby
d i
0 (x
1
;:::;x
n )=(x
1
;:::;x
i 1
;(0:1);x
i
;:::;x
n )
d i
1 (x
1
;:::;x
n )=(x
1
;:::;x
i 1
;(1:0);x
i
;:::;x
n ):
The urrentsweneed in order to integratethe form e
h(tr
n
(F)) are provided
bythefollowingresult.
Theorem 8.5([33℄, seealso[8℄and[15℄). There exists a family of urrents
f[W
n
℄g
n0
with [W
n
℄2D
n ((P
1
) n
;0) suhthat
1. [W
0
℄=1.
2. d
D [W
n
℄= n
X
i=1 ( 1)
i
(d i
0 )
[W
n 1
℄ (d i
1 )
[W
n 1
℄
.
Proof. Byequation6.9, weanwrite [W
1
℄=[1=2logtt℄. Letp
i :(P
1
) n
!
P 1
denote the projetion over the i-th fator. Let us write
i
= p
(1=2logtt).
Then
i
isaloallyintegrablefuntionover(P 1
) n
. LetI 0
andE 0
bethehomotopy
equivalenesintroduedattheend ofsetion5. Thenwewillwrite
W
n
=I 0
(E 0
(
1
)[[E 0
(
n )) (8.4)
whih is a loally integrable form. The urrent [W
n
℄ is the assoiated urrent.
Condition2is,formally,onsequeneoftheequation6.9andLeibnitzrule(see[8℄
Proposition6.7and[33℄). Expliitly,theformsW
n
aregivenby(ompare[15℄2.2)
W
n
= ( 1)
n
2n!
n
X
i=1 X
2S
n ( 1)
i 1
( 1)
log(t
(1) t
(1) )
dt
(2)
t
(2)
^^ dt
(i)
t
(i)
^ dt
(i+1)
t
(i+1)
^^ dt
(n)
t
(n) :
Definition 8.6. LetF beanemi-n-ube. Then then-thBott-Chernform of
F is
e
h
n
(F)=[W
n
℄(
e
h(tr
n (F)))
= 1
(2i) n
Z
(P 1
) n
W
n
^ e
h(tr
n (F)):
LetEbeasmallategoryofhermitianvetorbundlesoverX (seesetion 7).
Letus write e
ZC n
emi E=
e
ZC emi
n
E. Then e
ZC
emi
Eis aohomologialomplex. The
denition ofhigherBott-Chernformsinduesmaps
h: e
ZC n
emi
E !
M
p D
n
(X;p)[2p℄:
Proposition8.7. The induedmap
h: e
ZC
emi
E !
M
p D
(X;p)[2p℄:
isamorphism ofomplexes.
Proof. Thisproposition isadiretonsequeneof8.1and8.5.
ThemainresultonerninghigherBott-Chernformsis
Theorem 8.8([8℄). The ompositionmap
K
i (X)
Hurewiz
! H
i
( e
ZC
emi E) !
M
p H
2p i
D
(X;R(p))
agrees withBeilinson'sregulator map.
Remark 8.9. The onstrution of higher Bott-Chern an be made working
alwayswiththeThom-Whitneysimple. WedenetheChernharaterforminthe
Thom-Whitneyomplexas
e
h(F)
TW
=E 0
( e
h(F)):
TheanaloguesoftheformsW
n
aretheforms
(W ) =E 0
( )[[E 0
( ):
