THE INDEX THEOREM FOR QUASI-TORI
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THE INDEX THEOREM FOR QUASI-TORI
DISSERTATION
zu E langung
des DOKTORGRADES (DR. RER. NAT.)
de FAKULT¨
AT F¨
UR MATHEMATIK, PHYSIK UND INFORMATIK
de UNIVERSIT¨
AT BAYREUTH
o geleg on
TSZ ON MARIO CHAN
aus Hong Kong
1. Gu ach e : P o . D . Fab izio Ca anese
2. Gu ach e : P o . D . Philippe Eyssidieux
3. Gu ach e : P o . D . Ngaiming Mok
BAYREUTH
Tag de Ein eichung: 27. No embe , 2012
Tag de Kolloquiums: 15. Feb ua , 2013
E kl¨
a ung
Ich e siche e eidess a lich, dass ich diese A bei selbs ¨
andig e ass habe, und
ich keine ande en als die on mi angegebenen Quellen und Hil smi el benu z habe.
Ich bes ¨
a ige, dass Hil e on gewe blichen P omo ionsbe a e n bzw. - e mi le n
ode ¨
ahnlichen Diens leis e n wede bishe in Ansp uch genommen wu de noch
k¨
un ig in Ansp uch genommen wi d.
Ich bes ¨
a ige, dass ich keine ¨
uhe e P omo ions e suche gemach habe.
Un e sch i des Au o s
i
Acknowledgemen s
I is my pleasu e o exp ess he e my g a i ude o my supe iso P o . Fab izio
Ca anese o sugges ing me his esea ch p oblem and o his con inual guidance,
as well as sha ing his poin o iew abou Ma hema ics and a lo o his pe sonal
expe ience in li e.
My g a i ude also goes o P o . Ing id Baue o encou aging me o explo e
diffe en ields o Ma hema ics. Mo eo e , he ca e o me du ing my sickness made
me eel like home while I was s aying in a coun y dis an om mine.
Many hanks o all cu en and o me colleagues in he Leh s uhl Ma hema ik
VIII o Uni e si ¨a Bay eu h, in pa icula o Michael L¨onne, Fabio Pe oni, Masaaki
Mu akami, S ephen Coughlan, Ma eo Penegini, Wen ei Liu and Yi an Chen, o
hei help on my hesis, inspi ing discussions on Ma hema ical ideas, sha ing abou
he cul u es and li es yles o hei own coun ies, and, mos impo an ly, hei en-
cou agemen s which helped me o ge h ough he mos dep essing pe iod o my
Ph.D. s udy. Thanks also o ou sec e a y Leni Ros ock who helped o so ou all
he oubles du ing my s ay in Bay eu h, om ge ing he esidence pe mi o ind-
ing a medical doc o . Thanks o he , we ha e ne e missed he bi hday o anybody
in Leh s uhl VIII. Wish ha she would enjoy he li e a e e i emen .
Special hanks o my M.Phil. supe iso P o . Ngaiming Mok, who augh me he
basics abou he Bochne –Kodai a o mulas; and o Michael L¨onne, Flo ian Sch ack,
Sascha Weigl and Ch is ian Gleißne who helped me o ansla e he abs ac and
summa y in o Ge man.
I would also like o hank DAAD o hei suppo unde he Fo schungss ipen-
dien ¨
u Dok o anden.
Las ly, I would like o decla e ha I owe my iends ou side he Ma hema ics
communi y in bo h Hong Kong and Ge many a lo . Wi hou hei com o s and
encou agemen s, his hesis could ne e be inished. My deb s o hem can ne e be
ully edeemed. I am also badly indeb ed o my pa en s, who ha e gi en me eedom
o do wha e e I wish.
ii
Abs ac
The Index heo em o holomo phic line bundles on complex o i asse s ha
some cohomology g oups o a line bundle anish acco ding o he signa u e o he
associa ed he mi ian o m. In his a icle, his heo em is gene alized o quasi- o i,
i.e. connec ed complex abelian Lie g oups which a e no necessa ily compac . In
iew o he Remme –Mo imo o decomposi ion o quasi- o i as well as he K¨unne h
o mula, i suffices o conside only Cousin-quasi- o i, i.e. quasi- o i which ha e no
non-cons an holomo phic unc ions. The Index heo em is gene alized o holo-
mo phic line bundles, bo h linea izable and non-linea izable, on Cousin-quasi- o i
using L2-me hods coupled wi h he Kazama–Dolbeaul isomo phism and Bochne –
Kodai a o mulas.
iii
Zusammen assung
Ein Quasi-To us is eine zusammenh¨
angende komplexe abelsche Lie-G uppe
X=Cn/Γ, wobei Γ eine disk e e Un e g uppe on Cnis . Xheiß Cousin-Qua-
si-To us, wenn alle holomo phen Funk ionen au Xkons an sind. Is Xkompak ,
so is Xein komplexe To us.
Nach einem Sa z on Remme und Mo imo o ( gl. [Mo2] ode [CC1, P op. 1.1])
gib es ¨
u jeden Quasi-To us Xeine Ze legung X∼
=Ca×(C∗)b×X′, wobei
X′ein Cousin-Quasi-To us is . Das Ziel des o liegenden A ikels is , das Ve -
schwinden on Kohomologieg uppen on Ge adenb¨
undeln au Xzu un e suchen.
Die K¨
unne h o mel ( gl. [Kau]) besag , dass sich die Kohomologieg uppen on X
in di ek e Summen on opologischen Tenso p oduk en on Kohomologieg uppen
on Ca×(C∗)bund des Cousin-Quasi-To us X′ze legen lassen. Man wi d dadu ch
au den Fall ge ¨
uh , dass Xein Cousin-Quasi-To us is , da Ca×(C∗)bS einsch
is und somi alle h¨
ohe en Kohomologieg uppen (mi G ad ≥1) on koh¨
a en en
Ga ben e schwinden. Es wi d also im o liegenden A ikel angenommen, dass X
ein Cousin-Quasi-To us is .
Sei Fde maximale komplexe Un e aum on RΓ und m:= dimCF. Wie im
kompak en Fall kann jedem holomo phen Ge adenb¨
undel Leine he mi esche Fo m
Hau Cnzugeo dne we den, de en Imagin¨
a eil Im Hmi de e s en Che nklasse
c1(L) on Lassoziie is und ganzzahlige We e in Γ ×Γ annimm . Im Un e schied
zum kompak en Fall is Hnich eindeu ig. Lediglich die Einsch ¨
ankung on Im Hau
RΓ×RΓ, und somi H|F×F, is eindeu ig bes imm . Dies mach zumindes plausibel,
dass nu H|F×Fans elle on H ¨
u die Eigenscha en on L e an wo lich is . Die
o liegende Disse a ion widme sich dem Beweis des olgenden Sa zes:
Index-Sa z ¨
u Cousin-Quasi-To i. Sei X=Cn/Γein Cousin-Quasi-
To us, Fde maximale komplexe Un e aum on RΓ,Lein holomo phes Ge a-
denb¨
undel au Xund Heine mi Lassoziie e he mi esche Fo m au Cn×Cn. Sei
m:= dimCF. Die Einsch ¨
ankung H|F×Fhabe s−
Fnega i e und s+
Fposi i e Eigen-
we e. Dann gil
Hq(X, L) = 0 ¨
u q < s−
Fode q > m −s+
F.
Diese Sa z wi d zu ¨
uckge ¨
uh au den Index-Sa z ¨
u komplexe To i, wie e
on Mum o d [Mum], Kemp [Kem], Umemu a [U], Ma sushima [Ma] und Mu a-
kami [Mu ] ¨
u kompak e Xbewiesen wu de. Da Xs a k (m+ 1)- olls ¨
andig is
( gl. [Kaz1]; siehe auch §2.2), en h¨
al de Sa z auch einen Spezial all des Resul a s
on And eo i und G aue , das besag , dass Hq(X, F) = 0 is ¨
u alle q≥m+ 1
und ¨
u jede koh¨
a en e analy ische Ga be Fau X( gl. [AG ]).
Das Ve schwinden on Hq(X, L) kann un e Ve wendung de Dolbeaul -Isomo -
phismen au gewisse ∂-Gleichungen ¨
u L-we ige (0, q)-Fo men zu ckge ¨
uh we den.
Diese k¨
onnen mi L2-Me hoden gel¨
os we den. Man zeig zun¨
achs die Exis enz eine
o malen L¨
osung eine ∂-Gleichung in einem Hilbe aum, indem man die ben¨
o ig e
L2-Absch¨
a zung nachweis , und beweis dann die Gla hei de L¨
osung. Le z e es
i
ZUSAMMENFASSUNG
kann mi Hil e de Regula i ¨
a s heo ie on ∂-Ope a o en e ledig we den, also is de
en scheidende Sch i de Nachweis de ben¨
o ig en L2-Absch¨
a zungen. Diese kann
man du ch Anwendung de Bochne –Kodai a-Ungleichungen bekommen.
Jede Cousin Quasi-To us Xha eine Fase b¨
undels uk u ¨
ube einem komplexen
To us Tmi s einschen Fase n (siehe §2.1 und (eq 2.3)). Mi Hil e de Le ayschen
Spek alsequenz olg
Hq(X, L)∼
=Hq(T, p∗OX(L)) ¨
u alle q≥0,
wobei p:X→Tdie P ojek ion aus (eq 2.3) is . Die Idee is je z zu zeigen, dass de
Dolbeaul Komplex de Ga ben (A0,•
T⊗OTp∗OX(L), ∂), eine azyklische Au l¨
osung
on p∗OX(L) au Tis und das Ve schwinden de Kohomologie du ch L¨
osen de
∂-Gleichungen zu zeigen. Kazama [Kaz2] und Kazama–Umeno [KU2] geben ei-
ne leich e ¨
ande e Fo mulie ung, sie be ach en die Au l¨
osung on OX(L) du ch
einen Un e komplex (H0,•(L), ∂) on (A0,•
X(L), ∂)(siehe §2.3 ¨
u die De ini ion on
H0,q(L)). De Teilkomplex is eben alls eine azyklische Au l¨
osung on OX(L) au
Xund lie e dami den Kazama–Dolbeaul Isomo phismus ( gl. [KU2], siehe auch
Theo em 2.3.1). Le z e e Ansa z wi d hie au geg iffen. Das Ziel de Da s ellung
is dann die L¨
osung de ∂-Gleichung ∂ξ =ψ ¨
u ein gegebenes ψ∈Γ(X, H0,q(L))
mi ∂ψ = 0.
Jedes Ge adenb¨
undel Lau Xkann du ch ein Sys em on Au omo phie ak o en
de inie we den, die in eine zu Appell–Humbe -No mal o m analoge No mal o m
¨
ube ge ¨
uh we den k¨
onnen, die gegeben is du ch ( gl. [CC1,§2.2] und [V,§2])
ϱ(γ)eπH(z,γ)+ π
2H(γ,γ)+ γ(z)∀γ∈Γ,
wobei ϱein Halbcha ak e au Γ und { γ(z)}γ∈Γein addi i e Kozykel is ( gl.
[CC1,§2.2] und [V,§2], siehe auch (eq 2.8)). Wenn { γ(z)}γ∈Γein Ko and is ,
so wi d Lals linea isie ba bezeichne ; ande n alls als nich linea isie ba . Indem
man den T ick e wende , den Mu akami in [Mu ] ¨
u den kompak en Fall benu z
ha (siehe §3.3), n¨
amlich die Me ik gso abzu¨
ande n, dass de om linea en Teil
(dem zahmen Teil) on Lin den Basis ich ungen kommende K ¨
ummungs e m on
un en besch ¨
ank is , wenn qim gegebenen Be eich lieg , kann man die ben¨
o ig en
L2-Absch¨
a zungen e hal en, wenn Llinea isie ba is (siehe §4). Dies beweis den
Index-Sa z ¨
u linea isie ba e L(siehe Theo em 4.1.1).
Beim Nachweis de ben¨
o ig en L2-Absch¨
a zungen ¨
u nich linea isie ba e Lau
Xgib eine zus¨
a zliche echnische Schwie igkei , die on dem om nich linea en
Teil (dem wilden Teil) on Lkommenden K ¨
ummungs e m he ¨
uh . F¨
u diesen
wi d Takayama’s schwaches ∂∂-Lemma ([Taka2, Lemma 3.14]; siehe auch §5.1)
angewand , um den Te m au ela i kompak en Teilmengen on Xzu besch ¨
anken.
Dadu ch e h¨
al man die ben¨
o ig en L2-Absch¨
a zungen nich au X, sonde n lediglich
au de aussch¨
op enden Familie {Kc}c∈R>0 on pseudokon exen ela i kompak en
Teilmengen. Man e h¨
al dann eine Folge {ξν}ν≥1 on lokalen L¨
osungen, so dass
∂ξν=ψ|Kνis ¨
u ein gegebenes ψ∈Γ(X, H0,q(L)) ∩ke ∂und ¨
u alle ganzen
Zahlen ν≥1. Indem man ein A gumen im Beweis on Theo em B ¨
u S einsche
R¨
aume in [GR, Ch. IV, §5] nach ollzieh , speziell indem man eine App oxima ion
om Runge-Typ e wende , kann man die lokalen L¨
osungen ξνso ko igie en, dass
sie au jedem Kckon e gie en, was dann eine globale L¨
osung ¨
u alle qim gegebenen
Be eich lie e (siehe §5.4). De Beweis des Index-Sa zes is dami olls ¨
andig.
Con en s
E kl¨
a ung i
Acknowledgemen s ii
Abs ac iii
Zusammen assung i
Chap e 1. In oduc ion and he main heo em 1
1.1. The main heo em 2
1.2. Me hodology 3
Chap e 2. P elimina ies 4
2.1. A (C∗)n−m-p incipal bundle s uc u e on X4
2.2. An exhaus i e amily o pseudocon ex subse s 5
2.3. Kazama shea es and Kazama–Dolbeaul isomo phism 5
2.4. Holomo phic line bundles on X6
2.5. A he mi ian me ic on L7
2.6. An L2-no m, he L2-spaces L2
0,(q′,q′′)
c,χ and diffe en ial ope a o s 9
Chap e 3. L2es ima es 11
3.1. Exis ence o a solu ion o ∂ξ =ψ11
3.2. Bochne –Kodai a o mulas 16
3.3. Mu akami’s ick 20
Chap e 4. The linea izable case 27
4.1. P oo o Theo em 1.1.1 o linea izable L27
Chap e 5. The non-linea izable case 28
5.1. Bounds on he wild cu a u e e ms 28
5.2. Exis ence o weak solu ions on Kc29
5.3. A Runge- ype app oxima ion 30
5.4. P oo o Theo em 1.1.1 o gene al L32
Lis o Symbols 35
Bibliog aphy 36
i
CHAPTER 1
In oduc ion and he main heo em
Aquasi- o us is a complex abelian Lie g oup X=Cn/Γ, whe e Γ is a disc e e
subg oup o Cn.Xis said o be a Cousin-quasi- o us i all holomo phic unc ions
on Xa e cons an unc ions.1Xis he amilia complex o us when i is compac ,
i.e. when k Γ = 2n.
The s udy o quasi- o i da es back o he ea ly 20 h cen u y when Cousin s ud-
ied he iply pe iodic unc ions o wo complex a iables ([Cou]). The e he showed
he exis ence o 2-dimensional quasi- o i wi hou non-cons an holomo phic unc-
ions. He also ga e, among o he hings, a comple e desc ip ion o holomo phic line
bundles on quasi- o i o dimension 2 and hei sec ions using a me hod o asymp-
o ic coun ing o ze os o he sec ions. In he 60’s, Kop e mann ([Kop]) s udied
sys ema ically o oidal g oups o a bi a y dimensions wi h a iew o gene alize he
heo y o abelian unc ions on complex o i. He also ga e an example o a non-
compac o oidal g oup wi h no non-cons an me omo phic unc ions. Mo imo o
([Mo1] and [Mo2]) s udied Cousin-quasi- o us as he maximal o oidal subg oup
o a complex (no necessa ily abelian) Lie g oup, aiming o classi y non-compac
complex Lie g oups. He classi ied all 3-dimensional abelian complex Lie g oups. In
he ea ly 70’s, And eo i and Ghe a delli ga e semina s on quasi-abelian a ie ies,
i.e. Cousin-quasi- o i which possess s uc u es o quasi-p ojec i e algeb aic a ie ies
([AGh]). They showed ha , among o he hings, a Cousin-quasi- o us is a quasi-
abelian a ie y i and only i he Gene alized Riemann Rela ions a e sa is ied on
i . La e on, among o he con ibu o s, Kazama ([Kaz1] and [Kaz2]), Po he ing
([P]), He ez ([He ]), Vog ([V]), Hucklebe y and Ma gulis ([HM]), Abe ([Ab1]
and [Ab2]), Capocasa and Ca anese ([CC1] and [CC2]), and Takayama ([Taka2])
made some di ec con ibu ions o he heo y o quasi- o i and Cousin-quasi- o i.
A b ie exposi ion o he his o ical de elopmen o he Gene alized Riemann Rela-
ions can be ound in [CC1, p. 29], and he In oduc ion o [AK] desc ibes a b ie
ch onology o he s udy o o oidal g oups in gene al.
The cu en esea ch s ems om he s udy o Capocasa and Ca anese ( e . [CC1]
and [CC2]). In [CC1], hey ga e an affi ma i e answe o a long s anding p oblem o
whe he he exis ence o a non-degene a e me omo phic unc ion on a quasi- o us is
equi alen o he Gene alized Riemann Rela ions. In [CC2], hey mo ed on o p o e
he Le sche z ype heo ems on quasi- o i in he bes o m, based on a s a emen o
Abe wi h an e oneous p oo in [Ab3, Thm. 6.4] (see [CC2, Co olla y 1.2]).2Abe’s
s a emen is hen subs i u ed by a esul p o en by Takayama ([Taka1, Thm. 1.3 and
1A Cousin-quasi- o us is also called a o oidal g oup o (H, C)-g oup in li e a u e, whe e he
la e means ha all holomo phic unc ions a e cons an ( e . [AK, De . 1.1.1]).
2Th´eo `eme 6.4 in [Ab3] asse s ha , on a non-compac o oidal g oup X, he e exis s a
cons an c > 0 such ha , o any holomo phic line bundle Lwi h an associa ed he mi ian o m
Hon Cnsuch ha H|F×F> cIm(whe e Imis he m×m-iden i y ma ix and Fis he maximal
complex subspace o RΓ; see §2), H0(X, L) is non- i ial, and in ac in ini e-dimensional.
1
2 1. INTRODUCTION AND THE MAIN THEOREM
Thm. 6.1]).3These esul s cla i y some basic p ope ies o me omo phic unc ions
and global sec ions o holomo phic line bundles on quasi- o i. This a icle goes a
s ep u he in o he in es iga ion o he highe cohomology g oups o holomo phic
line bundles on quasi- o i. The aim is o gene alize he Index heo em on o i o
quasi- o i.
1.1. The main heo em
Deno e he C-span and R-span o Γ by CΓ and RΓ espec i ely. Le π:Cn→X
be he na u al p ojec ion. Then K:= π(RΓ) = RΓ/Γ is he maximal compac
subg oup o X, and F:= RΓ∩√−1RΓ is he maximal complex subspace in RΓ.
By a heo em o Remme and Mo imo o ( e . [Mo2], see also [CC1, P op. 1.1]),
i Xis a quasi- o us, he e is a decomposi ion X∼
=Ca×(C∗)b×X′, whe e X′is
Cousin. The aim o his a icle is o in es iga e he anishing o cohomology g oups
o holomo phic line bundles on X. The K¨unne h o mula ( e . [Kau]) asse s ha he
cohomology g oups on Xdecompose in o di ec sum o opological enso p oduc s
o cohomology g oups on Ca×(C∗)band he Cousin-quasi- o us X′. In iew o
his, since Ca×(C∗)bis S ein and hus all highe cohomology g oups (wi h deg ee
≥1) o cohe en shea es anish, one is educed o he case whe e Xis Cousin. In
wha ollows, Xis assumed o be a non-compac Cousin-quasi- o us unless o he wise
s a ed. In his case, CΓ = Cn, and k Γ = dimRRΓ = n+m o some in ege m
such ha 0 < m < n. No e ha mis he complex dimension o F.
Gi en a holomo phic line bundle Lon X, i is analogous o he compac case
ha he e is a he mi ian o m Hon Cn×Cnassocia ed o L, whose imagina y pa
Im H akes in eg al alues on Γ ×Γ and co esponds o he i s Che n class c1(L)
o L( e . [CC1]). Im His uniquely de e mined only on RΓ×RΓ, so His uniquely
de e mined only on F×F.
The ollowing heo em is a gene aliza ion o he Index heo em on complex o i
( e . [Mum, p. 150], [Mu ] o [BL,§3.4])4 o Cousin-quasi- o i, which is he main
esul o his a icle.
Theo em 1.1.1.Le X=Cn/Γbe a Cousin-quasi- o us, F he maximal complex
subspace o RΓ,La holomo phic line bundle on X, and Ha he mi ian o m on
Cn×Cnassocia ed o L. Le m:= dimCF. Suppose H|F×Fhas espec i ely s−
F
nega i e and s+
Fposi i e eigen alues. Then one has
Hq(X, L) = 0 o q < s−
Fo q > m −s+
F.
Le Ωp
Xbe he shea o ge ms o holomo phic p- o ms on X, and se Ωp
X(L) :=
Ωp
X⊗OXOX(L). Since he co angen bundle o Xis i ial, one has Ωp
X(L)∼
=
⊕(n
p)OX(L), and hus Hq(X, Ωp
X(L)) ∼
=⊕(n
p)Hq(X, L). The e o e, one has he
ollowing
3Theo em 1.3 and 6.1 in [Taka1] oge he asse s ha , o any posi i e line bundle Lon a non-
compac o oidal g oup X, he e exis s an explici ly gi en in ege µ0>0 such ha H0(X, L⊗µ) is
non- i ial o all µ≥µ0. Co olla y 1.2 in [CC2] holds ue by applying Takayama’s esul and
P oposi ion 1.1 in [CC2]. Takayama also gi es a diffe en p oo o a weake o m o Le sche z ype
heo ems in [Taka2].
4The Index heo em on complex o i was i s p o en by Mum o d [Mum] and Kemp [Kem]
in he algeb aic case, and la e by Umemu a [U], Ma sushima [Ma] and Mu akami [Mu ] in he
analy ic case.
2.6. AN L2-NORM, THE L2-SPACES L
2
0,(q′,q′′ )
c,χ AND DIFFERENTIAL OPERATORS 9
Wi h he chosen η and ηw, a he mi ian me ic on Lis de ined by
(eq 2.10) η(z) := η (z)ηw(z) = e−πH(z,z)−2 Re ℏδ(z).
The cu a u e o m o Lwi h espec o ηis hen gi en by
ΘT+ ΘW,
which ep esen s he class 2πc1(L) in H2(X, R) (while ΘT ep esen s 2πc1(L) in
2πH2(X, Z)).
2.6. An L2-no m, he L2-spaces L2
0,(q′,q′′)
c,χ and diffe en ial ope a o s
Le gbe a he mi ian me ic on X. Fix an ap coo dina e sys em. Fo he
pu pose o his a icle, gis chosen o be a ansla ional in a ian me ic such ha
he decomposi ion T1,0=T1,0
u⊕T1,0
is o hogonal. Deno e by ω:= −Im g he
associa ed (1,1)- o m as usual.
Fix any holomo phic line bundle L. Conside any 0 < c ≤ ∞ and 0 ≤q≤n.
Deno e he poin wise 2-no m on A0,q(Kc;L) induced om he he mi ian me ics g
and ηby |·|g,η. Le also eχ:R≥0→Rbe a smoo h unc ion and se χ:= eχ◦φ.
Fo he pu pose o his a icle, eχis always assumed o be a non-nega i e con ex
inc easing unc ion. In his case, χis plu isubha monic. Se |ζ|2
g,η,χ := |ζ|2
g,η e−χ.
Le µbe he measu e induced om he olume o m ω∧n
n!. De ine
∥ζ∥Kc,χ := √∫Kc|ζ|2
g,η,χ dµ o any ζ∈A0,q(Kc;L).
Then ∥·∥Kc,χ de ines an L2-no m wi h weigh e−χ(o simply χ) on A0,q
0(Kc;L), he
space o sec ions in A0,q(Kc;L) wi h compac suppo . To simpli y no a ion, dµ in
he in eg al is made implici in wha ollows. The inne p oduc co esponding o
∥·∥Kc,χ is deno ed by ⟨·,·⟩Kc,χ. The no m is w i en as ∥·∥Kc,g,η,χ o emphasize i s
dependence on gand ηwhen necessa y.
Deno e by L2
0,q
c,χ := L2
0,q
χ(Kc;L) he Hilbe space o (µ-)measu able L- alued
(0, q)- o ms ζon Kcsuch ha ∥ζ∥Kc,χ <∞. I is well known ha A0,q
0(Kc;L)⊂
L2
0,q
c,χ is a dense subspace unde he no m ∥·∥Kc,χ.
Fo any 0 ≤p′, q′≤n−mand 0 ≤p′′, q′′ ≤m, de ine
A(p′,p′′),(q′,q′′ ):= A(T∗p′,q′
u∧T∗p′′,q′′
),
i.e. a shea o ge ms o smoo h sec ions o T∗p′,q′
u∧T∗p′′,q′′
(de ined in §2.1). Fo
o he alues o p′, p′′, q′and q′′, se A(p′,p′′ ),(q′,q′′):= 0. No e ha , o 0 ≤p, q ≤n,
he e is a decomposi ion
(eq 2.11) Ap,q =⊕
p′+p′′=p
q′+q′′=q
A(p′,p′′),(q′,q′′ ).
This decomposi ion depends on he choice o he decomposi ion (eq 2.4). Since
he ib e and base di ec ions a e o hogonal o each o he wi h espec o g, he
decomposi ion is also o hogonal wi h espec o g. As only hose shea es wi h
p′+p′′ = 0 a e conside ed in wha ollows, se
A0,(q′,q′′):= A(0,0),(q′,q′′)
10 2. PRELIMINARIES
o no a ional con enience. No ice ha H0,q′′ (L) is a subshea o A0,(0,q′′)(L) o
0≤q′′ ≤m. Fo any c > 0, deno e also he space o sec ions in A0,(q′,q′′)(Kc;L)
wi h compac suppo by A0,(q′,q′′)
0(Kc;L). De ine
L2
0,(q′,q′′)
c,χ := L2
0,(q′,q′′)
χ(Kc;L) := A0,(q′,q′′)
0(Kc;L),
i.e. he closu e o A0,(q′,q′′)
0(Kc;L) in (L2
0,q′+q′′
c,χ ,∥·∥Kc,χ). No e ha he decomposi ion
(eq 2.12) L2
0,q
c,χ =⊕
q′+q′′=q
L2
0,(q′,q′′)
c,χ
induced om (eq 2.11) is also an o hogonal decomposi ion.
The ope a o ∂is decomposed in o ∂[u]+∂[ ]acco ding o he decomposi ion
(eq 2.4), whe e ∂[u]and ∂[ ]a e ope a o s such ha
∂[u]:A0,(q′,q′′)(Kc;L)→A0,(q′+1,q′′)(Kc;L) and
∂[ ]:A0,(q′,q′′)(Kc;L)→A0,(q′,q′′+1)(Kc;L).
Deno e he o mal adjoin s o ∂[u]and ∂[ ]abo e espec i ely by
ϑ[u]:A0,(q′+1,q′′)(Kc;L)→A0,(q′,q′′)(Kc;L) and
ϑ[ ]:A0,(q′,q′′+1)(Kc;L)→A0,(q′,q′′)(Kc;L)
(see, o example, [D1, Ch. VI, 1.5] o he de ini ion).
Some basic ac s abou diffe en ial ope a o s on Hilbe spaces a e ecalled he e.
Ex end he ac ion o hese ope a o s o L2
0,(q′,q′′)
c,χ in he sense o dis ibu ions (o
cu en s). Then, hey de ine closed (i.e. ha ing closed g aph) and densely de ined
linea ope a o s on L2
0,(q′,q′′)
c,χ (see, o example, [H¨o 2, Ch. 1] and [D2, P op. 4.9])
wi h domain gi en by
(eq 2.13) Dom(q′,q′′)
Kc,χ T(o Dom T) := {ζ∈L2
0,(q′,q′′)
c,χ :∥Tζ∥Kc,χ <∞},
whe e Tdeno es any o he abo e ope a o s. No e ha Tis densely de ined since
A0,(q′,q′′)
0(Kc;L)⊂Dom(q′,q′′)
Kc,χ T. An ope a o will be w i en as (T, Dom T) when
he domain is emphasized.
Gi en ∂[u]:L2
0,(q′,q′′)
c,χ →L2
0,(q′+1,q′′)
c,χ and ∂[ ]:L2
0,(q′,q′′)
c,χ →L2
0,(q′,q′′+1)
c,χ wi h domains
gi en as in (eq 2.13), hei Hilbe space adjoin s (also called Von Neumann’s ad-
join s, see o example [D1, Ch. VIII, §1] o a discussion on hem) a e deno ed
espec i ely by
∂∗
[u]:L2
0,(q′+1,q′′)
c,χ →L2
0,(q′,q′′)
c,χ and ∂∗
[ ]:L2
0,(q′,q′′+1)
c,χ →L2
0,(q′,q′′)
c,χ ,
which a e closed and densely de ined ope a o s on L2
0,(q′+1,q′′)
c,χ and L2
0,(q′,q′′+1)
c,χ espec-
i ely. Deno e also hei domains o de ini ion espec i ely by Dom(q′+1,q′′)
Kc,χ ∂∗
[u]and
Dom(q′,q′′+1)
Kc,χ ∂∗
[ ].
In gene al, one has Dom(q′+1,q′′)
Kc,χ ∂∗
[u]⊂Dom(q′+1,q′′)
Kc,χ ϑ[u]and ∂∗
[u]ζ=ϑ[u]ζ o all
ζ∈Dom(q′+1,q′′)
Kc,χ ∂∗
[u](see, o example, [D1, Ch. VIII, §3]). The same holds ue o
∂∗
[ ]and ϑ[ ].
CHAPTER 3
L2es ima es
3.1. Exis ence o a solu ion o ∂ξ =ψ
The aim o his sec ion is o show ha , o 0 ≤q≤m, gi en ψ∈H0,q(Kc;L)∩
L2
0,(0,q)
c,χ such ha ∂ψ = 0 on Kc, he e exis s a weak solu ion ξ∈L2
0,(0,q−1)
c,χ o he
∂-equa ion ∂ξ =ψp o ided ha an L2es ima e is sa is ied. When c=∞, he e
exis s a s ong solu ion which lies in H0,q−1(X;L).
Fi s ecall he ollowing classical heo ems o L2es ima es (see, o exam-
ple, [H¨o 3, Lemmas 4.1.1 and 4.1.2] o [D1, Ch. VIII, Thm. 1.2]). Le (H1,⟨·,·⟩1),
(H2,⟨·,·⟩2) and (H3,⟨·,·⟩3) be some Hilbe spaces, and le (S, Dom S) and (T, Dom T)
be wo closed (i.e. closed g aph) and densely de ined linea ope a o s wi h domains
Dom S⊂H2and Dom T⊂H1 espec i ely such ha
H1
T
//H2
S
//H3
and S◦T= 0, i.e. T(Dom T)⊂ke S:= {ζ∈Dom S:Sζ = 0}. Le S∗and
T∗deno e he Hilbe space adjoin s o Sand T espec i ely, which a e also closed,
densely de ined and sa is ies T∗◦S∗= 0 (see, o example, [D1, Ch. VIII, Thm. 1.1]).
Theo em 3.1.1 (see [H¨o 3, Lemmas 4.1.1 and 4.1.2]).I he e exis s a cons an
C > 0such ha
(eq 3.1) ∥Sζ∥2
3+∥T∗ζ∥2
1≥C∥ζ∥2
2 o all ζ∈Dom S∩Dom T∗,
hen
(1) o e e y ψ∈ke S, he e exis s ξ∈im T∗∩Dom Tsuch ha Tξ =ψand
∥ξ∥2
1≤1
C∥ψ∥2
2. In o he wo ds, ke S= im T(and hus im Tis closed as
ke Sis so);
(2) o e e y Ψ∈(ke T)⊥= im T∗, he e exis s Ξ∈im T∩Dom T∗such ha
T∗Ξ = Ψ and ∥Ξ∥2
2≤1
C∥Ψ∥2
1. In o he wo ds, im T∗= im T∗.
Rema k 3.1.2.By exchanging he oles o Sand T∗, one also ge s ke T∗= im S∗
and im S= im Si he L2es ima e (eq 3.1) is sa is ied.
When Xis compac , conside he complex
L2
0,q−1(X;L)∂
//L2
0,q(X;L)∂
//L2
0,q+1(X;L).
Mu akami [Mu ] shows ha he L2es ima es (eq 3.1) hold o q < s−o q > n−s+
by choosing he he mi ian me ic gsui ably. The L2es ima e on L2
0,q(X;L) implies
ha he ha monic L- alued (0, q)- o ms mus anish. Elemen s in Hq(X, L) a e
ep esen ed by ha monic o ms when Xis compac , so his p o es he anishing o
Hq(X, L) in he compac case.
In he cu en si ua ion, al hough elemen s in Hq(X, L) a e no ep esen ed
by ha monic o ms in gene al, he L2es ima e (eq 3.1) is s ill use ul in sol ing ∂-
equa ions which leads o he anishing o Hq(X, L) o sui able q’s acco ding o
Theo em 3.1.1 (1).
11
12 3. L2ESTIMATES
Due o he exis ence o non-linea izable line bundles, i u ns ou i is necessa y
o sol e ∂-equa ion on Kc o any 0 < c < ∞(see §5.1). The e o e, he aim now
is o sol e he ∂-equa ion ∂ξ =ψ|Kc o a gi en ψ∈H0,q(X;L) wi h ∂ψ = 0.
In iew o he ib e bundle s uc u e (eq 2.3), ins ead o conside ing he complex
L2
0,q−1
c,χ
∂
//L2
0,q
c,χ
∂
//L2
0,q+1
c,χ , i is na u al (see he discussion in §1.2) o conside he
subcomplex
(eq 3.2) L2
0,(0,q−1)
c,χ
Tq−1
//L2
0,q
c,χ <2>
T∗
q−1
oo
Sq
//L2
0,q+1
c,χ <3>
S∗
q
oo,
whe e Tq−1and Sqac as ∂on L2
0,(0,q−1)
c,χ and L2
0,q
c,χ <2> espec i ely, and T∗
q−1and S∗
q
a e hei Hilbe space adjoin s.1The Hilbe spaces in he complex a e de ined as
A0,q
<2>(Kc;L) := A0,(1,q−1) ⊕A0,(0,q)(Kc;L),
A0,q+1
<3>(Kc;L) := A0,(2,q−1) ⊕A0,(1,q)⊕A0,(0,q+1)(Kc;L) ;
L2
0,q
c,χ <2>:= A0,q
0<2>(Kc;L) = L2
0,(1,q−1)
c,χ ⊕L2
0,(0,q)
c,χ ,
L2
0,q+1
c,χ <3>:= A0,q+1
0<3>(Kc;L) = L2
0,(2,q−1)
c,χ ⊕L2
0,(1,q)
c,χ ⊕L2
0,(0,q+1)
c,χ .
Recall om (eq 2.11) and (eq 2.12) ha all he di ec sums on he igh hand sides
abo e a e o hogonal decomposi ions. Deno e he no ms on L2
0,(0,q−1)
c,χ ,L2
0,q
c,χ <2>and
L2
0,q+1
c,χ <3> espec i ely by ∥·∥1,∥·∥2and ∥·∥3, and hei inne p oduc s by ⟨·,·⟩ wi h
he co esponding subsc ip s.
W i e he Hilbe space adjoin o ∂:L2
0,q−1
c,χ →L2
0,q
c,χ as ∂∗. Le p : L2
0,q−1
c,χ →
L2
0,(0,q−1)
c,χ be he o hogonal p ojec ion. Fo la e use, (T∗
q−1,Dom T∗
q−1) is desc ibed
mo e explici ly.
P oposi ion 3.1.3.Wi h he no a ion desc ibed abo e, one has
Dom T∗
q−1= DomKc,χ ∂∗∩L2
0,q
c,χ <2>
= Dom(1,q−1)
Kc,χ ∂∗
[u]⊕Dom(0,q)
Kc,χ ∂∗
[ ].
Mo eo e , o any ζ=ζ′+ζ′′ ∈Dom T∗
q−1whe e ζ′∈Dom(1,q−1)
Kc,χ ∂∗
[u]and ζ′′ ∈
Dom(0,q)
Kc,χ ∂∗
[ ], one has T∗
q−1ζ= p ∂∗ζ=∂∗
[u]ζ′+∂∗
[ ]ζ′′.
P oo . De ine ope a o s (W1,Dom W1) and (W2,Dom W2) om L2
0,q
c,χ <2>in o
L2
0,(0,q−1)
c,χ such ha
Dom W1:= DomKc,χ ∂∗∩L2
0,q
c,χ <2>,
Dom W2:= Dom(1,q−1)
Kc,χ ∂∗
[u]⊕Dom(0,q)
Kc,χ ∂∗
[ ],
and
W1ζ:= p ∂∗ζ o ζ∈Dom W1,
W2ζ:= ∂∗
[u]ζ′+∂∗
[ ]ζ′′ o ζ=ζ′+ζ′′ ∈Dom W2.
1The symbol Tq−1( esp. Sq) is used ins ead o ∂so ha he domains and codomains o he wo
ope a o s can be dis inguished. Mo e p ecisely, i ι:L2
0,(0,q−1)
c,χ ,→L2
0,q−1
c,χ and p : L2
0,q
c,χ →L2
0,q
c,χ <2>
a e espec i ely he inclusion and p ojec ion, hen Tq−1= p ◦∂◦ι. The e o e, T∗
q−1and ∂∗a e
diffe en ope a o s.
3.1. EXISTENCE OF A SOLUTION OF ∂ξ =ψ13
These a e closed and densely de ined linea ope a o s on L2
0,q
c,χ <2>. Since ∥Tq−1ζ∥2
2=
∂ζ2
2=∂[u]ζ2
2+∂[ ]ζ2
2 o all ζ∈L2
0,(0,q−1)
c,χ , i ollows ha
Dom Tq−1= Dom ∂∩L2
0,(0,q−1)
c,χ
= Dom(0,q−1)
Kc,χ ∂[u]∩Dom(0,q−1)
Kc,χ ∂[ ].
Fi s is o show ha (T∗
q−1,Dom T∗
q−1) = (W1,Dom W1). No e ha , o any
∈L2
0,(0,q−1)
c,χ and any ζ∈Dom W1, one has
⟨ , W1ζ⟩1=⟨ , p ∂∗ζ⟩1=⟨ , ∂∗ζ⟩Kc,χ .
Fo any ˜
ζ∈L2
0,q
c,χ =L2
0,q
c,χ <2>⊕(L2
0,q
c,χ <2>)⊥, w i e ˜
ζ=ζ+ζ⊥whe e ζ∈L2
0,q
c,χ <2>
and ζ⊥∈(L2
0,q
c,χ <2>)⊥=⊕q
q′=2 L2
0,(q′,q−q′)
c,χ . No e ha ∂∗ζ⊥∈⊕q−1
q′=1 L2
0,(q′,q−1−q′)
c,χ =
(L2
0,(0,q−1)
c,χ )⊥, hus ⟨ , ∂∗ζ⊥⟩Kc,χ = 0 o any ∈L2
0,(0,q−1)
c,χ . The e o e, o any
∈L2
0,(0,q−1)
c,χ , one has
∈Dom W∗
1
:⇐⇒ ∃ C > 0: ∀ζ∈Dom W1,|⟨ , W1ζ⟩1|=⟨ , ∂∗ζ⟩Kc,χ≤C∥ζ∥2
⇐⇒ ∃ C > 0: ∀˜
ζ∈DomKc,χ ∂∗,
⟨ , ∂∗˜
ζ⟩Kc,χ=⟨ , ∂∗ζ⟩Kc,χ≤C˜
ζKc,χ
⇐⇒ ∈Dom ∂∩L2
0,(0,q−1)
c,χ = Dom Tq−1as (∂∗)∗=∂
( e . [D1, Ch. VIII, §1] o he de ini ion o he domain o Hilbe space adjoin s), and
hus Dom W∗
1= Dom Tq−1. I ollows ha ⟨ , W1ζ⟩1=⟨ , ∂∗ζ⟩Kc,χ =⟨∂ , ζ⟩2=
⟨Tq−1 , ζ⟩2 o any ∈Dom Tq−1and ζ∈Dom W1. As a esul , (Tq−1,Dom Tq−1) =
(W∗
1,Dom W∗
1), and hence (T∗
q−1,Dom T∗
q−1) = (W1,Dom W1) ( e . [D1, Ch. VIII,
Thm. 1.1]).
The p oo o (T∗
q−1,Dom T∗
q−1) = (W2,Dom W2) is simila . No ice ha ∥ζ∥2
2=
∥ζ′∥2
2+∥ζ′′∥2
2and hus ∥ζ′∥2+∥ζ′′∥2≤√2∥ζ∥2 o all ζ=ζ′+ζ′′ ∈L2
0,q
c,χ <2>. Then,
o any ∈L2
0,(0,q−1)
c,χ , one has
∈Dom W∗
2
:⇐⇒ ∃ C > 0: ∀ζ=ζ′+ζ′′ ∈Dom W2,
|⟨ , W2ζ⟩1|=⟨ , ∂∗
[u]ζ′+∂∗
[ ]ζ′′⟩1≤C∥ζ∥2
⇐⇒ ∃ C > 0: ∀ζ′∈Dom(1,q−1)
Kc,χ ∂∗
[u]and ∀ζ′′ ∈Dom(0,q)
Kc,χ ∂∗
[ ],
⟨ , ∂∗
[u]ζ′⟩1≤C∥ζ′∥2and ⟨ , ∂∗
[ ]ζ′′⟩1≤C∥ζ′′∥2
⇐⇒ ∈Dom(0,q−1)
Kc,χ ∂[u]∩Dom(0,q−1)
Kc,χ ∂[ ]= Dom Tq−1,
and hus Dom W∗
2= Dom Tq−1. No e ha ⟨ , W2ζ⟩1=⟨∂[u] , ζ′⟩2+⟨∂[ ] , ζ′′⟩2=
⟨∂[u] +∂[ ] , ζ′+ζ′′⟩2=⟨Tq−1 , ζ⟩2 o ∈Dom Tq−1and ζ∈Dom W2, since
14 3. L2ESTIMATES
L2
0,(1,q−1)
c,χ ⊥L2
0,(0,q)
c,χ . The e o e, one has (Tq−1,Dom Tq−1) = (W∗
2,Dom W∗
2), and
hus (T∗
q−1,Dom T∗
q−1) = (W2,Dom W2) ( e . [D1, Ch. VIII, Thm. 1.1]). □
Suppose now gi en 0 < c ≤ ∞ and ψ∈H0,q(Kc;L)∩L2
0,(0,q)
c,χ ⊂L2
0,q
c,χ <2>such
ha Sqψ=∂ψ = 0. Theo em 3.1.1 (1) asse s ha , i he L2es ima e (eq 3.1) is
sa is ied, hen he e exis s ξ∈im T∗
q−1⊂L2
0,(0,q−1)
c,χ such ha
(eq 3.3) Tq−1ξ=∂ξ =ψin L2
0,(0,q)
c,χ .
One can ha e a u he educ ion. When c=∞, since (X, g) is comple e in he
sense o Riemannian geome y, A0,q
0<2>(X;L) is dense in DomXT∗
q−1∩DomXSqunde
he abo e g aph no m (see, o example, [D1, Ch. VIII, Thm. 3.2]). The e o e, i
suffices o es ablish he equi ed L2es ima es (eq 3.1) o ζ∈A0,q
0<2>(X;L).
Suppose c < ∞. No e ha A0,q
<2>(Kc;L)⊂Dom Sq. Since ∂Kcis smoo h and
χis smoo h on a neighbo hood o Kc, using [H¨o 1, P op. 2.1.1] oge he wi h an
a gumen o pa i ion o uni y, i yields he ollowing
P oposi ion 3.1.4.A0,q
<2>(Kc;L)∩Dom T∗
q−1is dense in Dom T∗
q−1∩Dom Sq
unde he g aph no m √T∗
q−1ζ2
1+∥Sqζ∥2
3+∥ζ∥2
2.
P oo . No e ha he s a emen ollows om [H¨o 1, P op. 2.1.1] when T∗0,q
X
and La e bo h i ial by using a pa i ion o uni y. The aim now is o handle he
case when Lis non- i ial.
Take a locally ini e open co e {Uα}α∈Ao Xsuch ha e e y Uαis a coo dina e
cha o Xand Lis i ialized on each Uαwi h ansi ion unc ions σαβ ∈O∗
X(Uα∩
Uβ) o all α, β ∈A. Then, o any ζ∈L2
0,q
χ(X;L) wi h ζα ep esen ing ζo e Uα
unde he i ializa ion, one has ζα=σαβζβon Uα∩Uβ.
Fix any ζ∈Dom T∗
q−1∩Dom Sq. I suffices o show ha ζcan be app oxima ed
by a sequence {ζ(ν)}ν∈N⊂A0,q
<2>(Kc;L)∩Dom T∗
q−1unde he gi en g aph no m.
Ex end ζby ze o o a sec ion on X. Using a pa i ion o uni y which decomposes
ζin o a sum o ini ely many compac ly suppo ed sec ions, one can assume ha
ζis compac ly suppo ed in a coo dina e cha U:= U0∈ {Uα}α∈A. Then he
he mi ian me ic ηon Lcan be iewed as a unc ion eη:= η0on U=U0(unde
he gi en i ializa ion), and any L- alued o m ∈L2
0,q
g,η,χ(U;L) can be iewed
as a OX- alued o m e
:= 0∈L2
0,q
g,eη,χ(U). Le W:= U∩Kc. No e ha one has
e
W,g,eη,χ =∥ ∥W,g,η,χ,∂e
W,g,eη,χ =∂ W,g,η,χ and ∂∗e
W,g,eη,χ =∂∗ W,g,η,χ
o all ∈L2
0,q
g,η,χ(W;L). Then ζ∈Dom T∗
q−1∩Dom Sqimplies e
ζ∈DomW,g,eη,χ ∂∗∩
DomW,g,eη,χ ∂∩L2
0,q
g,eη,χ <2>(W). Since gand χa e ixed in wha ollows, subsc ip s o
hem a e omi ed om he no a ions below.
By [H¨o 1, P op. 2.1.1] (o applying [H¨o 1, P op. 1.2.4] di ec ly), he e exis s a
sequence {e
ζ(ν)}ν∈N⊂A0,q(W)∩DomW,eη∂∗such ha
∂∗(e
ζ(ν)−e
ζ)
2
W,e
η+∂(e
ζ(ν)−e
ζ)
2
W,eη+e
ζ(ν)−e
ζ
2
W,eη→0
as ν→ ∞ and supp e
ζ(ν)⋐U o all ν∈N. As e
ζ(ν)’s a e ob ained om con olu ions
be ween smoo hing ke nels and e
ζwhich do no change he ype o o ms, i ollows
ha e
ζ(ν)∈A0,q
<2>(W). The sec ions ζ(ν)∈A0,q
<2>(W;L) de ined by ζ(ν)
α:= 1
σ0αe
ζ(ν)
on Uα∩U=∅a e compac ly suppo ed in U(hence ζ(ν)∈A0,q
<2>(Kc;L)) and
3.1. EXISTENCE OF A SOLUTION OF ∂ξ =ψ15
sa is y g
ζ(ν)=e
ζ(ν). The e o e, one ob ains a sequence {ζ(ν)}ν∈N⊂DomKc,η ∂∗∩
A0,q
<2>(Kc;L) = Dom T∗
q−1∩A0,q
<2>(Kc;L) (see P oposi ion 3.1.3) such ha
T∗
q−1(ζ(ν)−ζ)2
1+Sq(ζ(ν)−ζ)2
3+ζ(ν)−ζ2
2
≤∂∗(ζ(ν)−ζ)
2
W,η +∂(ζ(ν)−ζ)2
W,η +ζ(ν)−ζ2
W,η
as T∗
q−1= p ∂∗
by P op. 3.1.3
=∂∗(e
ζ(ν)−e
ζ)
2
W,eη+∂(e
ζ(ν)−e
ζ)
2
W,eη+e
ζ(ν)−e
ζ
2
W,eη
→0 as ν→ ∞
as equi ed. □
As a esul , i suffices o es ablish he equi ed L2es ima es (eq 3.1) o ζ∈
A0,q
<2>(Kc;L)∩Dom T∗
q−1.
The abo e discussion is summa ized in he ollowing
P oposi ion 3.1.5.Suppose 0< c ≤ ∞. I he e exis s a cons an C > 0such
ha
(eq 3.4) ∥Sqζ∥2
3+T∗
q−1ζ2
1≥C∥ζ∥2
2
o all ζ∈{A0,q
<2>(Kc;L)∩Dom T∗
q−1when c < ∞,
A0,q
0<2>(X;L)when c=∞,
hen, o e e y ψ∈H0,q(Kc;L)∩L2
0,(0,q)
χ(Kc;L)such ha ∂ψ = 0, he e exis s
ξ∈L2
0,(0,q−1)
χ(Kc;L)such ha ∂ξ =ψin L2
0,(0,q)
χ(Kc;L).
Rema k 3.1.6.Le L2
0,q−1(Kc;L; loc) deno e he space o locally L2L- alued
(0, q −1)- o ms on Kc, which con ains L2
0,(0,q−1)
χ(Kc;L) as a subspace. I ollows
om he classical egula i y heo y o ∂-ope a o o ellip ic ope a o s ( e . [H¨o 3,
Thm. 4.2.5 and Co . 4.2.6] o [H¨o 2, Thm. 4.1.5 and Co . 4.1.2]) ha he exis ence o
ξ∈L2
0,q−1(Kc;L; loc) sa is ying he equa ion (eq 3.3) in L2
0,q(Kc;L; loc) implies ha
he e exis s ξ∈A0,q−1(Kc;L) (bu no necessa ily in A0,(0,q−1)(Kc;L)) sa is ying
he same equa ion in A0,q(Kc;L). In case c=∞, Theo em 2.3.1 implies ha he e
e en exis s a solu ion ξ∈H0,q−1(X;L) such ha ∂ξ =ψon X.
Rema k 3.1.7.W i e H0,q
L2(Kc;L) := H0,q(Kc;L)∩L2
0,(0,q)
c,χ . Following he
idea discussed in §1.2, i would be mo e na u al o conside he L2es ima e on
H0,q
c,χ := H0,q
L2(Kc;L) a he han L2
0,q
c,χ <2>, whe e he closu e is aken in L2
0,(0,q)
c,χ .
Howe e , he au ho aces he difficul y in ob aining he equi ed es ima e om he
Bochne –Kodai a inequali ies when H0,q
c,χ ins ead o L2
0,q
c,χ <2>is conside ed. W i e
∂∗
Hcas he Hilbe space adjoin o ∂=∂[ ]:H0,q
c,χ →H0,q+1
c,χ . I can be shown
ha ∂∗
Hc= p c◦∂∗
[ ]on Dom(0,q)
Kc,χ ∂∗
Hc, whe e p c:L2
0,(0,q)
c,χ →H0,q
c,χ is he o hogonal
p ojec ion. Se ð∗
⊥c:= ∂∗
[ ]−∂∗
Hc, hen ∂∗
Hcζand ð∗
⊥cζa e o hogonal o each o he
o all ζ∈Dom(0,q)
Kc,χ ∂∗
Hcand
∂∗
[ ]ζ
2
Kc,χ =∂∗
Hcζ
2
Kc,χ +ð∗
⊥cζ2
Kc,χ .
16 3. L2ESTIMATES
F om he Bochne –Kodai a inequali ies, one ob ains
∂ζ2
Kc,χ +∂∗
[ ]ζ
2
Kc,χ ≥∫Kc
Cu (ζ, ζ)
o all ζ∈H0,q
L2(Kc;L)∩Dom(0,q)
Kc,χ ∂∩Dom(0,q)
Kc,χ ∂∗
[ ], whe e ∫KcCu (ζ, ζ) is he
cu a u e e m a ising om he cu a u e o L. By choosing sui ably he me ics
gand η, he cu a u e e m can be bounded below by C∥ζ∥2
Kc,χ o some cons an
C > 0. The e o e, in o de o ob ain he desi ed es ima e ∂ζ2
Kc,χ +∂∗
Hcζ
2
Kc,χ ≥
C′′ ∥ζ∥2
Kc,χ o some cons an C′′ >0, one has o show ha ∥ð∗
⊥cζ∥2
Kc,χ ≤C′∥ζ∥2
Kc,χ
o some cons an C′>0 such ha C > C′. Howe e , he cons an C′depends on g
in gene al and one may no be able o make C′smalle han Cby al e ing g. Tha ’s
why he L2es ima e on L2
0,q
c,χ <2>ins ead o H0,q
c,χ is conside ed in his a icle.
3.2. Bochne –Kodai a o mulas
Le
∇:A(T∗•,•⊗L)→A(T∗C⊗T∗•,•⊗L),
whe e T∗C:= T∗1,0⊕T∗0,1, be he connec ion on T∗•,•⊗Linduced om he Che n
connec ions on he holomo phic he mi ian ec o bundles (T1,0, g) and (L, ηe−χ).
The e o e, ∇is compa ible wi h he poin wise no m |·|g,η,χ.
Unde a chosen ap coo dina e sys em, se ∂k:= ∂
∂zkand ∂k:= ∂
∂zk o 1 ≤k≤n.
These de ine global ec o ields on X. Se ∇k:= ∇∂kand ∇k:= ∇∂k o 1 ≤k≤n.
Se also ∇ j:= ∇n−m+j=∇∂
∂ jand ∇ j:= ∇n−m+j=∇∂
∂ j(and de ine ∂ jand ∂ j
simila ly) o 1 ≤j≤m o no a ional con enience. Since he he mi ian me ic gis
ansla ional in a ian on X, he Ch is offel symbols gi en om g anish and hus
one has locally
(eq 3.5) ∇k=∂k+∂klog (ηe−χ),
∇k=∂k
o 1 ≤k≤n. Fo la e use, no e ha he commu a o o ∇kand ∇ℓis gi en by
Θkℓ := [∇k,∇ℓ] = −∂k∂ℓlog (ηe−χ),
and he cu a u e o m o Lendowed wi h he me ic ηe−χis gi en by
(eq 3.6) Θ := −√−1∂∂ log (ηe−χ)=√−1
n
∑
k,ℓ=1
Θkℓ dzk∧dzℓ.
W i e he cu a u e enso associa ed o Θ as
R:=
n
∑
k,ℓ=1
Θkℓ dzk⊗dzℓ.
Since he base and ib e di ec ions a e o hogonal o each o he wi h espec
o g, he iden i ica ion be ween Ap,q and Ap,q =Aq,p := A(Tq,p) induced om g
espec s he decomposi ion (eq 2.4) (Ap,q he e means he complex conjuga e o Ap,q).
Fo la e use, se A(p′,p′′),(q′,q′′ ):= A(Tp′,q′
u∧Tp′′,q′′
)and A(p′,p′′),0:= A(p′,p′′ ),(0,0) o
0≤p′, q′≤n−mand 0 ≤p′′, q′′ ≤m. Fo any ζ∈Ap,0⊗A0,q, le ζ∨deno e he
image o ζin A0,p ⊗Aq,0 ia he isomo phism induced om g. Then, o example,
i ζ∈A0,(q′,q′′ ), one has ζ∨∈A(q′,q′′ ),0.
3.2. BOCHNER–KODAIRA FORMULAS 17
As a bilinea o m on A1,0⊗A1,0,Rcan be decomposed acco ding o he decom-
posi ion (eq 2.4) in o he sum o
Ruu := R|A(1,0),0⊗A(1,0),0,Ru := R|A(1,0),0⊗A(0,1),0,
R u := R|A(0,1),0⊗A(1,0),0,R := R|A(0,1),0⊗A(0,1),0.
Since Ris a he mi ian o m, i ollows ha Ruu =Ruu,R =R and Ru =R u.
Le T g:A0,q ⊗Aq,0→A0,0be he ace ope a o which is de ined in such a
way ha ζ⊗ξ7→ ξ∨⌟ζ, whe e ζ∈A0,q,ξ∈Aq,0and ξ∨⌟ζdeno es he comple e
con ac ion be ween ζand ξ∨. Deno e by T g,η he simila con ac ion o L- alued
o ms.
Fix any 0 <c<∞. Deno e he Hilbe space adjoin o ∂:L2
0,q−1
c,χ →L2
0,q
c,χ by
∂∗:L2
0,q
c,χ →L2
0,q−1
c,χ . Iden i y A1,1and A1,0⊗A0,1 ia he isomo phism dzk∧dzℓ7→
dzk⊗dzℓ o any 1 ≤k, ℓ ≤n. Le R∨(ζ⊗ζ) ( esp. (∂∂φ)∨(ζ⊗ζ)) deno es he
na u al con ac ion be ween R∨( esp. (∂∂φ)∨) and ζ⊗ζ. Le ∇=∇(1,0) +∇(0,1)
be he decomposi ion o ∇in o (1,0)- and (0,1)- ypes. The ∇-Bochne –Kodai a
o mula (c . [Siu, (2.1.4) and (1.3.3)]) is hen gi en by
(eq 3.7) ∂ζ2
Kc,χ +∂∗ζ
2
Kc,χ =∫∂Kc
e−χ
|dφ|gT g,η (∂∂φ)∨(ζ⊗ζ)
+∇(0,1)ζ2
Kc,χ +∫Kc
e−χT g,η R∨(ζ⊗ζ)
o all ζ∈A0,q(Kc;L)∩DomKc,χ ∂∗.
Rema k 3.2.1.No e ha he measu e o he bounda y in eg al is induced
om ((dφ)∨
|dφ|g
⌟ω∧n
n!)∂Kc
. In o de o compa e no a ions in [Siu, (2.1.4)] and hose in
(eq 3.7), w i e [x]Siu o mean he symbol xused in [Siu]. Then
[∇]Siu =∇(0,1) ,[∇]Siu =∇(1,0) ,[ρ]Siu =φ−c
|dφ|g
,[Rijkl]Siu = 0 ,
and [−Ωαβs ]Siu = componen s o R= Θkℓ .
No e ha [Rijkl]Siu = 0 as he Che n connec ion on (T1,0, g) is la . Also be awa e
o he ypos o he signs p eceding he cu a u e in eg als in ol ing [Ωs
αβ ]Siu and
[Rs
]Siu in [Siu, (2.1.4)]. The co ec signs can be ound in [Siu, (1.3.3)]. To see
ha he bounda y e m in (eq 3.7) coincides wi h he one in [Siu, (2.1.4)], no e ha
a e e y z∈∂Kc,
∂∂ (φ−c
|dφ|g)(z) = ∂∂φ
|dφ|g
(z)−∂φ ∧∂|dφ|g
|dφ|2
g
(z)−∂|dφ|g∧∂φ
|dφ|2
g
(z).
A e aking ∨and con ac ing wi h ζ⊗ζwhe e ζ∈A0,q(Kc;L)∩DomKc,χ ∂∗, he
las wo e ms on he igh hand side anish because, o ζ∈A0,q(Kc;L), (∂φ)∨⌟ζ=
0 on ∂Kci and only i ζ∈DomKc,χ ∂∗( e . [H¨o 1, pg. 101] o [Siu, (2.1.1)]). The
bounda y e ms he e o e coincides.
When he subcomplex (eq 3.2) is conside ed, he ∇-Bochne –Kodai a o mula
(eq 3.7) is es ic ed o ζ∈A0,q
<2>(Kc;L)∩DomKc,χ ∂∗=A0,q
<2>(Kc;L)∩DomKc,χ T∗
q−1
(see P oposi ion 3.1.3). The (0,1)-connec ion spli s in o ∇(0,1) =∇(0,1)
u+∇(0,1)
18 3. L2ESTIMATES
acco ding o he decomposi ion (eq 2.4). W i e ∇u:= ∇(0,1)
uand ∇ := ∇(0,1)
o
no a ional con enience. Le also p F:A0,q ⊗A0,s →A0,(0,q)⊗A0,(0,s)be he
canonical p ojec ion (whe e A0,s ( esp. A0,(0,s)) is he complex conjuga e o A0,s
( esp. A0,(0,s))). Se
(eq 3.8) Bd(ζ, ζ) := ∫∂Kc
e−χ
|dφ|gT g,η (∂∂φ)∨(ζ⊗ζ)
o no a ional con enience. Then (eq 3.7) gi es he ollowing
Lemma 3.2.2.Fo any ζ=ζ′+ζ′′ ∈A0,q
<2>(Kc;L)∩Dom T∗
q−1, whe e ζ′∈
A0,(1,q−1)(Kc;L)∩Dom ∂∗
[u]and ζ′′ ∈A0,(0,q)(Kc;L), one has
(eq 3.9)
∥Sqζ∥2
3+T∗
q−1ζ2
1= Bd(ζ, ζ) + ∂[u]ζ′′2
3+∂[ ]ζ′2
3
+∥∇uζ′∥2
Kc,χ +∥∇ ζ′′∥2
Kc,χ
+∫Kc
e−χT g,η p F(R∨(ζ⊗ζ)).
P oo . On DomKc,χ ∂∗, one has ∂∗=ϑ[u]+ϑ[ ]. Then, o all ζ=ζ′+ζ′′ ∈
Dom T∗
q−1= DomKc,χ ∂∗∩L2
0,q
c,χ <2>(see P oposi ion 3.1.3), one has
∂∗ζ=ϑ[u]ζ′+ϑ[u]ζ′′ +ϑ[ ]ζ′+ϑ[ ]ζ′′ =T∗
q−1ζ+ϑ[ ]ζ′,
as T∗
q−1ζ=∂∗
[u]ζ′+∂∗
[ ]ζ′′ (see P oposi ion 3.1.3) and ϑ[u]ζ′′ = 0. No e also ha
∇(0,1)ζ=∇uζ′+∇uζ′′ +∇ ζ′+∇ ζ′′, and ∂ζ =Sqζ. Since he decomposi ion
(eq 2.4) is o hogonal wi h espec o g, i ollows ha
∂∗ζ
2
Kc,χ =T∗
q−1ζ2
1+∥ϑ[ ]ζ′∥2
Kc,χ and
∇(0,1)ζ2
Kc,χ =∥∇uζ′∥2
Kc,χ +∥∇uζ′′∥2
Kc,χ +∥∇ ζ′∥2
Kc,χ +∥∇ ζ′′∥2
Kc,χ .
No e ha ∥∇uζ′′∥2
Kc,χ =∂[u]ζ′′2
3.
Following he a gumen in [H¨o 1, pg. 101] wi h ∂[ ]in place o ∂, i ollows ha ,
o any ζ∈A0,(q′,q′′)(Kc;L), ζ∈Dom(q′,q′′ )
Kc,χ ∂∗
[ ]i and only i (∂[ ]φ)∨⌟ζ= 0 on ∂Kc.
Since ∂[ ]φ= 0, i ollows ha A0,(q′,q′′ )(Kc;L)⊂Dom(q′,q′′ )
Kc,χ ∂∗
[ ]. In pa icula , ζ′∈
Dom(1,q−1)
Kc,χ ∂∗
[ ] o all ζ′∈A0,(1,q−1)(Kc;L). Then, since he decomposi ion (eq 2.4)
is o hogonal wi h espec o g, by aking he analogy be ween he decomposi ions
A =⊕p+q= Ap,q and Ap,q =⊕p=p′+p′′
q=q′+q′′
A(p′,p′′),(q′,q′′ )and pu ing ∂[ ]in place o
∂, one can ollow he de i a ion o (eq 3.7) as in [Siu,§1 and §2] o ob ain
∂[ ]ζ′2
Kc,χ +∥ϑ[ ]ζ′∥2
Kc,χ =∫∂Kc
e−χ
|dφ|gT g,η (∂[ ]∂[ ]φ)∨(ζ′⊗ζ′)
+∥∇ ζ′∥2
Kc,χ +∫Kc
e−χT g,η R∨
(ζ′⊗ζ′)
o any ζ′∈A0,(1,q−1)(Kc;L). The bounda y e m anishes as ∂[ ]∂[ ]φ= 0. The e-
o e, combining he abo e esul s wi h (eq 3.7) yields
∥Sqζ∥2
3+T∗
q−1ζ2
1= Bd(ζ, ζ) + ∂[u]ζ′′2
3+∂[ ]ζ′2
3+∥∇uζ′∥2
Kc,χ +∥∇ ζ′′∥2
Kc,χ
+∫Kc
e−χT g,η R∨(ζ⊗ζ)−∫Kc
e−χT g,η R∨
(ζ′⊗ζ′).
3.3. MURAKAMI’S TRICK 25
P oo . Fo q= 0, i ollows om (eq 3.18) ha
π∫Kc
e−χT g,η p F((e
H(M))∨(ζ⊗ζ))=πM ∥ζ∥2
2≥π
4M∥ζ∥2
2,
so his case is done.
Assume q= 0. Since H∨
u is a bounded linea ope a o on L2
0,(1,0)
c,χ ⊗L2
0,(0,1)
c,χ
(whe e L2
0,(0,1)
c,χ he e means he complex conjuga e o L2
0,(0,1)
c,χ ), i ollows ha he e is
a bounded linea ope a o N:L2
0,(0,q)
c,χ →L2
0,(1,q−1)
c,χ such ha
∫Kc
e−χT g,η H∨
u (ζ′⊗ζ′′) = ⟨ζ′,Nζ′′⟩2
o all ζ′∈L2
0,(1,q−1)
c,χ and ζ′′ ∈L2
0,(0,q)
c,χ . In ac , a e a linea change o coo dina es
such ha gbecomes he Euclidean me ic while keeping he decomposi ion (eq 2.4)
o hogonal, one has
T g,η H∨
u (ζ′⊗ζ′′) = η∑′
Jq−1
n−m
∑
i=1
m
∑
j=1
ζ′
iJq−1(H u)ji ζ′′
jJq−1,
whe e ∑′
Jq−1deno es summa ion o e all o de ed mul iindices Jq−1such ha 1 ≤
j1<··· < jq−1≤m, and (H u)ji’s a e he componen s o H u =Hu . The e o e,
unde such coo dina es,
(Nζ′′)iJq−1=
m
∑
j=1
(H u)ji ζ′′
jJq−1.
Mo eo e ,
|Nζ′′|2
g,η =η∑′
Jq−1
n−m
∑
i=1
m
∑
j=1
(H u)ji ζ′′
jJq−1
2
≤η∑′
Jq−1
n−m
∑
i=1 (m
∑
j=1 (H u)ji2)(m
∑
j=1 ζ′′
jJq−1
2)by Cauchy–
Schwa z ineq.,
=|H u|2
g·q|ζ′′|2
g,η =|Hu |2
g·q|ζ′′|2
g,η as Hu =H u .
Since bo h Hu and ga e ansla ional in a ian o ms, |Hu |2
gis a cons an . Se
ν:= √q|Hu |g. Then, one has
(∗ν)∥Nζ′′∥2≤ν∥ζ′′∥2
o all ζ′′ ∈L2
0,(0,q)
c,χ . No e ha νdepends only on q,Hu and g. I is independen o
HEin pa icula .
Since he decomposi ion (eq 2.4) is o hogonal wi h espec o g,gcan be de-
composed in o gE+gFsuch ha gEis a he mi ian me ic on T1,0
uand gFis ha on
T1,0
. Choose a eal numbe λ > 0 such ha
(∗λ)λ≥max {M
2,2ν2
M,4ν}.
Since νis independen o HE, by a ying he eal pa o he ma ix o HEunde
he chosen ap coo dina es acco ding o P oposi ion 2.4.2, HEcan be chosen such
ha
HE≥λgE,
26 3. L2ESTIMATES
and he e o e, ∫Kc
e−χT g,η H∨
E(ζ′⊗ζ′)≥λ∥ζ′∥2
2
o all ζ′∈A0,(1,q−1)(Kc;L).
I ollows om (eq 3.18) ha , o any ζ=ζ′+ζ′′ ∈A0,q
<2>(Kc;L),
∫Kc
e−χT g,η p F(e
H(M))∨(ζ⊗ζ)
≥λ∥ζ′∥2
2+ 2 Re ⟨ζ′,Nζ′′⟩2+M∥ζ′′∥2
2
=λζ′+1
λNζ′′
2
2−1
λ∥Nζ′′∥2
2+M∥ζ′′∥2
2by comple ing squa e ,
≥λζ′+1
λNζ′′
2
2−ν2
λ∥ζ′′∥2
2+M∥ζ′′∥2
2by (∗ν),
≥M
2(ζ′+1
λNζ′′
2
2
+∥ζ′′∥2
2)by (∗λ), hus ν2
λ≤M
2,
=M
2ζ+1
λNζ′′
2
2
as L2
0,(1,q−1)
c,χ ⊥L2
0,(0,q)
c,χ .
Fu he mo e, since
ζ+1
λNζ′′2≥ ∥ζ∥2−1
λ∥Nζ′′∥2
≥ ∥ζ∥2−ν
λ∥ζ′′∥2by (∗ν),
≥(1−ν
λ)∥ζ∥2as ∥ζ′′∥2≤ ∥ζ∥2,
≥3
4∥ζ∥2≥0 by (∗λ),
one has
M
2ζ+1
λNζ′′
2
2≥M
2·(3
4)2
∥ζ∥2
2≥M
4∥ζ∥2
2.
This comple es he p oo . □
CHAPTER 4
The linea izable case
4.1. P oo o Theo em 1.1.1 o linea izable L
The p oo o Theo em 1.1.1 o linea izable Lis gi en he e so ha one can
see clea ly how he p oo wo ks wi hou ha ing o handle addi ional echnicali y
equi ed o he case o non-linea izable line bundles.
Theo em 4.1.1.Suppose Lis linea izable and q < s−
Fo q > m −s+
F. Then,
o any ψ∈H0,q(X;L)such ha ∂ψ = 0, he e exis s ξ∈H0,q−1(X;L)such ha
∂ξ =ψon X. (In case q= 0 < s−
F, his means ψ= 0.) In o he wo ds, by i ue o
Theo em 2.3.1, Hq(X, L) = 0 o any qin he gi en ange.
P oo . Fix any ψ∈H0,q(X;L)∩ke ∂.
An L2-no m ∥·∥X,χ is chosen as ollows. Since Lis linea izable, one can ake
ℏ= 0 (see §2.5 o he de ini ion o ℏ). Then, choose δ= 0 and hus ℏδ=ℏ−δ= 0.
Choose he ansla ional in a ian he mi ian me ic go he o m as desc ibed in
he p oo o Lemma 3.3.2 o q > m −s+
Fo Lemma 3.3.4 o q < s−
F, wi h M= 1.
Fo he he mi ian o m Hassocia ed o L, choose HE:= H|E×Eas desc ibed in
he p oo o Lemma 3.3.6. A he mi ian me ic ηon Lis hen de ined as in §2.5.
Choose a con ex inc easing smoo h unc ion eχ( hus χ:= eχ◦φis plu isubha monic,
i.e. √−1∂∂χ ≥0) such ha ∥ψ∥X,χ <∞. An L2-no m ∥·∥X,χ is hen ixed and
ψ∈L2
0,(0,q)
χ(X;L).
No e ha e e y ζ∈A0,q
0<2>(X;L) is con ained in A0,q
0<2>(Kc;L) o some suffi-
cien ly la ge bu ini e c > 0. Consequen ly, he conclusion o Co olla y 3.3.3 when
q > m−s+
Fo Co olla y 3.3.5 when q < s−
F, as well as ha o Lemma 3.3.6, holds o
all ζ=ζ′+ζ′′ ∈A0,q
0<2>(X;L), whe e ζ′∈A0,(1,q−1)
0(X;L) and ζ′′ ∈A0,(0,q)
0(X;L).
Since ℏδ= 0, W(ζ, ζ) (see (eq 3.14)) and W′
F(ζ′′, ζ′′) (see (eq 3.19)) bo h anish o
all ζ=ζ′+ζ′′ ∈A0,q
0<2>(X;L).
Since χis plu isubha monic on Xand ∂[ ]χ= 0 = ∂[ ]χ, one can choose a e e y
poin z∈X he coo dina es such ha bo h gand √−1∂[u]∂[u]χa e simul aneously
diagonalized while keeping he decomposi ion (eq 2.4) o hogonal, and see ha
T g,η p F((∂∂χ)∨(ζ⊗ζ))= T g,η (∂[u]∂[u]χ)∨(ζ′⊗ζ′)≥0.
The e o e, w (ζ, ζ)≥0 (see (eq 3.14)).
As a esul , combining Lemma 3.3.6 as well as he abo e ac s abou W,W′
F
and w wi h Co olla y 3.3.3 o Co olla y 3.3.5, one ob ains
∥Sqζ∥2
3+T∗
q−1ζ2
1≥π
4∥ζ∥2
2
o all ζ∈A0,q
0<2>(X;L). This is he equi ed L2es ima e. P oposi ion 3.1.5 and
Rema k 3.1.6 hen asse ha he e exis s ξ∈H0,q−1(X;L) such ha ∂ξ =ψon
X.□
27
CHAPTER 5
The non-linea izable case
Fo a non-linea izable line bundle L, he wild cu a u e e ms W(see (eq 3.14))
and W′
F(see (eq 3.19)) a e no iden ically ze o. In o de o ge he es ima es o
hese e ms, Takayama’s Weak ∂∂-Lemma ( e . [Taka2, Lemma 3.14]) is in oked.
One is hen o ced o es ic a en ion o each o he Kc’s and ob ain he equi ed
L2es ima es he e. Wha hen emains is o show ha he exis ence o a solu ion o
he ∂-equa ion ∂ξ =ψon e e y Kcimplies he exis ence o a global solu ion. The
a gumen o his la e pa is essen ially he same as he one in [GR, Ch. IV, §1,
Thm. 7].
An ap coo dina e sys em is ixed h oughou his sec ion.
5.1. Bounds on he wild cu a u e e ms
Takayama p o es in [Taka2] he ollowing Weak ∂∂-Lemma.
Weak ∂∂-Lemma 5.1.1 (c . [Taka2, Lemma 3.14]).Le ωbe a posi i e eal
(1,1)- o m on X, and le θbe a smoo h eal 1- o m on Xsuch ha θ=β+β o
some smoo h (0,1)- o m β, and dθ is o ype (1,1). Then o e e y posi i e numbe
εand e e y ela i ely compac open subse Wo X, he e exis s a smoo h unc ion
δon Xsuch ha
−εω < dθ −2√−1∂∂ Re δ < εω on W .
Mo eo e , i β∈H0,1(X), hen δcan be chosen such ha δ∈H(X).
In he cu en si ua ion, he ole o βin Lemma 5.1.1 is aken by √−1∂ℏ
( he e o e dθ = 2√−1∂∂ Re ℏ), and ha o Wby Kc.
Rema k 5.1.2.In Takayama’s o mula ion, he asse ion o he Weak ∂∂-Lemma
is ha he e exis s a smoo h eal alued unc ion εW := 2(Im 0+ Im ΨM0) on X
such ha −εω < dθ −√−1∂∂ εW < εω on W, in which 0is a smoo h unc ion
on Xsuch ha β=ϕ+∂ 0 o some eal analy ic (0,1)- o m ϕin H0,1(X), and
ΨM0is some eal analy ic unc ion in H(X). The e o e, he smoo h unc ion δhe e
is gi en by δ:= −√−1( 0+ ΨM0) in Takayama’s no a ion. I β∈H0,1(X), hen
one has 0∈H(X) as ∂[u] 0= 0, so δ∈H(X) also.
Rema k 5.1.3.As a side ema k, ollowing he cons uc ion o δin [Taka2,
Lemma 3.14], ∂ℏδ=∂ℏ−∂δ is eal analy ic on X, so ℏδis eal analy ic on Cn. I
ollows ha he he mi ian me ic ηon Lis eal analy ic.
Sui able es ima es o he wild cu a u e e ms Wand W′
Fa e ob ained by
choosing a p ope δ∈H(X) acco ding o he Weak ∂∂-Lemma.
Lemma 5.1.4.Suppose a he mi ian me ic gon Xand a choice o HEa e ixed.
Then, on e e y Kcwhe e 0< c < ∞, gi en any eal numbe εw>0and o any
28
5.2. EXISTENCE OF WEAK SOLUTIONS ON Kc29
q≥0, one can choose δc∈H(X)which yields a he mi ian me ic ηcon Lsuch
ha , o any gi en weigh χ,
|W(ζ, ζ)| ≤ εwq∥ζ∥2
Kc,ηc,χ
(eq 5.1)
|W′
F(ζ′′, ζ′′)| ≤ εwm∥ζ′′∥2
Kc,ηc,χ ≤εwm∥ζ∥2
Kc,ηc,χ
(eq 5.2)
o all ζ=ζ′+ζ′′ ∈A0,q
<2>(Kc;L)whe e ζ′∈A0,(1,q−1)(Kc;L)and ζ′′ ∈A0,(0,q)(Kc;L).
P oo . Fi s he es ima e o Wis conside ed. Recall ha ωis he (1,1)- o m
associa ed o g. The Weak ∂∂-Lemma asse s ha , o any εw>0, he e exis s
δc∈H(X) such ha
(eq 5.3) −2εwω < 2√−1∂∂ Re ℏδc<2εwωon Kc.
Such δcyields a he mi ian me ic ηcon Lgi en he ixed choice o HE. Then, i
ollows om (eq 3.14) ha , o any weigh χ,
−εw∫Kc
e−χT g,ηcp F(g∨(ζ⊗ζ))≤W(ζ, ζ)≤εw∫Kc
e−χT g,ηcp F(g∨(ζ⊗ζ))
o any ζ=ζ′+ζ′′ ∈A0,q
<2>(Kc;L) (εwins ead o 2εwin he bounds because o he
ac o 1
2in ω=−Im g=√−1
2∑k,ℓ gkℓdzk∧dzℓ). No e ha
∫Kc
e−χT g,ηcp F(g∨(ζ⊗ζ))=∥ζ′∥2
Kc,ηc,χ +q∥ζ′′∥2
Kc,ηc,χ ≤q∥ζ∥2
Kc,ηc,χ
when q≥1. When q= 0, he in eg al on he le hand side is ze o, so he abo e
inequali y is s ill alid. As a esul , one ob ains
−εwq∥ζ∥2
Kc,ηc,χ ≤W(ζ, ζ)≤εwq∥ζ∥2
Kc,ηc,χ
and hence (eq 5.1).
Fo he es ima e o W′
F, no e ha (eq 5.3) implies
−2εwp Fω < 2√−1∂[ ]∂[ ]Re ℏδc<2εwp Fωon Kc.
Then, one has −εwm < 2 T g∂[ ]∂[ ]Re ℏδc< εwmwi h he same εwand δcas abo e.
The e o e, i ollows om (eq 3.19) ha
−εwm∥ζ′′∥2
Kc,ηc,χ ≤W′
F(ζ′′, ζ′′)≤εwm∥ζ′′∥2
Kc,ηc,χ
o any ζ′′ ∈A0,(0,q)(Kc;L), and hence (eq 5.2). □
5.2. Exis ence o weak solu ions on Kc
Wi h he bounds gi en in §5.1 o he wild cu a u e e ms, i is easy o ollow
he p oo o Theo em 4.1.1 and ge he ollowing
P oposi ion 5.2.1.Suppose Lis a holomo phic line bundle on X(which can
possibly be non-linea izable), and suppose q < s−
Fo q > m−s+
F. Then, he e exis s a
sui able he mi ian me ic gon Xsuch ha he ollowing holds: o any 0< c < ∞,
a he mi ian me ic ηcon Lcan be chosen such ha , gi en any plu isubha monic
weigh χ, he L2es ima e
∥Sqζ∥2
Kc,ηc,χ +T∗
q−1ζ2
Kc,ηc,χ ≥π
4∥ζ∥2
Kc,ηc,χ
o all ζ∈A0,q
<2>(Kc;L)∩DomKc,ηc,χ T∗
q−1is sa is ied.
30 5. THE NON-LINEARIZABLE CASE
P oo . Choose he ansla ional in a ian he mi ian me ic gas desc ibed in
he p oo o Lemma 3.3.2 o q > m −s+
Fo Lemma 3.3.4 o q < s−
F, wi h M= 2.
Fo he he mi ian o m Hassocia ed o L, choose HEas desc ibed in he p oo o
Lemma 3.3.6. These choices a e independen o c.
Conside Kc o some ixed 0 < c < ∞. Take any εw>0 such ha
(∗)εw(q+m)≤π
4
and choose δc∈H(X) acco ding o Lemma 5.1.4 such ha , o any gi en weigh
χ, he inequali ies (eq 5.1) and (eq 5.2) hold unde he induced L2-no m ∥·∥Kc,ηc,χ.
By he choices o he me ics, he conclusion o Co olla y 3.3.3 when q > m −s+
F
o Co olla y 3.3.5 when q < s−
F, as well as ha o Lemma 3.3.6, holds o all ζ=
ζ′+ζ′′ ∈A0,q
<2>(Kc;L)∩DomKc,ηc,χ T∗
q−1, whe e ζ′∈A0,(1,q−1)(Kc;L)∩Dom(1,q−1)
Kc,ηc,χ ∂∗
[u]
and ζ′′ ∈A0,(0,q)(Kc;L).
Since χis plu isubha monic, w (ζ, ζ)≥0 o all ζ∈A0,q
<2>(Kc;L) as in he p oo
o Theo em 4.1.1.
As a esul , om Co olla y 3.3.3 o 3.3.5 as well as Lemma 3.3.6, one ob ains
∥Sqζ∥2
Kc,ηc,χ +T∗
q−1ζ2
Kc,ηc,χ
≥{π
2∥ζ∥2
Kc,ηc,χ +W(ζ, ζ) o q > m −s+
F
π
2∥ζ∥2
Kc,ηc,χ +W′
F(ζ′′, ζ′′) + W(ζ, ζ) o q < s−
F
≥π
2∥ζ∥2
Kc,ηc,χ −εw(m+q)∥ζ∥Kc,ηc,χ
by (eq 5.1) and (eq 5.2),
and εwq < εw(m+q)
≥π
4∥ζ∥2
Kc,ηc,χ by (∗).
This gi es he equi ed L2es ima e. □
Since, o any ψ∈H0,q(X;L), one has ψ|Kc∈L2
0,(0,q)(Kc;L) (unweigh ed) o
any 0 < c < ∞, i ollows he ollowing co olla y o P oposi ions 3.1.5 and 5.2.1.
Co olla y 5.2.2.Conside he exhaus i e sequence {Kν}ν∈N>0o ela i ely
compac open subse s o X. Suppose q < s−
Fo q > m −s+
F. Then one can choose
a sui able he mi ian me ic gon Xand a sequence o he mi ian me ics {ην}ν∈N>0
on Las in P oposi ion 5.2.1 such ha , o any ψ∈H0,q(X;L)∩ke ∂, he e exis s
a sequence o solu ions {ξ′
ν}ν∈N>0such ha ξ′
ν∈L2
0,(0,q−1)
ην(Kν;L)(unweigh ed) and
∂ξ′
ν=ψ|Kνin L2
0,(0,q)
ην(Kν;L).
Rema k 5.2.3.Since χhas o be smoo h on a neighbo hood o Kc(as equi ed
by [H¨o 1, P op. 2.1.1] so ha A0,q
<2>(Kc;L)∩Dom T∗
q−1is dense in Dom T∗
q−1∩Dom Sq
unde he sui able g aph no m), i ψ∈H0,q(Kc;L), he e may no exis such χsuch
ha ∥ψ∥Kc,χ <∞. To a oid echnical difficul y, he au ho does no a emp o
sol e he ∂-equa ion o any ψ∈H0,q(Kc;L) such ha ∂ψ = 0 by means o L2
es ima es di ec ly.
5.3. A Runge- ype app oxima ion
This sec ion is de o ed o p o ing a Runge- ype app oxima ion which is equi ed
o cons uc a global solu ion o he equa ion ∂ξ =ψ om he solu ions on Kν’s
gi en in Co olla y 5.2.2.
5.3. A RUNGE-TYPE APPROXIMATION 31
In wha ollows, qis assumed o be 0 < q < s−
Fo q > m−s+
F, and he he mi ian
me ic gas well as he amily o he mi ian me ics {ηc}c>0as asse ed by P oposi ion
5.2.1 is ixed. Then, acco ding o he choices o he ηc’s in he p oo o P oposi ion
5.2.1, o any c′, c > 0, one has
ηc=ηc′e2 Re(δc′−δc)=: ηc′eδc′c.
No e ha eδc′c>0 on X. I is unde s ood ha he he mi ian me ic ηcon Lis cho-
sen when he L2-no m on Kcis conside ed, so w i e L2
0,(0,q)
ηc,χ (Kc;L) as L2
0,(0,q)
χ(Kc;L),
⟨·,·⟩Kc,ηc,χ as ⟨·,·⟩Kc,χ and so on o simpli y no a ion. When he weigh χis absen
om he no a ion, e.g. L2
0,(0,q)(Kc;L) o ⟨·,·⟩Kc, i is unde s ood ha he co e-
sponding objec is unweigh ed, i.e. χ= 0.
Fo any ini e c′> c > 0 and o any Ψ ∈L2
0,(0,q−1)(Kc;L), i Ψ is ex ended by
ze o o a sec ion in L2
0,(0,q−1)(Kc′;L), hen i ollows ha
(eq 5.4) ⟨ζ, Ψ⟩Kc=⟨ζ, Ψeδc′c⟩Kc′
o any ζ∈L2
0,(0,q−1)(Kc′;L).
De ine (ke Kc′Tq−1)Kc o be he image o ke Kc′Tq−1unde he es ic ion map
L2
0,(0,q−1)(Kc′;L)→L2
0,(0,q−1)(Kc;L). No e ha Tq−1commu es wi h he es ic ion
map (as c > 0), so one has(ke Kc′Tq−1)Kc⊂ke KcTq−1.
The ollowing p oo o he equi ed Runge- ype app oxima ion is an analogue o
he one o s ongly pseudocon ex mani olds gi en in [H¨o 3, Lemma 4.3.1].
P oposi ion 5.3.1.Suppose 0< q < s−
Fo q > m −s+
F, and gand ηc’s a e
chosen acco ding o P oposi ion 5.2.1. Then, o any ini e c′> c > 0, he closu e o
(ke Kc′Tq−1)Kcin L2
0,(0,q−1)(Kc;L)is ke KcTq−1. In o he wo ds, (ke Kc′Tq−1)Kc
is dense in ke KcTq−1.
P oo . By i ue o he Hahn-Banach heo em, i suffices o show ha o e e y
Ψ∈L2
0,(0,q−1)(Kc;L), i he induced bounded linea unc ional
L2
0,(0,q−1)(Kc;L)∋ζ7→ ⟨ζ, Ψ⟩Kc
anishes on (ke Kc′Tq−1)Kc, hen i also anishes on ke KcTq−1.1
Suppose ha Ψ ∈L2
0,(0,q−1)(Kc;L) sa is ies he abo e assump ion. Ex end Ψ by
ze o o Kc′as a sec ion in L2
0,(0,q−1)(Kc′;L). Now i suffices o show ha he e exis s
Ξ∈L2
0,q
<2>(Kc′;L) such ha Ξ ≡0 on Kc′ Kcand
(†)⟨ζ, Ψeδc′c⟩Kc′=⟨Tq−1ζ, Ξ⟩Kc′
o any ζ∈DomKc′Tq−1, which hen implies ha
(‡)⟨ζ, Ψ⟩Kc=⟨Tq−1ζ, Ξe−δc′c⟩Kc
o any ζ∈DomKc′Tq−1due o (eq 5.4). The equali y (‡) holds ue o ζ∈
A0,(0,q−1)
0(Kc′;L) in pa icula , and A0,(0,q−1)(Kc;L) is dense in DomKcTq−1un-
de he g aph no m √∥ζ∥2
Kc+∥Tq−1ζ∥2
Kcby [H¨o 1, P op. 2.1.1], so (‡) also holds
1I he e exis s ζ∈ke KcTq−1which does no lie in he closu e o (ke Kc′Tq−1)Kc
in
L2
0,(0,q−1)(Kc;L), hen he Hahn-Banach heo em asse s ha he e is a bounded linea unc ional
Λ such ha (ke Kc′Tq−1)Kc⊂ke Λ and Λζ= 1.
32 5. THE NON-LINEARIZABLE CASE
ue o ζ∈DomKcTq−1. I ollows ha
⟨ζ, Ψ⟩Kc=⟨Tq−1ζ, Ξe−δc′c⟩Kc= 0
o all ζ∈ke KcTq−1⊂DomKcTq−1as equi ed. I emains o show he exis ence
o such Ξ.
Take a sequence o smoo h con ex inc easing unc ions eχν:R→Rsuch ha
eχν(x) = 0 o all x≤c, and eχν(x)↗+∞as ν→ ∞ o e e y x>c. No e ha
eχν≥0 o any ν≥0 by such choice. Se χν:= eχν◦φas be o e. A sequence o
weigh ed no ms ∥·∥c′,ν := ∥·∥Kc′,χνon Kc′is hen de ined. Le he co esponding
inne p oduc s, Hilbe spaces and Dom also be dis inguished by using he subsc ip s
c′, ν, and he co esponding adjoin o Tq−1by T∗,ν
q−1.
Fo any qin he gi en ange, he L2es ima e in P oposi ion 5.2.1 holds unde
each o he abo e weigh ed no ms wi h T∗
q−1 eplaced by T∗,ν
q−1. Since ⟨ζ, Ψeδc′ceχν⟩c′,ν =
⟨ζ, Ψeδc′c⟩Kc′and he igh hand side anishes o all ζ∈ke Kc′Tq−1= ke c′,ν Tq−1
by he assump ion on Ψ, i ollows ha
Ψeδc′ceχν∈(ke c′,ν Tq−1)⊥= imc′,ν T∗,ν
q−1.
Gi en he L2es ima e, Theo em 3.1.1 (2) hen asse s ha he e exis s e
Ξν∈
Domc′,ν T∗,ν
q−1such ha T∗,ν
q−1e
Ξν= Ψeδc′ceχν. The e o e, one has
⟨ζ, Ψeδc′ceχν⟩c′,ν =⟨ζ, T∗,ν
q−1e
Ξν⟩c′,ν
=⟨Tq−1ζ, e
Ξν⟩c′,ν =⟨Tq−1ζ, e
Ξνe−χν⟩Kc′
o all ν∈Nand o all ζ∈Domc′,ν Tq−1= DomKc′Tq−1. By de ining Ξν:= e
Ξνe−χν,
one ob ains
(∗)⟨ζ, Ψeδc′c⟩Kc′=⟨Tq−1ζ, Ξν⟩Kc′.
Mo eo e , no ice ha he cons an in he L2es ima e is independen o ν(which is
chosen o be π
4in P oposi ion 5.2.1). The es ima e on he solu ion e
Ξν om Theo em
3.1.1 (2) hen implies ha
(∗∗)π
4∫Kc′|Ξν|2
g,ηc′eχν≤∫Kc′Ψeδc′c2
g,ηc′eχν=∫Kc|Ψ|2
g,ηceδc′ceχν,
whe e he las equali y is due o he ac ha Ψ anishes on Kc′ Kc. Since eχν(φ)
is independen o νwhen φ≤c, he in eg al on he igh hand side is independen
o ν, so he le hand side is a bounded sequence in ν. This in u n implies ha
he e exis s a subsequence o {Ξν}ν∈Nwhich con e ges o some Ξ ∈L2
0,q
<2>(Kc′;L)
(unweigh ed) in he weak opology. F om (∗∗), since eχν(φ)↗+∞ o φ > c, i
ollows ha Ξ ≡0 when φ > c, i.e. on Kc′ Kc. Mo eo e , om (∗) i ollows ha
(†) holds o all ζ∈DomKc′Tq−1. This is wha is desi ed. □
5.4. P oo o Theo em 1.1.1 o gene al L
Fi s no ice ha , i q= 0 < s−
F, hen he L2es ima e in P oposi ion 5.2.1 holds
when he me ics a e chosen sui ably, and hus o any ψ∈H(X;L)∩ke ∂one
has
0 = ∂ψ2
Kc≥π
4∥ψ∥2
Kc
5.4. PROOF OF THEOREM 1.1.1 FOR GENERAL L33
(no e ha T∗
−1ζ= 0 o all ζ∈A(Kc;L)). This means ha ψ|Kc= 0 o any c > 0,
and hus ψ= 0 on X. The e o e, one has he ollowing
Theo em 5.4.1.I s−
F>0, one has H0(X, L) = 0.
Assume 0 < q < s−
Fo q > m −s+
Fin wha ollows. The me ics gand
ην’s om Co olla y 5.2.2 a e ixed o his sec ion. Again, w i e L2
0,(0,q)
ην,χ (Kν;L) as
L2
0,(0,q)
χ(Kν;L) and so on, and no a ions like L2
0,(0,q)(Kc;L) o ∥·∥Kca e unde s ood
as unweigh ed objec s, i.e. χ= 0.
Fo e e y in ege ν≥1, as δν+1 −δνis smoo h on Xand Kν+1 is compac , he e
exis s a cons an M′
ν+1 ≥1 such ha
(eq 5.5) ∥ζ∥Kν≤M′
ν+1 ∥ζ∥Kν+1
o all ζ∈L2
0,(0,q)(Kν+1;L). De ine also M1:= 1 and Mν:= ∏ν
k=2 M′
k o ν≥2.
P oposi ion 5.3.1 is used o comple e he p oo o Theo em 1.1.1. The ollowing
a gumen is adop ed om [GR, Ch. IV, §1, Thm. 7].
Theo em 5.4.2.Suppose 0< q < s−
Fo q > m−s+
F. Then one has Hq(X, L) = 0
o any qin he gi en ange.
P oo . Gi en any ψ∈H0,q(X;L)∩ke ∂, Co olla y 5.2.2 p o ides a sequence o
local solu ions {ξ′
ν}ν≥1such ha ξ′
ν∈L2
0,(0,q−1)(Kν;L) and ∂ξ′
ν=ψ|Kν o all in ege s
ν≥1. Fi s a sequence o local solu ions {ξν}ν≥1such ha ξν∈L2
0,(0,q−1)(Kν;L),
∂ξν=ψ|Kνand
(∗)∥ξν+1 −ξν∥Kν<1
Mν2ν
o all ν≥1 is de ined induc i ely as ollows. Se ξ1:= ξ′
1. Suppose ξ1, . . . , ξν
a e de ined o some ν≥1. Le γ′
ν:= ξ′
ν+1|Kν−ξν. No ice ha γ′
ν∈ke KνTq−1⊂
L2
0,(0,q−1)(Kν;L). P oposi ion 5.3.1 hen implies ha he e exis s γν∈ke Kν+1 Tq−1⊂
L2
0,(0,q−1)(Kν+1;L) such ha
∥γ′
ν−γν∥Kν<1
Mν2ν.
Se ξν+1 := ξ′
ν+1 −γν. Then one has ∂ξν+1 =∂ξ′
ν+1 =ψ|Kν+1 and he inequali y (∗)
is sa is ied. The equi ed sequence {ξν}ν≥1is he e o e de ined.
No ice ha , o e e y ν≥1, he sequence {ξµ|Kν}µ≥νcon e ges in L2
0,(0,q−1)(Kν;L).
Indeed, o any µ≥ν≥1 and o any in ege k > 0,
∥ξµ+k−ξµ∥Kν≤
k−1
∑
=0 ∥ξµ+ +1 −ξµ+ ∥Kν
≤
k−1
∑
=0
Mµ+
Mν∥ξµ+ +1 −ξµ+ ∥Kµ+ by (eq 5.5) ,
≤1
Mν
k−1
∑
=0
1
2µ+ by (∗),
≤1
Mν2µ−1,
34 5. THE NON-LINEARIZABLE CASE
which ends o 0 as µ→ ∞, so {ξµ|Kν}µ≥νis a Cauchy sequence in L2
0,(0,q−1)(Kν;L).
Le ξ(ν)be he limi o {ξµ|Kν}µ≥νin L2
0,(0,q−1)(Kν;L). Since ∂ξµ|Kν=ψ|Kν o all
µ≥ν, and ∂is a closed ope a o , one has ∂ξ(ν)=ψ|Kν o all ν≥1. Now no ice
ha es ic ion om Kν+1 o Kνis con inuous by (eq 5.5), so
ξ(ν+1)|Kν−ξ(ν)= lim
µ≥ν+1
µ→∞
(ξµ|Kν−ξµ|Kν) = 0
in L2
0,(0,q−1)(Kν;L). On e e y Kν, diffe en choices o δν∈H(X) yield equi alen
no ms. The e o e, by ixing one δ∈H(X), one can conside L2
0,q−1(X;L; loc),
he space o locally L2L- alued (0, q −1)- o ms on X, and he e exis s ξ′∈
L2
0,q−1(X;L; loc) such ha
ξ′|Kν=ξ(ν) o all ν≥1,and
∂ξ′=ψin L2
0,q−1(X;L; loc) .
Rema k 3.1.6 hen assu es ha he e exis s ξ∈H0,q−1(X;L) such ha ∂ξ =ψon
X.
Since ψ∈H0,q(X;L)∩ke ∂is a bi a y, his shows ha Hq(X, L) = 0. This
comple es he p oo . □