Ngo Thi Kim Quy Tap chi KHOA HOC & CONG NGHE 103(03). 133- 139
D A N G T O N G Q U A T C U A B I N H L Y T H A C T R I E N H A R T O G S D O I V d l C A C A N H X A C H I N H H I N H T A C H B I E N
Trudng Dgi hqc Kmh le i
Ngo Thj Kim Quy a Quan In kinh doanh - DH Thdi Nguyen
TOM T A T
Muc dich chinh ciia bai bao la dua ra mgt dang tong quat cua djnh ly thac trien Hartogs not tieng vdi cac ham chinh hinh tach. Su dyng cac kgt qua phai trien gdn day cua ly thuygt Polelsky tren cac dTa, bai bao chung minh ket qua sau- Gia sir X. Y la 2 da lap phuc, Z la khong gian giai lich phirc c6 tinh chal thac trien Hartogs. Gia sir A (tuong ung B) la tap con khong da cue dja phuong cua X (luang ung Y). Khi do mgi anh xa chinh hinh tach / : W " ( / J X } ' ) U ( A ' X B ) - > Z dgu thac uign tdi anh \ a chinh hinh / tren W .^\{z,w)& X KY .Q}[z,A,X) + a{w,B,y)<\\ sao cho / = / tren »•'f)H-', trong dd d)(...\.X\ (tuong ung d)(w,B,y) la do do da dieu hoa dudi cua A (tuong img B) tuong doi vdi X (tuong img Y).
Sy tong quai hoa cua ket qua nay doi vdi chii ihap N la cung duoc dua ra.
Tir kh6a: Da ctrc dia phmrng, do do da dieu hoa dudi. chir thdp N Id, chinh hinh lach, tinh chdl thac trien Hartogs
KIEN T H U C C H U A N BI
H a m d a dieu h o a d i r d i , t a p d a c u e , d a chinh q u y d i a p h i r o n g
Ham dieu hod dur&i
Gia su D la mpt tap con m d t r o n g IR" Ham
« ; / ) - > [ - 0 0 , - 1 - 0 0 ) , u ^ - ^ trgn mpi thanh phan hen thdng ciia D d u g c ggi la dieu hod dudi trong D ngu U t h o a man hai digu kien sau:
i) U la nira lign tuc trgn trong D, tire la l i m s u p w ( z ) < u ( z o ) VCTi V z ^ e D -
ii) Vdi mdi tap con m d c o m p a c t t u o n g ddi G cua D, vdi mdi h a m _ / i : G - > K digu hoa trong G va lign tyc tren G : neu u<h trgn dC thi u<h tren G -
Hdm da dieu hod du&i
Gia su Q. la mot tap con m d t r o n g C" H a m
^ : Q _ > f _ o o , - K o ) d u g c ggi la da diiu hod t/ifij/trgng Q. ngu:
i) (p la nira lign tyc trgn trgng Q va ^ ?t -00 yen mgi thanh phan I ien thdng ciia O. • ii) Vdi mdi digm z^&Q. va moi d u d n g thing phirc l(^) = ^o + ^ 4 ^^ 1 " ^ ^0 ^^,^°
£ u e C " , ^ e C ) . han chg (p tren d u d n g t h i n g
Tel. 0917 333725, Email kimquyktln@gmad com
nay, tire la ham g}ollA hoac la digu hoa dudi hoac ddng nhat bang -00 tren mgi thanh phan lien thdng ciia tap m d [^ e C : / ( ^ ) e O} • Hdm da dieu hod dudi tren khong gian phur Gia su X la k h d n g gian phirc Mot ham da dieu hoa d u d i trgn X la ham
^ : X - > [ - c o , + o o ) t h o a man: Vdi moi xeX tdn tai lan can U ciia X va mdt anh xa song chinh hinh h:U ->V, vai V \a mgt khdng gian con phirc d d n g ciia mdt mien G nao dd trong C " va tdn tai mdt ham da dieu hoa d u d i ^ : G - > [ - » , + 0 0 ) sao c h o pp =^oh.
Tap da cue
T a gia thigt tdt ca cac da tap p h u c la huu han chigu dja p h u o n g ( t u c la chigu ciia mdi thanh phan lign t h d n g ciia da tap la hiru han) va tat ca cac k h d n g gian giai tich phirc xet trong luan van dgu gia thiet la d u g c thu ggn, bat kha quy va hiru han chieu.
G i a su A4 la da tap phirc va A la tap con ciia A 4 . D a t
h^j^:=sup{u:u&VSH{M),ii<\ tren J^, u < 0 trgn A}
trong d o •pSH(M) ' ^ ki hieu ndn cua tat ca cac ham da dieu hoa dudi trgn M
+) T a p A d u o c ggi la da cue trong M ngu cd ueVSHiM) sao c h o u k h d n g ddng nhal
133
Ngo Thi Kim Quy Tap chf KHOA HQC & CONG NGHE 103(03): 133- 139 bang —GO irgn mgi thanh phan licii thdng ciia
-VI va ^ c { r c . V 1 : t / ( z ) = -OTj. ^ +) Tap A dugc ggi la da cue itja phiarng irong A4 neu vdi mdi Z G A, cd mgt lan can md V ciia z sao cho AOV la da eye trong V.
+) Tap .1 dugc gpi la khdng da aa' (tuong img khdng da cue dia phmrng) neu nd khdng da eye (lirang img khdng da eye dja phuong).
Theo nipt ket qua cd dien ciia Joscfson va Bedford (xem [4], [8]), neu A4 la mien Riemann tren mpt da tap Stein thi A cz A4 ik da eye dja phuang neu va cbi ngu nd da eye.
Tap da chinh quy dia phmrng
+') Cho ham h:M^U, ham h' M -> ^ dugc xac djnh bdi:
/ / ( r ) = limsup/;((c), zeM duac ggi la hdm chinh quy hod nua lien tiJC Iren ciia h.
-(-) Tap hap ,4 c M l3 i^<^ chinh quy dia phuong tgt mdi diem as A neu H ^ (a) - 0 vdl mgi lan can md 6'ciia a
+) Tap A duoc goi ta da chinh quy dia phuang neu nd da chinh quy dja phuang tai mgi diem a G A.
Ta ki higu A' = A' la tap hop tat ca cac diem a^'A ma tai dd A la da chinh quy dja phuang.
Neu A khdng da eye dja phuong thi mdt ket qua cd dien ciia Bedford va Taylor (xem [4], [5]) chi ra A khdng da cue dja phuang va A A' la da eye dja phuang. Hon nua, A la dia phuong kigu g,^ (tuc la vdi mdi ae A' ,QC\
mdt lan can md L'ciia a thoa man A'C\U 'a giao dgm dugc ciia cac tap md) va /I* la da chinh dia phuong (tuc la (/!'] = .4').
Tinh chat thac tricn Hartogs
Dinh nghia 1.2.1. Cho sd nguygn p>2. Vdi 0 < r < I, lap hgp
H^X'-)--l(^\z^,)^E'':\\z'\\<r hoac
K.|>1-/-}
dugc goi la luae dd Hartogs p chtiu.
Trong do E la dTa don vi trong C va
z' = [z,,..z^_),\\z\=maxjz\.
Dinh nghia 1.2.2. Khdng gian giai tich phuc Z dugc ggi la cd tinh chat thdc irlen Hartogs
vaip chtiu ngu mgi anh xa / e C ' ( / / , ( r ) , Z ) deu thac trien tdi anh xa fsO(E'',Z)- Hon nila, Z dugc gpi la cd tinh chat thac trign Hartogs neu nd cd tinh chat thac trien Hartogs vdi mpi chieu p>2.
Hg do da dieu hoa duoi va chinh hinh tach Djnh nghta 1.3.1. Dp do da dieu hod dudi ciia .4 luang ddi vdi M la ham dugc xac djnh bdi:
d){z,A,M):=h]. Jz), zeM-
Chii y rang d)UA,M)eVSH{M) va
0<Q9(z,A,M)<\, zeM.
Djnh nghia 1.3.2.
Gia su A' e^ I J. A' > 2 va 0^ A (Z D , trong dd D ladalapphuc, y = |,...,A'. Ta djnh nghTa chir thdp N la:
X:=X{A,. .A.,.D,, . .DJ
^\jA^x...xA^_,xD^xA^,,K.. xA^
Theo Alehyane - Zenachi [3], ta djnh nghTa phdn chinh quy X' ciia A'nhu sau
X' =X{A„...,A^.,;D„...,D^)
= X(A;,..,A[.D„...,D^}
-^\JA;X . X 4'_^xD x.4\,x...xA[- Hon nua, dat:
a>(:).= t<'>i= •-•'•DI .- = (.-,. .zJ^D,x...xD,
Vdi chir thap A'ta A':^X(.-(.. ..-l,;£j Dj-^at .V = X(.4,. ..f,:©,, . , D j
= {(r, . , r , ) e D , x . . . x D , ' 6 J ( z ) < l } Khidd. tacd X' czX.
Dinh nghia 1.3.3. Gia sir Z la khdng gian giai tich phuc.
Ta ndi rang anh xa f.X^Z 'a chinh hinh /(if/; va viel r^oix z)'^^^ '*'°'' "^oi j e{\, .,M]
''^a\a")€{A,x...xA,^)x(A^^,x...xA.,) ^^^ "^
thu hep f{a\.,a")\ , la chinh hinh tren Dj '' Cho da tap phiic Al \a khdng gian giai tich phirc Z, ki hieu 0{M,Z) la tap tat ca cac anh xa chinh hinh tir M vao Z.
Ngo Thj Kim Quy Tap chi KHOA HOC & CONG NGHE 103(03)' 133- 139
DINH LY THAC TRIEN HARTOGS DOI VCil CAC ANH XA CHINH HINH TACH BIEN Md dau
Nam 2001, Alehyane - Zeriahi da dua ra dang tdng quat ciia djnh ly thac trign Hartogs ddi vdi cac ham chinh hinh tach, trong trudng hgp bao chinh hinh ciia tap chii- thap bat ky la tich cac mign con cua cac da tap Stein ciia dg dg da dieu hoa dudi
nhu sau: i Binlily 2.1.1. (Alehyane - Zeriahi [3, dinh ly
2.2.4])^
Gia sir X, la da tap Stein. D czX la mgt mien, A r^D ^^ 'dP con khong da cue, j = 1.
.. N va Z la khong gian gidi tich phirc cd linh chdt thdc triefi Hartdgs. Khi do, vdi mdi dnhxg f &0 [X,Z\ ^^f 'on tai duy nhdt dnh
^dJ^o[x,z] ^aocho f = f tren xfiX- ' "
Vi du sau (xem[3]) chi ra rang gia thigt Z la khong gian giai tich phiiq cd linh chat thac trien Hartogs la can thiel^ Xet anh xa / C^ ^ P ' cho bdi:
|[(i-i-«;)^(—.<-)•] khiU^u-)^ (0,0) 1[1:I], f. khi {z,w) = {0,Q) th' f&0^{X{Z.Z.2.Z).I?') rihung f khdng lien tyc tai (0,0). .
Cau hdi nay sinh mgt pach ty nliign rang dinh ly trgn cdn diing khdng neu o . khdng nhat thiet la mign con ciia da'tap Stein, y = 1, . , N. Dg tra loi cau hdi \t&n bai bao nay dua ra t6ng quat Jioa dinh ly ciia Alehyane - Zeriahi cho tap chu thap la tich cac da tap phirc tuy y Trong ehirng minh kgl qua nay, chii yeu sir dung ly thuyet Poletsky ve cac dTa (xem [ 12], [13]), djnh ly cua Rosay trgn cac dTa chinh hinh (xem [14]). Ngoai ra, ky thuat quan' Irgng khac,la su dyng cae tap miic ciia dg do da dieu hoa dudi.
Cac ket qua chinh
Binh ly'A. Gid su o Id da tap phdc vd A' c D let Idp coii khdng da cue dia phuang, J = I. ..., N; Z la khong gian gidi tich ^
fUM:=
phirc CO linh chdt thdc trien Harlogs Khi do, vai moi dnh xg f&O^X.Z) cd duy nhdt dnh
^d 'fso[x.z) -^"^ ^^^ ,? = ./• ^^t'" -vriA- Han nira. neu Z^C va \f\ <rj thi
|/(r)j<]/j';"|/|','" ^e-V
Dmh ly A cd mpt hg qua quan Irgng. Trudc khi dira ra linh chat nay, ta can gidi thieu mdt thuat ngu. Da tap phuc A4 dugc gpi la da tap Liguville neu VSHiM) khdng chua bat ki ham bj chan tren khac hang nao.Ta thay Idp da tap Liouville chira lop cac da tap compact lign thdng.
He qud B. Gid su jy Id da lap phirc vd A <z D la lap con khong da cue dia phmrng.
j -• /, . A^,- Z la khdng gian giai tich phirc cd linh chdl thdc trien Hartogs Gia su them rdng D Id eta lgp Liouville. f = 2, , N thi vdi moi dnh xa f^OiX.Z) co duy nhdt dnh xa f'eO(D,x. .xD^.Z) saocho f^f tren X.
He qua B : suy ra tryc tiep tir dinh ly A vi a}[.,A^,D)^QJ = 2,...,N •
Hudng ehirng minh djnh ly A nhu sau Budc mdt, ta Chirng minh cac trudng hgp dac biet ma mdi A 'a mdt tap md y = |....V Budc hai, ta criLrng minh djnh ly A trong trudng hgp tdng quat.
Trgng budc mgt, dg chiing minh djnh ly A ta ap dung ly,thuyet Poletsky vdi cac dTa va dinh ly ciia Rosay trgn cac dTa chinh hinh. V^i vay, ta cd thg xay dung anh xa thac trien / trgn X, Dg chii'ng minh / la chinh hinh, ta diing dinh ly chu' thap cd dien (djnh ly 2 1.1).
Trong budc hai ta quy trudng hop tdng quat ve trudng hgp dac biet tren. KT thuat quan trgng la sir dyng cac lap mirc ciia do do da dieu hoa dudi. Chinh xac ban, ta thay mdi D, bdi cac tap rniic ciia do do da dieu hoa dudi Q)[.,A^,D) tii'clabdi
D,,:=\z^^ D^- d)[z ,.A .D]<\- S]
' ( 0 < ^ < l ) .
Vdi phuang phap nhu vay, ta thay the tap hgp A bdi tap hgp A ,. sao cho trong mdt so
Ngo Thj Kim Quy Tap chf KHOA HQC & CONG NGHg 103(03): 133- 139
trudng hgp cgi ,:,(,, j ) nhu dj[.,A^,D^) k h i ( ^ - > 0 * . .
Ap dung djnh ly 2.1.1 va djnh ly 1.4.1, la cd the md rpng chinh hinh tach ciia f tdi Anh xa f^ xac dinh Ircn tap chii' thap
.\\,:=X(A,,....,A,.y,D,^ l\ _,).
Ap dyng ket qua ciia budc mgt ta thu dugc anh xa "/ g o [ ^ z] '-^^'''^^ f ? ^ ] la Ihu dugc anh xa thac trign /
Phan 1 cua chiirni^ minh djnh ly A Myc dich ciia phan nay la chiing minh djnh ly A trong trudng hgp dac bipt sau:
Dinh ly 2.3.1. Cho D la da lap phirc, G la da lgp phdc md song chinh hinh tai lap md trong C(q e N) Gid su-A la tap con md cua D, B Id tgp con khdng da circ dia phuang ciia G vd Z la khdng gian gidi tich phirc co tinh chdl thdcJriiii Hartogs. Dat X •.^X{^A,B;D,G) vd X •.~X(A,B:D,G)- Khi do, vai mdi dnh xa f^OlX.Z) co^ duy nhdt dnh xg
7 e O ( x , Z ) i"oc/jo / = / tren XQX' Chu)r2.3.2'.Va\ gia thiel trgn cd:
X{}X' ={AxG){}[Dx(B[^B')]
Trong ehirng minh dudi day, ta gia sir rang G la mien trong C. Vdi gia sii nay ta chiing minh dugc djnh ly.
Chung mmh
Ta bat dau chirng minh vdi bd dg dudi day.
Bd de 2.3.3. Vdn gid thiet nhu dinh ly 2.3.1.
Vai y e {1,2}, gid su ^^^OCE.D) let mot dia chinh hinh va dat I e E sao cho
!*,(',) = !*. ( ' 2 )
0 Vth / E { 1 , 2 ) , iinh xa (/,w)l->/(«((<),TO) Ihuoc
ci,(x((*;'(,-i)n£,s.£.a),z)
irongdo f'(A):=\t^l: .,liy)^A].
ii) V&i je[\,2], gio-^ir f la anh Xa duy nhdt trong
o[x(t,;'(A)nE.B:E,G).z)
sao cho
/ , ( / , « ; ) = / ( ( * , ( / ) , w ) , {t,w)^X(f/{A){^E,B(^B'•,E,G)
Khi do, theo khdng dfnh 1). chu y 2.3.2 vd dp dung dinh ly 2.1 I, la
vdi VWGG sao cho (l^,w)eX{,^;'{A)r\E,B;E,G)j ^{12]., Chiing minh dfnh ly 2.3.1.
Budc I: Xay dyng anh xa thac t r i e n / tren x.
Budc 2. Chirng minh dang thirc f = f tren
xnx'.
Budc 3. Chiing minh rang fsO{x,z]- Phan 2 cua chirng minh dinh ly A Muc dich chinh ciia phan nay la chirng minh djnh ly A trong trudng hgp dac biet sau:
Dinh ly 2.4.1. Gid sir D. G la cdc da lap phirc, A cD, B (zG la cdc tap con md vd Z Id khong gian gidi tich phuc co tinh chdl thdc triin Hartogs. Ddt X •.= X{A,B;D,G) va X-.= X(A,B;D^G).
Khi do, vdi moi dnh xa f eO(X,Z) co duy nhdt dnh xg 'f^o{xz\ ^^^ ^^^
J = f trenX.
Chuy 2.4.2. Vdi gia thiet tren, ta cd -. X' =X.
Chimg minh dinh ly 2.4.1 Ta sir dung bd dg sau:
Bd de 2.4.3. Vdn giir gid thiit nhu dinh ly 24 1 Vdl j^[\,2}.gidsinf/eOrE,G')io dia chinh hinh vd .- e £ sao cho
i) Vai j^[\,2],anhxg[z,T)^f(z,ii/j{t]) thugc 0^{\[A,IV',\B)^E;D,E),Z), trongdo ^ ( / ; ' { 5 ) : = { r e £ : ^ ^ ( r ) € 5 } . ii) Vai y e l l , 2 } , gici sir f Id dnh xg duy nhdl trong
Ngo Thi Kim Quy Tap chi KHOA HQC & CONG NGHE 103(03): 133-139
0[f{A,tp-\B)nE;D,E),z)
saocho U'^')=f('^1^i{')\
{z,T)e\(A,ii/]\B)nE',D,E) Khi do, theo phdn i), chu y 2.4.2 vd dp dung dinhly2.3.1, tacd: /^[Z,T^) = f^[z,T^) vdi moi ZG D sao cho
( z , r J e X ( / i , ( / / ; ' ( 5 ) n £ ; 0 , £ ) , vdi je{\,2]
Phan 3 ciia chirng minh dinh ly A cho truong hffp jV = 2
Muc dich chinh cua phan nay la chimg minh djnh ly A trong trudng hpp N=2.
Binh ly 2.5.1. Gid sir D. G la cac da tgp phirc, A cD, B cG Id cdc tap con khdng da cyx: dia phuang vd Z la khong gian gidi tich phirc CO tinh chat thdc trien Hartogs. Dal
X:=X{A,B;D,G) vdX:=X[A,B',D,G).
Khidd, vdi moi dnh xa f eOAX,Z) coduy nhdl dnh xg f eOix,z] ^'^ci cho J-f tren X^X'.
Dg chirng minh ta can mdt sd kgl qua sau:
Vdi mdi a e ^ ' (tugng ling b&B'), co dinh mgt lan can md t / ciia a (tuong ling V^ ciia b) sao cho U^ (lucmg irng V^) la song chinh hinh tdi mdt mien trong C''" (tuong irng trong C'''), Uong dd d^ (lucmg ling d^) la s6 chigu cua D (tuong irng G) tai a (tuong irng b). Vdi moi 0 < ^ < 1 ta djnh nghTa:
" 2
V^,'=[z^V^:(b{z,AV\V^,lJ^)<5\, a^A{\A\
y,,:={w&V,:(b{w,BV\V„V,)<5\, b€Br\B\
^^•= U ^-.s^ ^s-= U ^*.^'
Di:={zeD:m{z,A,D)<\-S],
Gg:={w€G: d>{w, B,G)<\-S].
Bd de 2.5.2. Vdn giir cdc ki hiiu nhu tren thi ta cd:
d){z,A,D)-S<d){z,A^,D)<d}{z,A,D), zsD.
Dinh nghia 2.5.3. Gia sir At la da tap phirc
va Y la khdng gian phiic. Gpi ty \ la hp cac tap con md ciia fA\k tj-\ la ho cac anh xa saocho /^ GC?((y^,7).Tandi {f\ \aho ddn dupe neu vdi mdi j,k^J ta cd f - firen U f) L'^ . Anh xa chinh hinh duy nhat j-.i 1^ Y xac dinh bdi f'.— f tren If , j^j dugc ggi la dnh xg ddn lgi ciia hg
Ba de 2.5.4. Vdn gid gid thiit nhu dinh ly 2.5.1 vd ky hieu tren. Han nira, gid sic vai mdi aeAf]A', cd duy nhdt dnh xg 7 „ e O ( x ( / i n i 7 „ 5 ; f / „ G ) , z ) saocho ?„(2>'«') = /(2.w).
{z.w)eX(Af]A' f]U^,Bf]B';U,„G) thi hg f 'y \ ^ Id ddn duac.
Bd de 2.5.5. Gid sir 22 vd ff Id hai da lap phdc, {^Ag) ^.^1 (iifcmg dng {B^\^^^\_) ^^ ^o cdc tap con khong da cue dia phuang ciia V (tuang ung 0 vd {p^^^^^^^Jtuang ung
(QS). i) Id hg cdc tap con md cua 22 (tuang dng 0 vdi cdc tinh chdt sau:
(i) ^ ^ c A , ^ ^ . , <=^^, "^
B,^(^B,^czG,^ czg,. vdl Q<5,<5,<-.
(ii) Co mot hp cdc dnh xg chinh hinh
\ f x\ I ^^c) cho
\ ° h<s<-- J,^0^{A„B„V„Q,),Z]
"'' ^^' 0<S,<S,<-.
137
Ngo Thj Kim Quy Tap chi KHOA HQC & CONG N G H f 103(03): 133- 139
(Hi) Co mgt tap con md U (lirang irng V) eua t> (tuang img Q vd mpi so ^ ^ \_ .sao cho
d){^z.A,.D_,')^ dj{^z.B,,ggJ)< 1 vdi mpi (z. iv) eUxV vd 0<d< S,y
Khi do f^^i^z.(!•)-- 7 A ( - • " ' ) ^^' "'^'' {z.if)eUxV vd 0<S<d„
M^nh de 2.5.6. Voi cdc gid lliiel nhu djnh ly 2.5.1. Gid .sir them rdng G Id song chinh hinh tdi mdt mien trong £•' Lj e N) 'hi kit lugn cua dinh Iv 2 5 1 van ditng.
P h a n 4: M d r u n ^ eiia c h i i n g m i n h d i n h ly A T r o n g phan nay. la chirng minh djnh ly A vdi mgi A ^ > 3 .
Ta chia c h u n g minh thanh hai phan: „ Chirng minh sutdn ttii vd duy nhat cua f Chung minh ddnh gid trong dinh ly A Chia phan nay thanh hai b u d c Buac 1.
Chiing minh bat d a n g tl Buac 2. Chirng minh bat d bat dang'-thuc
[m\AV\fi '(--)
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[3], O. Alehyane el A. Zeriahi (2001), "Une nouvelle version du iheoreme d'extension de Harlogs pour les applications separement
holomorphes enire espaces analytiques", Ann.
Polon. Math. 7 6 , 2 4 5 - 2 7 8
[4]. E. Berford (1982), "The operator (dd')" on complex spaces", Semin. P. Leiong - H. Skoda, Analyse, Annees 1980/81, L e d . Notes Math. 919.
294-323.
[5]. E. Bedford and B. A. Taylor (1982), "A new capacity for plurisubharmonic functions", Ada Math. 149, I - 4 0 .
(6). S. M. Ivashkovich (1997), T h e Hartogs phenomenon for holomorphically convex Kahler manifolds". Math. USSR - Izv. 29. 2 2 5 - 2 3 2 . [7]. M Jamicki and P. Pflug (2000), Extension of holomorphic Functions, de Gruyter Expositions in Mathematics 34, Walter de Gruyter.
[8]. B Josefson (1978), "On the equivalence between polar and globally polar sets for plurisubharmonic functions on C"", Ark. Mai. 16 , 1 0 9 - 115.
(9] Nguyen VSn Khue - Le Mau Hai, Hdm bien phirc, Nxb DH Quoc gia Ha Npi.
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Ng6 Thi Kim Quy Tap chi KHOA HOC & CONG NGHE 103(03): 133- 139
S U M M A R Y
G E N E R A L F O R M O F T H E T H E O R E M W A T E R F A L L D E V E L O P M E N T H A R T O G S M A P P I N G F O R O R T H O P E D I C S E P A R A T I O N O F V A R I A B L E S
Ngo Thi Kim Quy*
College of Economics and Business Adminislralion - Ti\'L' The main purpose of this article is to give a general version ofthe well-known Harrlogs extension theorem for separately holomorphic hmclions. Using recent development in Poletsky theory of discs, we prove the following result: Let X, Y be lo complex manifolds, lel Z be a complexanalylic space which possesses the Hartogs extension property, lel A (resp. B) be a non locally pluripolar subset of X (resp. Y). We show that every separately holomorphic m a p p i n g / : W . ^ ( , 4 x > ' ) U { ^ x S ) - > Z extends lo a holomorphic mapping / onW:=[(z,u>)eXxY-.&{z.A.X) + d){w,B,Y)<\] such that / = / on ivr\IV, where d){,A,X) (resp. d){w,B,y) is the plurisubharmonic measure of A (resp. B) relative to X (resp. Y).
Generalizations of this resuH for an N-foId cross are also given.
Key words; Local pluripolarily. plurisubharmonic measured N - fold cross, separately holomorphic, Hartogs extension property.
Ngdynhgnbdi- 15/1/2013, ngdyphdnbiin 22/2/2013. ngdy duyet dang. 26/3/2013
Tel 0917 333725, Email kimquyklln@gmail com
