1 0ò1 0Ï ê Æ Æ Vol.0, No.0 201xc 1 ACTA MATHEMATICA SINICA, CHINESE SERIES Jan., 201x Ù?Ò: (20xx)0x-0xxx-0x zi è: A ";/+" ½n Ú«] oa ŒÆêÆ ^ ÆÆ

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1 1 ò1 Ï ê Æ Æ Vol., No. 21xc 1 ACTA MATHEMATICA SINICA, CHINESE SERIES Jan., 21x Ù?Ò: (2xx)x-xxx-x zi è: A ";/+" ½n Ú«] oa ŒÆêÆ ^ ÆÆ Ñ yÿ oaœæêææ!ôêæ % Ñ ` Department of Mathematics, University of California Irvine CA Á 3, Ñ «AÛf;/+½Â, ѽf;/ + â., y²;f;/+;/+g!;"f;/+ " Ú; Lagrangian f;/+ Lagrangian 35. ' c f;/+; ;/+G ; "f;/+; " ; Lagrangian MR(21)ÌK a 57R18; 57R17; 57N4 ã a O186.1 Symplectic neighborhood theorem for symplectic orbifold groupoids Cheng-Yong DU School of Mathematics, Sichuan Normal University, Chengdu 6168, China Bohui CHEN School of Mathematics and Yangtze Center of Mathematics, Sichuan University, Chengdu 6164, China Rui WANG Department of Mathematics, University of California, Irvine, CA, , USA Abstract In this paper, we give a geometric definition of sub-orbifold groupoid, and a criterion for determining a sub-orbifold groupoid. Then we prove the existence of ÂvFÏ: 2x-xx-xx; ÉFÏ: 2x-xx-xx Ä7 8: I[g, ÆÄ7]Ï 8( ; ; ); oaž e]ï ï 8(15ZB27) ÏÕŠö: yÿ

2 2 ê Æ Æ ò orbifold tubular neighborhoods of compact sub-orbifold groupoids, the existence of symplectic neighborhoods of compact symplectic sub-orbifold groupoids in symplectic orbifold groupoids, and, the existence of Lagrangian neighborhoods of compact Lagrangian sub-orbifold groupoids. Keywords sub-orbifold groupoid; orbifold tubular neighborhood; symplectic suborbifold groupoid; symplectic neighborhood; Lagrangian neighborhood MR(21) Subject Classification 57R18; 57R17; 57N4 Chinese Library Classification O Úó ;/ orbifold 6/ I. Satake [12] cú\. ;, Hausdorff ÿà m X þ;/ Ik orbifold chart o (Ũ, G, φ, U). Ù Ũ îª m R n m8; G k +, 1wk/Š^3 Ũ þ; U X m 8; π : Ũ U ëyn, p Ó Ũ/G U. X þ ;/ãþ orbifold atlas x v ½ƒN5^ ;/ Ik U = {(Ũi, G i, φ i, U i ) i I}. ;/ãþ daò X ;/( c, Moerdijk-Pronk [8] uyœ±^;/+ orbifold groupoid 5ïÄ;/. y²;/ k effective ;/+ƒméa'x, = l;/ãþœ±k;/+, ƒlk;/+œ±;/ãþ. d;/ãþéau Morita d;/+. ù (J Œ± ^;/+5ïÄ ;/AÛ ÿà. 3 VÐ Chen-Ruan [4, 5] EÑ";/ Gromov-Witten nø. ù nø 8c ISþš~¹ïÄ+. ²Lù 5cuÐ, < uy;/+';/ãþ Ü^ 5ïÄ;/, AO 3ïÄ;/ Gromov-Witten nøžÿ. ïä;/ Gromov- Witten nø, I á Ù ";/+Ä"ÿÀ5Ÿ. ùò Ñu:. l;/+ý5ïä;/aû. ÄkÄf;/+Vg. ; /+ aaï+, Ïd Œ±lX ê*:5äf;/+½â ë [6, 7]. ØL;/+Š ;/éaaûn, X ê½âøuéðêüù éa. Ï d ATk AÛf;/+Vg. IO. ;/m", ù éðë. ddñué Ünf;/+½Â ½Â 3.2. ½ ;/+ G = (G, G 1 ), uy äkƒó; mf;/+ñ Morita d ½n 3.9!íØ 3.11., z ùf;/+ Morita dap k Œ, = f;/+ ½Â 3.8. dd Ñä G 1w6/UÄû½ f;/+äok ½n 3.13 Ú½n 3.14., Äf;/+G. uy, éu A½;/+ G Ú f;/+ H, = f;/+ ; =; m H ;, Ã{ n Ž+G, =ˆ? Ý +G ~ 5.3. ØL, oué Morita df;/+ K G K knž+g ½n 5.4. Ä";/+Ú "f;/+, y²3 Morita d Âe"f;/+" 35 ½n 7.6,

3 Ï Ú«]! yÿ!`: ";/+" ½n 3 Ú Lagrangian f;/+ Lagrangian 35 ½n 7.8. ù(j3";/+ "ÃâÚ Gromov-Witten nø ò k A^. (Xe. 31 2! Á;/+½ÂÚ5Ÿ, 31 3! Ñf;/+ ½Â,?Ø 5Ÿ, 31 4! Á;/+þ þm, ;/+ƒm, Úf;/ +{mvg., 31 5!y²;f;/+;/+G 35, 3 1 6! Ñ;/+G A^, ;/+þóôúª. ^1 5!6!(J31 7!?Ø;"f;/+" 35Ú; Lagrangian f;/+ Lagrangian ;/+! e;/+½â!5ÿ±9 ƒm Morita d'x. ùvgú5 ŸŒ [1, 6 9]. ½Â groupoid G = (G, G 1 ) Æ, Ù kñœ_. k±e 5 (N: 1 N s : G 1 G 1 Ú8IN t : G 1 G ; 2 EÜN m : G 1 (s,g,t) G 1 G 1 ; 3 ü N u : G G 1 ; 4 _N i : G 1 G 1. XJ g Hom(x, y) G 1, K s(g) = x, t(g) = y. òùp g : x y, ƒ Þ. XJ s(g) = t(h), P m(g, h) = gh. éu x G P u(x) = 1 x. é g G 1, P i(g) = g 1. ù 5 (N v±eún: s(1 x ) = t(1 x ) = x, s(g 1 ) = t(g), t(g 1 ) = s(g); s(gh) = s(h), t(gh) = t(g), g(hk) = (gh)k; g1 x = 1 y g = g, g 1 g = 1 x, gg 1 = 1 y, g : x y. ½Â 2.2 G o+ Lie groupoid, XJ G Ú G 1 Ñ 1w6/, k (NÑ 1wN, s, t Ñ ìv submersion. ½Â 2.3 o+ G ;/+ orbifold groupoid, XJN (s, t) : G 1 G G T proper, s, t Ñ ÛÜ Ó. dž dim G = dim G 1. ù ê G ê. ½Â 2.4 ½ ;/+ G = (G, G 1 ) Ú? : x G, G x s 1 (x) t 1 (x) x ÛÜ+. ù k +. ½ o+ G = (G, G 1 ), 3 G þk 'X/ ½Â x y, XJ3 g G 1 g : x y. ù d'x. ½Â 2.5 G o m coarse space ½ö; m orbit space ½Â G G /. Pl G G Ý π : G G. G ;, XJ G ;ÿà m.

4 4 ê Æ Æ ò ~ 2.6 ~X, k + G 1wŠ^3 6/ M þ, Œ± ;/+: G M (M, G M). NÚ8IN s(g, m) = m, t(g, m) = g m. Ù 3 (N w,. y=ù ;/+. ½Â 2.7 b G Ú H ü o+. φ : H G d 2 1wN φ : H G Ú φ : H 1 G 1 ¼f, = ˆg(N. XJ φ, φ 1 Ñ Ó, Ò φ = (φ, φ 1 ) Ó, P φ : H = G. ü o+ƒm φ : H G d, XJ i ½Â36/n È G 1 (s,g,φ ) H {(g, y) g G 1, y H, s(g) = φ (y)} þn tπ 1 : G 1 (s,g,φ ) H G ìv. ii Nµ H 1 φ 1 G 1 (s,t) (s,t) φ φ H H G G n È. Ún 2.8 ([1]) XJ φ : G H φ ÛÜ, p ; mþó φ : G H. ½Â 2.9 ü o+ G Ú H Morita d, XJ31 3 o+ K Úü ψ o+d G K φ H. ù o+ƒmd'x, Œ 3;/+þ ;/+ƒm d'x. ØL3`ü ;/+ Morita dž #N m1 3 o+. 3 f;/+ 3ù!?Øf;/+½ÂÚ5Ÿ. 3.1 f;/+½â Äkw Ún. Ún 3.1 G = (G, G 1 ) ;/+, H G { k 1wf 6/. XJ H 1 s 1 (H ) t 1 (H ) G 1 { k G (N 3 H, H 1 þ, ;/+ H = (H, H 1 ). y² y² ¹üÚ. 1 H +. Ï H 1 = s 1 (H ) t 1 (H ), Œ±ò s, t : G 1 G 3 H 1 þ s = s H1 : H 1 H Ú t = t H1 : H 1 H. ÓŒ±ò u 3 H þ

5 Ï Ú«]! yÿ!`: ";/+" ½n 5 u = u H : H H 1, i 3 H 1 þ i = i H1 : H 1 H 1. Ï H 1 H H 1 G 1 G G 1, k m : H 1 H H 1 H 1. y=, þù(n, H = (H 1 H ) +. 2 H ;/+. db Iy H NÚ8IN s, t ÛÜ Ó.? : g H 1, k g U g G 1 s : U g U x Ó, Ù x = s(g). Ï s ÛÜ Ó, ± s 1 (H ) G 1 { k 1wf6/. H 1 s 1 (H ) äk{ ê k, ± Œ±Â U g v U g s 1 (H ) H s : U g s 1 (H ) U x H Ò Ó. ± s : H 1 H ÛÜ Ó. ÓnŒy t : H 1 H ÛÜ Ó. y.. ½Â 3.2 Ún 3.1 ;/+ H = (H, H 1 ) G = (G, G 1 ) f;/ + sub-orbifold groupoid d½âœ, XJ H G x H, 3 G Û Ü+ G x Ú 3 H ÛÜ+ H x v G x = H x. ù ½Â Ï~f+ Š f Æ ½ÂØÓƒ?. ~ 3.4 XJ H = {x}, K x (x, x G x ) G f;/+. G :. ~ 3.5? G m1wf6/ U, U = ( U, U 1 = s 1 (U ) t 1 (U ) ) G f ;/+. ò P G U. G mf;/+. mf;/+ ;/+ÛÜ.k'. K 3.6 ([1, 8]) ½ ;/+ G 9? : x G, o3 x m U x G s 1 (U x ) = t 1 (U x ) = g G x W g, s, t : W g U x Ñ Ó, G Ux = Gx U x. 3.2 f;/+ f;/+ k aaïf;/+. ½Â 3.7 G f8 H G-µ4, XJ s 1 (H ) = t 1 (H ). XJ H G-µ4 G { k s, t ÛÜ Ó, H 1 = s 1 (H ) = t 1 (H ) G 1 { k 1wf6/, l H = (H, H 1 ) G = (G, G 1 ) f;/+. ½Â 3.8 G f;/+ H = (H, H 1 ) f;/+ full sub-orbifold groupoid, XJ H G-µ4. ½n 3.9 b H G f;/+. XJ K G f;/+, v π(h ) = H = π(k ) = K G, K K H, K 1 H 1. Pùü ¹N i, i 1, K i = (i, i 1 ) : K H ;/+ƒmd. y² b x K, π(x) = x K = y H π(y) = y = x. Ïd3 G, 3 Þ g G 1 g : x y. Ï H G-µ4, ± x H, g H 1. Ïd K H. ùl² K 1 s 1 (K ) s 1 (H ) = H 1. w, i (s(g)) = s(g) = s(i 1 (g)), i (t(g)) = t(g) = t(i 1 (g)). Ïd i = (i, i 1 ) : K H. e y² i d.

6 6 ê Æ Æ ò 5 H 1 (s,i) K = s 1 (K ) 1w6/. N tπ 1 : H 1 K K H = t : s 1 (K ) H. ùw,.? g s 1 (K ) g : x g m U g G 1 s : U g s(u g ) Ú t : U g t(u g ) Ñ Ó. Ïd t : U g s 1 (K ) t(u g s 1 (K )) Ó. du dim H = dim K, K 1 H 1, ÏL U g k t(u g s 1 (K )) = t(u g ) H. Ïd t : s 1 (K ) H Œ± y n È. y.. 8Ü ÛÜ Ó. l tπ 1 : H 1 K K H ìv. (s,t) K 1 i 1 H 1 (s,t) K K i i H H Œ± ^+n È ë [1] ½Â95Ÿ5y²ù (Ø. kxeíø íø 3.11 ½Â 3.12 M(H, G). 3.3 ½f;/+ G käkƒó; mf;/+ñ pƒ Morita d. b H G f;/+, P G käk; m H f;/+ e?ø: éu ;/+ G = (G, G 1 ),? ½ G 1wf6/ H, ÄU f;/+. f;/+ +Š^ØCf6/k ƒ'x. ½n 3.13 b H G H = ( H, s 1 (H ) t 1 (H ) ) G = (G, G 1 ) f;/+ =éu? : x H, ±9 /X K 3.6 m U x G, H U x U x 3 G x Š^eØCf6/. y² 5. ÛÜþd K 3.6 k G Ux = Gx U x = (U x, G x U x ). d [9], Œ±ÏL IC G x 3 U x R n þš^ 5Š^. - V x = U x H. XJk, g G x g V x V g U x ( s 1 (V x ) t 1 (V x ) ) ( g U x ) = (g Vx ) (g (g 1 V x )) H êø. ù H f;/+gñ I y² H 1 s 1 (H ) t 1 (H ) G 1 1wf6/.? g H 1, g : x y, x, y H, Œ± O x, y Ú g m U x, U y Ú U g s : U g U x, t : U g U y Ó, V x = H U x Ú V y = H U y O U x, U y ˆg3 G x, G y Š^eØCf6/. t s 1 : U x U y g 1 ( )g : G x G y k Ïd H 1 U g = s 1 (V x ) = t 1 (V y ) s 1 (V x ) = t 1 (V y ) U g. f6/, { ê H. ± H G f;/+. y.. C. Ïd U g 1wf6/. d g? 5 H 1 G 1 1w

7 Ï Ú«]! yÿ!`: ";/+" ½n 7 ½n 3.14 ½ G 1wf6/ H. H = ( H, s 1 (H ) t 1 (H ) ) G f; /+ = π 1 ( H ) G 1wf6/. Ù H = π(h ). y² - K π 1 ( K G-µ4, H K b H G f;/+. e y² K G 1wf6/.? : x K \ H. I y²3 G m U x K U x U x f6/. ÄkŒ± y H 3 Þ g : y x. o x /X K 3.6 m U U x Œ±é y /X K 3.6 U y ±9 ÏL g Ó t s 1 : U y U x. d Ó g( )g 1 : G y G x K U x = H U y U x 1wf6/. 5. XJ K G K = (K, K 1 = s 1 (K )) G f;/+. s 1 (H ), t 1 (H ) Ñ K 1 { f6/. ± H 1 = s 1 (H ) t 1 (H ) H kƒó{ G 1 1wf6/. Ïd H = (H, H 1 ) G f;/+. y.. 4 ;/+þ þm ù! Ä;/+þ þm. SNŒë [1]. 4.1 ;/m ½ ;/+ G = (G, G 1 ) Ú G þ þm π : E G, Œ±ü G 1 þ þm π s : s E G 1, π t : t E G 1. ½Â 4.1 ½ m Hom(s E, t E ) G 1 σ. ` (E, σ) G þ ;/ þm ;/m, XJé? g G 1, σ(g) : E s(g) E,t(g) 5Ó, σ(g 1 g 2 ) = σ(g 1 )σ(g 2 ). l σ Œ± E 1 {(v, w) s E t E σ(π s (v))(v) = w} = {(v, σ(g)) v E,s(g) }. ù 1w þm E 1 G. kw,n s : E 1 E, s(v, w) = v, Ú8I N t : E 1 E, t(v, w) = w ±9Ù 3 (N. Œ± y þù(n E = (E, E 1 ) ;/+. k E = (E, E 1 ) G = (G, G 1 ). Ïd;/mÒ é v ½ƒN5 þm. 3 rn σ ž ÒòmP (E, σ). ½Â 4.2 ½ ;/m E = (E, E 1 ) G = (G, G 1 ). XJkü þm E i G i ξ i v ξ 1 = s ξ = t ξ, Ò ξ = (ξ, ξ 1 ) E G. ùdu` ξ σ -ØC. P E m ½Â 4.3 Γ(E). w, G E -. ½ü ;/m E 1 = (E 1, E 1 1) G, E 2 = (E 2, E 2 1) G. ;/mn f = (f, f 1 ) : E 1 E 2 ;/+ f i : Ei 1 E2 i 6/þmN. ;/m Ng,p - ƒm;/ f : G G. ;/m f = (f, f 1 ) : E 1 E 2 ;/móxj f, f 1 Ñ mó f = id G. ½Â 4.4 ½ ;/m E = (E, σ) G ±9 E G fm. XJ σ U Hom(s F, t F ) G 1 σ F, Ò (F, σ F ) E f;/m, P

8 8 ê Æ Æ ò F = (F, σ F ). l F = (F, σ F ) Œ± F = (F, F 1 ). Ún 4.5 F = (F, F 1 ) E = (E, E 1 ) f;/+. Ïdf;/m, «w{ò E G, E 1 G 1 ü fm F, F 1. E f;/+ F = (F, F 1 ). ùò F = (F, F 1 ) E f;/m. XJ F E = (E, σ) σ ± F E. Ïd km Hom(s (E /F ), t (E /F )) G 1 [σ]. ù [σ] Ú E /F Ò/ ;/m. ½Â 4.6 (E /F, [σ]) E 'uf;/m F û;/m, Pƒ E/F. w, k E/F = (E /F, E 1 /F 1 ). 4.2 ƒmú{m e?ø;/+ƒm, f;/+{m. ½Â 4.7 b G = (G, G 1 ) ;/+. ƒm TG G ½Â T G 1 ds T G dt Ù ds(g, v) = (s(g), ds g (v)), dt(g, v) = (t(g), dt g (v)). ƒam Hom(s T G, t T G ) G 1 σ TG (g)(g, v) = (g, dt g ds 1 g (v)), g G 1 Ú v T s(g) G. ½Â 4.8 b H G f;/+. 3 G {m½â N H ( N 1,H = T G 1 H1 /T H 1 [ds] [dt] N,H = T G H /T H ) = TG H /TH, Ù [d g s]([v]) = [d g s(v)], [d g t]([v]) = [d g t(v)]. ƒam Hom(s N,H, t N 1,H ) H 1 σ NH (g)(g, [v]) = (g, [d g t d g s 1 (v)]) = (g, [d g t] [d g s] 1 ([v])), g G 1 Ú [v] N s(g) H. 5 +G ½n Äk Ñ;/+ G þiùýþ½â. ½Â 5.1 G þ iùýþ ½Ýþ G þ G 1 -ØCÝþ η, = s η = t η G 1 þiùýþ. d`, G þýþ é G Ú G 1 þýþ (η, η 1 ), v s η = t η = η 1. Pflaum-Posthuma-Tang [1] é a ½n 5.2 ([1]) e5 Ä+G. 2o+E aýþ. (JL²? ;/+ G þo3iùýþ. b H G f;/+. Ð+G AT ve 5Ÿ. = `, é? g : x y, x, y H, d 3 x Ú y? ÝAT, = `+G Ý3 H z :;þat ~Š., / H f;/+ž, Ã{ ù :. ØLÏLé G H df;/+ Œ± ù :.

9 Ï Ú«]! yÿ!`: ";/+" ½n 9 ~ D i R 2 Œ» 1 i, i N 1. - G = D i, G 1 = D i D j. i 1 i,j 1 kw,n s : i,j D i D j i D i ò D i D j N D i, Ú8IN t : i,j D i D j i D i ò D i D j N D j, ±9Ù{ 3 w,(n (G, G 1 ) ;/+. þ, ù ;/+L«1w6/: D 1. IOîªÝ þ Ñù ;/+þ iùýþ. P D i : i. K H = i { i} G 1 -µ4 G 1wf6/. Ñ G f;/+ H = (H, H 1 = { i, j }). i,j w,ù f;/ ¹ ;,,ù ;þ: vk Ý. Œ±, f;/+ K = ({ 1 }, { 1, 1 }). w, K M(H, G). K ; ¹k :, Ïd käk Ý. Œ±òù (Jí2 œ¹. ½n 5.4 (;/+G ½n) b H G { ê k 1 ;f;/+, η = (η, η 1 ) G ½3 f;/+ K M(H, G),, 3 G 3 K m U, ±9 Ýþ Ó φ : D ɛ N K U. (5.1) Ù, ɛ ~ê, D ɛ N K K {m N K Œ» ɛ m. k  N ρ : U K 9 ã D ɛ N K φ U (5.2) π ρ K. y² P H 3 G {m N H. Ú6/œ/aq, Œ±ÏLÝþ η = (η, η 1 ) ò Óu E = TG H fm, = TH Ö. þ, - N i {v T G i η i (v, w) =, w T H i }, i =, 1. Kd s η = t η = η 1 N (N 1 N ) E fm, Óu{m N H. Œ±ò E þ Ýþ 3 N þ, Pƒ η N = (η,n, η 1,N ). ò N (N, σ N η,n Ò σ N -Ø C. w, Œ±ò H Óu N -. Œ±ò η 3 H þ H þ Ýþ.? : x H, Œ±é ~ê ɛ x > Ú H Œ» ɛ x ÿ/ V x, ±9 m x U x G, exp : B x D ɛx (N Vx ) U x V x (5.3) Ó, Ù B x D ɛx (N Vx ) m N Vx Œ» ɛ x m.

10 1 ê Æ Æ ò du G ;/+, d K 3.6,é? x H, o3 G U x, x ÛÜ+ G x Œ±Š^3 U x þ, G U x = Gx U x. ÏdÏL (5.3) ɛ x, V x Ú U x, Œ±bù ɛ x, V x, U x v (1) exp : B x D ɛx (N Vx ) U x V x Ó, (2) G x 1wŠ^3 U x þ, G Ux = Gx U x. - V x H ± x %, ɛ x /2 Œ»ÿ/, B x N V x Œ» ɛ x /2 m. P B x 3 ên exp e U x U x. P; mý x H, ù Ò é x π : G G.? ; m : x H, Œ± x V x U x. Ïd k H mc X: { V x = π(v x)}, ±9 H 3 G m CX { U x = π(u x)}. db, H ;, Ïd H k mcx V = { V x α α A} Ú 3 G k - mcx U = { U x α α A}. ùp #A <. K = ɛ = min{ɛ xα /2 α A}. ( α A U x α ) H K G 1wf61 K = ( K, K 1 = s 1 (K ) t 1 (K ) ) G f;/+. lþ E K = H, Ïd K M(H, G). du U k mcx, k α A V x α. #π 1 x K <, #s 1 (π 1 x ) K 1 <, x H = K. y3 òm N H 3 K þ. m= K 3 G {m N K. ò N 3 K þ. òmp N = (N, N 1) = (N, σ K ). K N = N K. dþ E ên exp Ñ N - D ɛ N = {(x, v) N η (v, v) < ɛ, x K } K 3 G, U Ó. U α A U x α. - U 1 s 1 (U ) t 1 (U 3.5 U = (U, U 1 ) G mf;/+.? g K 1, g : x y, N 1 3 g þn N 1,g = {v T g G 1 η 1 (v, w) =, w T g K 1 = T g H 1 }. Ä N 1 Œ» ɛ m D ɛ N 1 {v N 1,g η 1 (v, v) < ɛ, g K 1 }. P N NÚ8IN s N, t N. Ï G U x = Gx U x, Ýþ η σ K -ØC, ± D ɛ N 1 = s 1 N (D ɛ N ) t 1 N (D ɛ N ). Ïd D ɛ N = (D ɛ N, D ɛ N 1) N f;/+. d EŒ ên3 D ɛ N Ú D ɛ N 1 þñk½â. yœ exp(s(g), d g s(w)) = s(exp(g, w)), exp(t(g), d g t(w)) = t(exp(g, w)).

11 Ï Ú«]! yÿ!`: ";/+" ½n 11 l ên exp : D ɛ N U. ù Ó. Œ± D ɛ N K. D ɛ N Ó. PdÓ f : D ɛ N K D ɛ N. Ó φ exp f : D ɛ N K U. kmý p : N K K Ú ¹'X D ɛ N K N K, l k p : D ɛ N K l φ ρ = p φ 1 : U K. Ïd ã (5.2). y.. ½Â 5.5 XþEÑ K 3 G ;/ + +G. 6 ÓÔúª ½ ;/+ G = (G, G 1 ). Œ±½Â {ƒm T G. G {ƒ m T G G, ÚÓm Hom(s T G, t T G ) G 1 σ T G(g) = (t g) 1 s G {ƒm T G = (T G, σ T G). aq, éu k dim G, Œ±½Â Èm k T G. E,kXeÓ k T G = ( k T G 1 k (s 1 ) k (t 1 ) k T G ). ^ Ω k (G) 5Pm k T G m. p ƒ ;/ k-/ª. ;/ k-/ª éï~ k-/ª (ω, ω 1 ) Ω k (G ) Ω k (G 1 ), v s ω = t ω = ω 1. ù ω G-ØC k-/ª. Ïd Ω k (G) = {(ω, ω 1 ) Ω k (G ) Ω k (G 1 ) s ω = t ω = ω 1 Ω k (G 1 )} = {ω Ω k (G ) s ω = t ω }. Ï~ Žf d Œ±Š^3 Ω (G) þ: d : Ω k (G) Ω k+1 (G). k d 2 =. G de Rham þón+½â H k dr(g) ker(d : Ωk (G) Ω k+1 (G)) Im(d : Ω k 1 (G) Ω k (G)) ;/œ/óôúª 6/œ/aq. b H G f;/+, k;/+g U = (U, U 1 ), =3½ Ýþ, 3 ~ê ɛ U = D ɛ N H. ½n 6.1 XJ ω = (ω, ω 1 ) U þ4 k-/ª, = dω i =, i =, 1. ω(x) =, x = (x, G x ) ω T, = `, 3 µ Ω k 1 (U) ω = dµ., Œ±À µ µ(x) =, x H.

12 12 ê Æ Æ ò ùp µ(x) = du µ (x) =. Ï d G-ØC5, g,k µ 1 (g) =, g U 1. y² du U = D ɛ N H, I 3 D ɛ N H þäù K. w,d n?1 =Œ D ɛ N H = NH. Ïd e 3 N H = (N, N 1 ) þäù K. P N H N Ú8IN s Ú t. P;/mÝ π = (π, π 1 ) : N H H. P H Š N H - ¹N i = (i, i 1) : H N H. 5, k x;/+ó ρ τ = (ρ τ, ρ τ 1) : N H N H, (v, v 1 ) (τv, τv 1 ), τ 1. Ù ρ τ l i π id N ½ H ÓÔ, ρ τ 1 l i 1 π 1 id N1 ½ H 1 ÓÔ. ùp ρ τ ÓÔŽf Q (η ) 1 ρ τ, (ι v τ η )dτ, η Ω k (N ), Ù v τ d ρ τ (p) ) þ, = v τ (p) = d dt ρt (p) t=τ. ù k dq (η ) + Q (dη ) = ρ 1, η ρ, η = η i, η. (6.1) - µ = Q (ω ). du ω 3 H, µ 3 H þ. ò µ \ (6.1). du dω =, ω 3 H, dµ = dq (ω ) = dq (ω ) + Q (dω ) = ω i, ω = ω. e y² µ N H -ØC. 5 ρ τ 1 Q 1 (η 1 ) 1 ÓÔŽf ρ τ, 1 (ι v τ 1 η 1)dτ, η 1 Ω k (N 1 ) Ù v τ 1 d v τ 1 (p) = d dt ρt 1(p) t=τ û½. Ï ρ τ ;/+, k Ïd aqœ Ï s µ = = 1 1 ω N-ØC, Ïd s ρ τ, (ι v τ ω )dτ = ρ τ, 1 s (ι v τ ω )dτ = ds(v τ 1 ) = v τ = dt(v τ 1 ). 1 1 (ρ τ s) (ι v τ ω )dτ = 1 ρ τ, 1 (ι v τ 1 s ω )dτ = Q 1 (s ω ). t µ = Q 1 (t ω ). s µ = t µ = Q 1 (ω 1 ). (s ρ τ 1) (ι v τ ω )dτ ± µ û½ N þ k 1-/ª µ = (µ, µ 1 ), µ 1 = Q 1 (ω 1 ), dµ = ω. d µ ½  µ(x) =, x H. y.. 7 " ½nÚ Lagrangian ½n 3ù! Ä";/+, ò"6/œ/" ½nÚ Lagrangian ½ ní2";/+þ. 7.1 ";/+

13 Ï Ú«]! yÿ!`: ";/+" ½n 13 ½Â 7.1 ½ ;/+ G. G-ØC 2-/ª ω G þ"/ª, XJ G þ"/ª. ù Œ G 1 þ"/ª ω 1 = s ω = t ω. d /, G þ"/ª ;/ 2-/ª ω = (ω, ω 1 ) Ω 2 (G), ω, ω 1 O G, G 1 þ "/ª. ;/+ G þ "/ª ω = (ω, ω 1 ) = ";/+. ò P (G, ω), ½ (G, ω ), ½{P G. ü ";/+ƒm φ : (G, ω) (H, ρ) ", XJ φ (ρ) = ω. ½Â 7.2 b H (G, ω) f;/+. XJ ω H Ω 2 (H) H þ"/ª, Ò (H, ω H ) G "f;/+. XJ ω H = dim H = dim G 2, Ò H G Lagrangian f;/+, ½{ Lagrangian. ½ (G, ω) "f;/+ H, Œ±ò {mó TH ω, = TH 3 TG 'u"/ª"zü. Xd 5, TH ω n þò kg,"/ª. ùm ";/m. ";/mƒm±n "/ª;/m ";/m. K 7.3 XJ φ : G H ;/+d, (G, ω) H þk p"/ª ω, φ : (G, ω) (H, ω ) ". y² dún 2.8, φ dž φ : G H ÛÜ Ó, φ : G H Ó. Ïd, Œ±ò ω í φ (G ) H þ 2-/ª ω. w, ω 3 φ (G ) þšòz 4. e ò ω òÿ H.? : x H \ φ (G ), Ï φ d, ½3 : y φ (G ) 9 Þ h H 1 h : y x. H ;/+, ± s H, t H Û Ü Ó, Ïd s H,h Ú t H,h 3{ƒ mþñ Ó. ½Â ω (x) = (t H,h) 1 s H,h(ω (y)). y=ù *Ü h ÀÃ', ω H þ"/ª. Ï Ïd ω G-ØC, s Hω = t Hω. ± H þ"/ª ω = (ω, ω 1 = s H ω ). w,dž φ ". y " ½n e5 Ä" ½n. 3c ²w, ;/+G Ø ½3. Ïdùp Ä;"f;/+œ/. ýkb;/+g 35. Ä", ÄkI Ä;/+þ þ =ƒm 6. b ξ = (ξ, ξ 1 ) Γ(TG) G þ þ. Ïd ds(ξ 1 ) = dt(ξ 1 ) = ξ. ÄXe G þ ÜÐ Š~ d dτ φ (x, τ) = ξ (φ (x, τ)), φ (x, ) = x, τ R, x G, Ú G 1 þ ÜÐ Š~ d dτ φ 1(g, τ) = ξ 1 (φ 1 (g, τ)), φ 1 (g, ) = g, τ R, g G 1.

14 14 ê Æ Æ ò? ½ g G 1, g : x y. b γ(τ) ξ 1 ± g å :È, = γ(τ) = φ 1 (g, τ) : ( ɛ, ɛ) G 1, γ() = g. ù G ^L x η(τ) = s(γ(τ)), η() = x. d dτ η(τ) = d γ(τ)s( d dτ γ(τ)) = d γ(τ)s(ξ 1 (γ(τ))) = ξ (s(γ(τ))) = ξ (η(τ)). Ïd η(τ) X ± x Ð :È. aq/, t(γ(τ)) X ± y Ð :È. Ïd k Xd y²e K s φ 1 (g, τ) = φ (s(g), τ), t φ 1 (g, τ) = φ (t(g), τ). K 7.4 é τ, φ τ = (φ,τ, φ 1,τ ) G gó. ½n 7.5 (ƒé Moser ½n) +G U G. XJk G þü "/ª ω Ú ω 1 b H G { k 1 ;f;/+, k;/ ω TGx = ω 1 TGx, x K M(H, G), K H ±9ü K ;/+G U, U 1 U ±9 Ó φ : U U 1, φ ω 1 = ω. U φ U i i 7 K y² Ø b,3 G þ, Ýþ η e U Óu H {m N Œ» ɛ m D ɛ N. ùp Eò N Óu TH 3 TG Ýþ η eö. db, ω ω 1 U þ4/ª, ω (x) ω 1 (x) =, x H. ÏddÓÔúª ë ½n 6.1, 3 1-/ª µ Ω 1 (U) dµ = ω ω 1 µ(x) =, x H. Ä x U þ4 2-/ª ω τ = (1 τ)ω + τω 1 = ω τdµ. db, H ;, x H, ω TGx = ω 1 TGx, Ïd3 < ɛ < ɛ 3 U = (U, U 1) = D ɛ N þù x 2-/ ª ω τ Ñ "/ª. Ïd3 U þ þ ξ τ = (ξ τ, ξ τ 1 ) ι ξ τ ω τ = µ. AO/, Ï 3 H þ µ =, ±3 H þ ξ τ =. y3éz x H, Ñ3 U x U G Ú V x H 3 ~ê < ɛ x < ɛ 9 Ó exp : D ɛx N Vx U x, ξ τ 3 U x þè éžm τ 1 Ñ3. Œ± U x G Ux = G x U x. ù ξ τ 1 3 s 1 (U x ) t 1 (U x ) þè éžm τ 1 Ñ3. ù { V x x H } / H mcx. du H ;, Œ± k CX { V xα α A}. - ɛ K = min{ɛ xα α A}. f

15 Ï Ú«]! yÿ!`: ";/+" ½n 15 y3 K = α A V xα. K k K = ( K, s 1 (K ) t 1 (K ) ) M(H, G), K H. Œ±ò N 3 K þ, 2 ɛ K Œ»m D ɛk N K. ÏL ên, Œ± U = exp(d ɛk N K ) U. dž þ ξ τ U 3 U þkž 1 È ρ τ, τ 1. - φ = ρ 1, U 1 = ρ 1 (U ). KkÓ φ : U U 1. φ ± K ØÄ, φ ω 1 = ω. y.. ½n 7.6 (" ½n) é j =, 1, - (G j, ω j ) ü ";/+, ˆg { 2k 2 ;"f;/+ H j. ò {m OP N j = (TH j ) ωj. b k ";/mó Φ : N 1 N 2, p - ;/+ ";/+Ó φ : (H 1, ω 1 H 1) (H 2, ω 2 H K j M(H j, G j ), K j H j, ±9 ˆg3 G j m U j Ú ";/Ó ψ : (U 1, ω 1 ) (U 2, ω 2 ) ψ K 1 = φ K 1, 3 N 1 K 1 þk dψ = Φ. y² Äk O3 G j þˆ iùýþ η j. Ï H 1 ;f;/+, d½ n 5.4 Œ±é X 1 M(H 1, G 1 ), X 1 H 1 k;/+g WX 1 9Ó exp 1 : D ɛ 1 X N 1 X 1 WX 1, Ù ɛ1 X ~ê. Ï φ : H 1 H 2 Ó, P X 2 = φ(x 1 X 2 M(H 2, G 2 ), X 2 H 2., Œ±é Y 2 M(H 2, G 2 ), Y 2 X 2 H 2, k; /+G WY 2 9Ó exp2 : D ɛ 2 Y N 2 Y 2 WY 2,Ù ɛ2 Y ~ê., - Y 1 = φ 1 (Y 2 ). K Y 1 M(H 1, G 1 ), Y 1 X 1 H 1, k;/+g VY 1 = exp1 (D ɛ 1 X N 1 Y 1) WX 1. ÏL ɛ 1 X k;/+i\ ϕ = exp 2 Φ (exp 1 ) 1 : VY 1 WY 2 ϕ WY 2 mf;/+, ϕ V1 Y Ó. d êne ϕ Y 1 = φ Y 1, 3 N 1 þ dϕ = Φ. y3 ò ω 2 ^ ϕ. VY 1 þ ϕ ω 2. Ï Φ ";/mó, k ω 1 (x) = ϕ ω 2 (x), x Y 1. Ïd 5½n 7.5 œ/. Ïd Œ±é K 1 M(H 1, G 1 ), K 1 Y 1 H 1, ±9 2 +G U 1, V 2 VY 1 Ú ± K 1 ØÄÓ ρ : U 1 V 2 3 U 1 þ ρ (ϕ ω 2 ) = ω 1. y3 U 2 = ϕ(v 2 ), K kó ψ = ϕ ρ : (U 1, ω 1 ) (U 2, ω 2 ), ψ K 1 = ϕ K 1 = φ K 1. qï 3 K 1 þ ω 1 (x) = ϕ ω 2 (x), ±d½n 7.5 y²œ3 3 N 1 K 1 þƒ dρ = id. Ïd3 N 1 K 1 þ k dψ = Φ. y AÏœ/Ò H = {x} G SþÒ C Darboux ½n. 7.3 Lagrangian ½n e ò Weinstein Lagrangian ½ní2;/+þ. ½n 7.8 (Lagrangian ½n) b H G Œ ê;f;/+. ¹NP i : H G. XJ ω Ú ω 1 G þü "/ª i ω = i ω 1 =, = H O (G, ω ) Ú (G, ω 1 ) Lagrangian.

16 16 ê Æ Æ K M(H, G), K H Ú ü ;/+G Ó φ : U U 1, φ ω 1 = ω. U φ U i i 7 K U U 1 ±9 ;/+ y² Äk3 G þ iùýþ. é? : x H, - V x = TG x = (T G,x, T G 1,Gx ), U x = TH x = (T H,x, T H 1,Hx ). 5 ùpdf;/+½âk G x = H x. W x = U x U x 3 V x Ö. Ï i ω = i ω 1 =, U x (V x, ω x ) (V x, ω 1 x ) 5 Lagrangian f m. dio" 5 ê3 IO 5Ó L x : TG x TG x v L x THx = Id THx Ú L xωx 1 = ωx. ùpï iùýþ G-ØC, Ïd" 5 ê Ñ T G,x gó U G x -ØC, Ïd U L x. Ï z, l L x 'u x 1wCz., iùýþ 1w, ± U x 'u x 1wC Ïd 1w;/mN L : T G H T G H, 3fm T H þ ð, v L ω 1 = L ω. Ïdd½n 7.5 Ú½n 7.6 E, 3 K M(H, G), K H, ü ;/+G U Ú U 1, ±9 ;/+Ó ψ : U U 1 ψ 3 K þ ð, éu? x K 3 T G x þk (ψ ω 1 )(x) = ω 7.5, 3 K M(H, G), K K H, ü ;/+G U U, ±9 ± K ØC;/+ Ó ϕ : U U ϕ ω = ψ ω 1, Ù U U. y3 -U 1 = ψ ϕ 1 (U φ : U U 1 ± K ØC;/+Ó, y.. φ ω 1 = (ϕ 1 ) ψ ω 1 = ω þ, 3E" Ú Lagrangian ž, Œ±ÏLØ G :# G M(G, G), G G, þ ½n E K G f;/+. ë z [1] Adem A., Leida J., Ruan Y., Orbifolds and stringy topology, Cambridge University Press, Cambridge, 27. [2] Cannas da Silva A., Lectures on symplectic geometry, Springer-Verlag, Berlin Heidelberg, 28. [3] Chen B., Hu S., A de Rham model for Chen-Ruan cohomology ring of abelian orbifolds, Math. Ann., 26, 336(1): [4] Chen W., Ruan Y., A new cohomology theory of orbifold, Comm. Math. Phys., 24, 248(1): [5] Chen W., Ruan Y., Orbifold Gromov-Witten theory, Cont. Math., 22, 31: [6] Mackenzie K. C. H., Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, Cambridge, [7] Mackenzie K. C. H., General theory of Lie groupoids and Lie algebroids, Cambridge University Press, Cambridge, 25.

17 Ï Ú«]! yÿ!`: ";/+" ½n 17 [8] Moerdijk I., Pronk D., Orbifolds, sheaves and groupoids, K-theory, 1997, 12 (1): [9] Moerdijk I., Pronk D., Simplcial cohomolgy of orbifolds, Indag. Math. (N.S.), 1999, 1(2): [1] Pflaum M. J., Posthuma H., Tang X., Geometry of orbit spaces of proper Lie groupoids, J. Reine Angew. Math., 214, 214 (694): [11] Warner F. W., Foundations of differentiable manifolds and Lie groups, Springer-Verlag, New York, [12] Satake I., On a generalization of the notion of manifold, Proc. Natl. Acad. Sci. USA, 1956, 42 (6):