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 cyd9966@hotmail.com yÿ oaœæêææ!ôêæ % Ñ bohui@cs.wisc.edu ` Department of Mathematics, University of California Irvine CA ruiw1@math.uci.edu 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 cyd9966@hotmail.com Bohui CHEN School of Mathematics and Yangtze Center of Mathematics, Sichuan University, Chengdu 6164, China bohui@cs.wisc.edu Rui WANG Department of Mathematics, University of California, Irvine, CA, , USA ruiw1@math.uci.edu 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):