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\u6570\u5b66\u3067\u306f\u3001\u8907\u7d20\u5fae\u5206\u5f62\u5f0f(complex differential form)\u306f\u3001\u8907\u7d20\u6570\u4fc2\u6570\u3092\u6301\u3064\u591a\u69d8\u4f53\uff08\u901a\u5e38\u306f\u8907\u7d20\u591a\u69d8\u4f53\uff09\u4e0a\u306e\u5fae\u5206\u5f62\u5f0f\u3067\u3042\u308b\u3002 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\u500b\u306e\u5916\u5fae\u5206\u306e\u30a6\u30a7\u30c3\u30b8\u7a4d\u3067\u3042\u308b\u3002(p,q)-\u5f62\u5f0f\u306e\u96c6\u5408\u306f\u3001\u57fa\u672c\u7684\u7814\u7a76\u5bfe\u8c61\u3067\u3042\u308a\u3001k-\u5f62\u5f0f\u4ee5\u4e0a\u306b\u3001\u591a\u69d8\u4f53\u306e\u5e7e\u4f55\u5b66\u7684\u69cb\u9020\u3092\u3088\u308a\u3088\u304f\u53cd\u6620\u5b9a\u3059\u308b\u3002\u305f\u3068\u3048\u3070\u3001\u30db\u30c3\u30b8\u7406\u8ad6\u304c\u9069\u7528\u53ef\u80fd\u306a\u5834\u5408\u306f\u3001\uff08k-\u5f62\u5f0f\u3088\u308a\u3082\uff09\u826f\u3044\u591a\u69d8\u4f53\u306e\u69cb\u9020\u304c\u5b58\u5728\u3059\u308b\u3002 \u8907\u7d20\u591a\u69d8\u4f53\u4e0a\u306e\u5fae\u5206\u5f62\u5f0f[\u7de8\u96c6] M \u304c\u8907\u7d20\u591a\u69d8\u4f53\u3067\u3042\u308b\u3068\u3059\u308b\u3068\u3001n \u500b\u306e\u8907\u7d20\u5909\u6570\u51fd\u6570 z1,…,zn 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differential form)\u306f\u3001\u8907\u7d20\u6570\u4fc2\u6570\u3092\u6301\u3064\u591a\u69d8\u4f53\uff08\u901a\u5e38\u306f\u8907\u7d20\u591a\u69d8\u4f53\uff09\u4e0a\u306e\u5fae\u5206\u5f62\u5f0f\u3067\u3042\u308b\u3002\u8907\u7d20\u5fae\u5206\u5f62\u5f0f\u306f\u3001\u5fae\u5206\u5e7e\u4f55\u5b66\u306b\u304a\u3044\u3066\u5e83\u304f\u5fdc\u7528\u3055\u308c\u3066\u3044\u308b\u3002\u8907\u7d20\u591a\u69d8\u4f53\u4e0a\u3067\u306e\u4ee3\u6570\u5e7e\u4f55\u5b66\u3084\u30b1\u30fc\u30e9\u30fc\u5e7e\u4f55\u5b66\u3084\u30db\u30c3\u30b8\u7406\u8ad6\u306e\u591a\u304f\u3067\u3001\u8907\u7d20\u5fae\u5206\u5f62\u5f0f\u306f\u91cd\u8981\u306a\u57fa\u672c\u3068\u3057\u306a\u3063\u3066\u3044\u308b\u3002\u8907\u7d20\u591a\u69d8\u4f53\u3067\u306a\u3044\u5834\u5408\u3067\u3082\u3001\u8907\u7d20\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\u6982\u8907\u7d20\u69cb\u9020\u3084\u30b9\u30d4\u30ce\u30eb\u306e\u7406\u8ad6\u3084CR\u69cb\u9020\u306e\u7814\u7a76\u3067\u91cd\u8981\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u3066\u3044\u308b\u3002\u5178\u578b\u7684\u306b\u306f\u3001\u8907\u7d20\u5fae\u5206\u5f62\u5f0f\u306f\u5bb9\u6613\u306b\u671f\u5f85\u3055\u308c\u308b\u5206\u89e3\u3092\u6301\u3064\u8003\u3048\u3089\u308c\u3066\u3044\u308b\u3002\u305f\u3068\u3048\u3070\u3001\u8907\u7d20\u591a\u69d8\u4f53\u4e0a\u3067\u306f\u3001\u4efb\u610f\u306e k-\u5f62\u5f0f\u304c\u4e00\u610f\u306b (p,q)-\u5f62\u5f0f\u306b\u5206\u89e3\u3059\u308b\u3002(p,q)-\u5f62\u5f0f\u3068\u306f\u3001\u5927\u307e\u304b\u306b\u306f\u3001\u6b63\u5247\u5ea7\u6a19\u306e p \u500b\u306e\u5916\u5fae\u5206\u3068\u3001\u305d\u306e\u8907\u7d20\u5171\u5f79\u306e q \u500b\u306e\u5916\u5fae\u5206\u306e\u30a6\u30a7\u30c3\u30b8\u7a4d\u3067\u3042\u308b\u3002(p,q)-\u5f62\u5f0f\u306e\u96c6\u5408\u306f\u3001\u57fa\u672c\u7684\u7814\u7a76\u5bfe\u8c61\u3067\u3042\u308a\u3001k-\u5f62\u5f0f\u4ee5\u4e0a\u306b\u3001\u591a\u69d8\u4f53\u306e\u5e7e\u4f55\u5b66\u7684\u69cb\u9020\u3092\u3088\u308a\u3088\u304f\u53cd\u6620\u5b9a\u3059\u308b\u3002\u305f\u3068\u3048\u3070\u3001\u30db\u30c3\u30b8\u7406\u8ad6\u304c\u9069\u7528\u53ef\u80fd\u306a\u5834\u5408\u306f\u3001\uff08k-\u5f62\u5f0f\u3088\u308a\u3082\uff09\u826f\u3044\u591a\u69d8\u4f53\u306e\u69cb\u9020\u304c\u5b58\u5728\u3059\u308b\u3002\u8907\u7d20\u591a\u69d8\u4f53\u4e0a\u306e\u5fae\u5206\u5f62\u5f0f[\u7de8\u96c6]M \u304c\u8907\u7d20\u591a\u69d8\u4f53\u3067\u3042\u308b\u3068\u3059\u308b\u3068\u3001n \u500b\u306e\u8907\u7d20\u5909\u6570\u51fd\u6570 z1,…,zn \u304b\u3089\u306a\u308b\u5c40\u6240\u5ea7\u6a19\u5909\u63db\u304c\u5b58\u5728\u3057\u3001\u3042\u308b\u70b9\u306e\u8fd1\u508d\u304b\u3089\u5225\u306e\u70b9\u306e\u8fd1\u508d\u3078\u306e\u5ea7\u6a19\u5909\u63db\u304c\u8907\u6570\u306e\u5909\u6570 zi \u306e\u6b63\u5247\u51fd\u6570\u3068\u306a\u308b\u3002\u8907\u7d20\u5fae\u5206\u5f62\u5f0f\u306e\u7a7a\u9593\u306f\u3001\u8c4a\u304b\u306a\u69cb\u9020\u3092\u6301\u3063\u3066\u3044\u3066\u3001\u57fa\u672c\u7684\u306b\u306f\u3001\u5ea7\u6a19\u5909\u63db\u306e\u51fd\u6570\u304c\u6ed1\u3089\u304b(smooth)\u3067\u3042\u308b\u3053\u3068\u3088\u308a\u3082\u6b63\u5247\u3067\u3042\u308b\u3053\u3068\u306b\u4f9d\u5b58\u3057\u3066\u3044\u308b\u30021-\u5f62\u5f0f[\u7de8\u96c6]1-\u5f62\u5f0f\u306e\u5834\u5408\u304b\u3089\u306f\u3058\u3081\u308b\u3002\u6700\u521d\u306b\u3001\u305d\u308c\u305e\u308c\u306e j \u306b\u3064\u3044\u3066\u8907\u7d20\u6570\u306e\u5ea7\u6a19\u3092\u5b9f\u90e8\u3068\u865a\u90e8 zj = xj + iyj \u3078\u5206\u89e3\u3059\u308b\u3002dzj=dxj+idyj,dz\u00afj=dxj\u2212idyj,{displaystyle dz^{j}=dx^{j}+idy^{j},quad d{bar {z}}^{j}=dx^{j}-idy^{j},}\u3068\u304a\u304f\u3068\u3001\u8907\u7d20\u6570\u4fc2\u6570\u3092\u6301\u3064\u3059\u3079\u3066\u306e\u5fae\u5206\u5f62\u5f0f\u306f\u3001\u548c\u2211j=1nfjdzj+gjdz\u00afj.{displaystyle sum _{j=1}^{n}f_{j}dz^{j}+g_{j}d{bar {z}}^{j}.}\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u304c\u5206\u304b\u308b\u3002\u03a91,0 \u3092 dz{displaystyle dz} \u306e\u307f\u3092\u542b\u3080\u8907\u7d20\u5fae\u5206\u5f62\u5f0f\u306e\u7a7a\u9593\u3068\u3057\u3001\u03a90,1 \u3092 dz\u00af{displaystyle d{bar {z}}} \u306e\u307f\u3092\u542b\u3080\u7a7a\u9593\u3068\u3059\u308b\u3068\u3001\u30b3\u30fc\u30b7\u30fc\u30fb\u30ea\u30fc\u30de\u30f3\u306e\u65b9\u7a0b\u5f0f\u306b\u3088\u308a\u7a7a\u9593 \u03a91,0 \u3068 \u03a90,1 \u306f\u6b63\u5247\u5ea7\u6a19\u5909\u63db\u306e\u4e0b\u3067\u4e0d\u5909\u3067\u3042\u308b\u3002\u8a00\u3044\u63db\u3048\u308b\u3068\u3001\u7570\u306a\u308b\u6b63\u5247\u5ea7\u6a19\u7cfb wi \u3092\u9078\u3093\u3067\u3082\u3001\u03a90,1 \u306e\u5143\u306b\u3088\u308b\u5909\u63db\u3068\u3068\u3082\u306b\u3001\u03a91,0 \u306e\u5143\u3082\u30c6\u30f3\u30bd\u30eb\u7684\u306b\u5909\u63db\u3059\u308b\u3002\u3053\u306e\u3088\u3046\u306b\u7a7a\u9593 \u03a90,1 \u3068 \u03a91,0 \u306f\u8907\u7d20\u591a\u69d8\u4f53\u4e0a\u306e\u8907\u7d20\u30d9\u30af\u30c8\u30eb\u5834\u3092\u5b9a\u7fa9\u3059\u308b\u3002\u9ad8\u6b21\u306e\u5f62\u5f0f[\u7de8\u96c6]\u8907\u7d20\u5fae\u5206\u5f62\u5f0f\u306e\u30a6\u30a7\u30c3\u30b8\u7a4d\u306f\u3001\u5b9f\u5f62\u5f0f\u3068\u540c\u69d8\u306a\u65b9\u6cd5\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002p \u3068 q \u3092\u975e\u8ca0\u306a\u6574\u6570 \u2264 n \u306e\u30da\u30a2\u3068\u3059\u308b\u3068\u3001(p,q)-\u5f62\u5f0f\u306e\u7a7a\u9593 \u03a9p,q \u306f\u3001\u03a91,0 \u306e p \u500b\u306e\u5143\u3068 \u03a90,1 \u306e q \u500b\u306e\u5143\u306e\u30a6\u30a7\u30c3\u30b8\u7a4d\u306e\u7dda\u578b\u7d50\u5408\u306b\u3088\u308a\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u8a18\u53f7\u3067\u66f8\u304f\u3068\u3001\u03a9p,q=\u03a91,0\u2227\u22ef\u2227\u03a91,0\u2227\u03a90,1\u2227\u22ef\u2227\u03a90,1{displaystyle Omega ^{p,q}=Omega ^{1,0}wedge dotsb wedge Omega ^{1,0}wedge Omega ^{0,1}wedge dotsb wedge Omega ^{0,1}}\u3067\u3042\u308a\u3001\u3053\u3053\u306b \u03a91,0 \u306e p \u500b\u306e\u8981\u7d20\u3001\u03a90,1 \u306e q \u500b\u306e\u8981\u7d20\u304c\u5b58\u5728\u3059\u308b\u3002\u307e\u3055\u306b\u30011-\u5f62\u5f0f\u306e 2\u3064\u306e\u7a7a\u9593\u304c\u3001\u5ea7\u6a19\u306e\u6b63\u5247\u306a\u5909\u63db\u306e\u4e0b\u3067\u5b89\u5b9a\u3067\u3042\u308b\u306e\u3067\u3001\u30d9\u30af\u30c8\u30eb\u30d0\u30f3\u30c9\u30eb\u3092\u6c7a\u5b9a\u3059\u308b\u3002Ek \u3092\u5168\u6b21\u6570 k \u306e\u5168\u8907\u7d20\u5fae\u5206\u5f62\u5f0f\u306e\u7a7a\u9593\u3068\u3059\u308b\u3068\u3001Ek \u306e\u5404\u3005\u306e\u5143\u306f\u4e00\u610f\u306a\u65b9\u6cd5\u3067 p + q = k \u3067\u3042\u308b\u7a7a\u9593 \u03a9p,q \u306e\u5143\u306e\u7dda\u578b\u7d50\u5408\u3067\u8868\u308f\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3088\u308a\u7c21\u6f54\u306b\u8a00\u3046\u3068\u3001\u76f4\u7a4d\u5206\u89e3Ek=\u03a9k,0\u2295\u03a9k\u22121,1\u2295\u22ef\u2295\u03a91,k\u22121\u2295\u03a90,k=\u2a01p+q=k\u03a9p,q{displaystyle E^{k}=Omega ^{k,0}oplus Omega ^{k-1,1}oplus dotsb oplus Omega ^{1,k-1}oplus Omega ^{0,k}=bigoplus _{p+q=k}Omega ^{p,q}}\u3068\u306a\u308b\u3002\u76f4\u7a4d\u5206\u89e3\u306f\u6b63\u5247\u5ea7\u6a19\u5909\u63db\u306e\u4e0b\u306b\u5b89\u5b9a\u3067\u3042\u308b\u304b\u3089\u3001\u76f4\u7a4d\u5206\u89e3\u306f\u30d9\u30af\u30c8\u30eb\u30d0\u30f3\u30c9\u30eb\u306e\u5206\u89e3\u3092\u3082\u6c7a\u5b9a\u3059\u308b\u3002\u7279\u306b\u3001\u5404\u3005\u306e k = p + q \u3067\u3042\u308b p \u3068 q \u306b\u5bfe\u3057\u3001\u30d9\u30af\u30c8\u30eb\u30d0\u30f3\u30c9\u30eb\u306e\u6a19\u6e96\u7684\u306a\u5c04\u5f71\u03c0p,q:Ek\u2192\u03a9p,q{displaystyle pi ^{p,q}:E^{k}rightarrow Omega ^{p,q}}\u304c\u5b58\u5728\u3059\u308b\u3002\u30c9\u30eb\u30dc\u30fc\u4f5c\u7528\u7d20[\u7de8\u96c6]\u901a\u5e38\u306e\u5916\u5fae\u5206\u306f\u3001\u5207\u65ad\u306e\u5199\u50cf d:Ek\u2192Ek+1 \u3092\u5b9a\u7fa9\u3059\u308b\u3002\u3053\u306e\u5199\u50cf\u3092 \u03a9p,q \u306e\u5207\u65ad\u306b\u9650\u5b9a\u3059\u308b\u3068\u3001\u5b9f\u969b d:\u03a9p,q\u2192\u03a9p+1,q + \u03a9p,q+1 \u3067\u3042\u308b[\u8981\u8aac\u660e] \u5916\u5fae\u5206\u306f\u591a\u69d8\u4f53\u306e\u3088\u308a\u53b3\u5bc6\u306a\u8907\u7d20\u69cb\u9020\u3092\u53cd\u6620\u306f\u3057\u306a\u3044\u3002d \u3068\u524d\u306e\u30b5\u30d6\u30bb\u30af\u30b7\u30e7\u30f3\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u3053\u3068\u3092\u4f7f\u3046\u3068\u3001\u30c9\u30eb\u30dc\u30fc\u4f5c\u7528\u7d20(Dolbeault operators)\u2202=\u03c0p+1,q\u2218d:\u03a9p,q\u2192\u03a9p+1,q,\u2202\u00af=\u03c0p,q+1\u2218d:\u03a9p,q\u2192\u03a9p,q+1{displaystyle partial =pi ^{p+1,q}circ d:Omega ^{p,q}rightarrow Omega ^{p+1,q},quad {bar {partial }}=pi ^{p,q+1}circ d:Omega ^{p,q}rightarrow Omega ^{p,q+1}}\u3068\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u308c\u3089\u306e\u4f5c\u7528\u7d20\u3092\u5c40\u6240\u5ea7\u6a19\u3067\u8868\u308f\u3059\u305f\u3081\u3001\u03b1=\u2211|I|=p,|J|=q\u00a0fIJdzI\u2227dz\u00afJ\u2208\u03a9p,q{displaystyle alpha =sum _{|I|=p,|J|=q} f_{IJ},dz^{I}wedge d{bar {z}}^{J}in Omega ^{p,q}}\u3068\u304a\u304f\u3002\u3053\u3053\u306b I \u3068 J \u306f\u8907\u6570\u306e\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u3092\u6301\u3063\u3066\u3044\u308b\u3002\u3059\u308b\u3068\u3001\u2202\u03b1=\u2211|I|,|J|\u2211\u2113\u2202fIJ\u2202z\u2113dz\u2113\u2227dzI\u2227dz\u00afJ{displaystyle partial alpha =sum _{|I|,|J|}sum _{ell }{frac {partial f_{IJ}}{partial z^{ell }}},dz^{ell }wedge dz^{I}wedge d{bar {z}}^{J}}\u2202\u00af\u03b1=\u2211|I|,|J|\u2211\u2113\u2202fIJ\u2202z\u00af\u2113dz\u00af\u2113\u2227dzI\u2227dz\u00afJ{displaystyle {bar {partial }}alpha =sum _{|I|,|J|}sum _{ell }{frac {partial f_{IJ}}{partial {bar {z}}^{ell }}}d{bar {z}}^{ell }wedge dz^{I}wedge d{bar {z}}^{J}}\u304c\u6210\u308a\u7acb\u3064\u3002\u6b21\u306e\u6027\u8cea\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u5206\u304b\u308b\u3002d=\u2202+\u2202\u00af{displaystyle d=partial +{bar {partial }}}\u22022=\u2202\u00af2=\u2202\u2202\u00af+\u2202\u00af\u2202=0.{displaystyle partial ^{2}={bar {partial }}^{2}=partial {bar {partial }}+{bar {partial }}partial =0.}\u3053\u308c\u3089\u306e\u4f5c\u7528\u3068\u6027\u8cea\u306f\u3001\u30c9\u30eb\u30dc\u30fc\u30b3\u30db\u30e2\u30ed\u30b8\u30fc\u306e\u57fa\u790e\u3068\u30db\u30c3\u30b8\u7406\u8ad6\u306e\u69d8\u3005\u306a\u9762\u3092\u4e0e\u3048\u308b\u3002\u6b63\u5247\u5f62\u5f0f[\u7de8\u96c6]\u5404\u3005\u306e p \u306b\u5bfe\u3057\u3001\u6b63\u5247 p-\u5f62\u5f0f\u306f\u30d0\u30f3\u30c9\u30eb \u03a9p,0 \u306e\u6b63\u5247\u5207\u65ad\u3067\u3042\u308b\u3002\u5c40\u6240\u5ea7\u6a19\u3067\u306f\u3001\u6b63\u5247 p-\u5f62\u5f0f\u306f\u3001\u03b1=\u2211|I|=pfIdzI{displaystyle alpha =sum _{|I|=p}f_{I},dz^{I}}\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u3053\u306b fI \u306f\u6b63\u5247\u51fd\u6570\u3067\u3042\u308b\u3002\u540c\u3058\u3053\u3068\u3067\u3042\u308b\u304c\u3001(p,0)-\u5f62\u5f0f \u03b1 \u304c\u6b63\u5247\u3067\u3042\u308b\u3053\u3068\u3068\u3001\u2202\u00af\u03b1=0.{displaystyle {bar {partial }}alpha =0.}\u306f\u540c\u5024\u3067\u3042\u308b\u3002\u6b63\u5247 p-\u5f62\u5f0f\u306e\u5c64\u306f\u3001\u3088\u304f \u03a9p \u3068\u8868\u308f\u3055\u308c\u308b\u304c\u3001\u6df7\u4e71\u3092\u6642\u3005\u62db\u304f\u306e\u3067\u3001\u4ee3\u308f\u308a\u306e\u8a18\u6cd5\u3092\u4f7f\u3046\u3088\u3046\u306b\u306a\u3063\u3066\u304d\u3066\u3044\u308b\u3002\u53c2\u7167\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki4\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki4\/2020\/08\/03\/%e8%a4%87%e7%b4%a0%e5%be%ae%e5%88%86%e5%bd%a2%e5%bc%8f-wikipedia\/#breadcrumbitem","name":"\u8907\u7d20\u5fae\u5206\u5f62\u5f0f – Wikipedia"}}]}]