{"id":2460,"date":"2019-03-05T18:27:11","date_gmt":"2019-03-05T09:27:11","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=2460"},"modified":"2020-05-16T22:07:48","modified_gmt":"2020-05-16T13:07:48","slug":"hartogs-extension-theorem","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/hartogs-extension-theorem\/","title":{"rendered":"\ud558\ub974\ud1a0\ud06c\uc758 \ud655\uc7a5 \uc815\ub9ac"},"content":{"rendered":"

\ud558\ub974\ud1a0\ud06c\uc758 \ud655\uc7a5 \uc815\ub9ac(Hartog’s extension theorem)\ub294 \\(X\\)\uac00 \\(\\mathbb{C}^n\\)\uc758 \uc5f4\ub9b0\ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(n\\ge 2\\)\uc774\uba70 \\(K\\subseteq X\\)\uac00 \ucef4\ud329\ud2b8\uc774\uace0 \\(X\\setminus K\\)\uac00 \uc5f0\uacb0\uc9d1\ud569\uc77c \ub54c \\(X\\setminus K\\)\uc5d0\uc11c \ud574\uc11d\uc801\uc778 \ud568\uc218\ub294 \\(X\\)\uc5d0\uc11c \ud574\uc11d\uc801\uc778 \ud568\uc218\ub85c \uc720\uc77c\ud558\uac8c \ud655\uc7a5\ub420 \uc218 \uc788\ub2e4\ub294 \uc815\ub9ac\uc774\ub2e4. \uba3c\uc800 \ub2e8\uc21c\ud55c \uacbd\uc6b0\ubd80\ud130 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

\uc9d1\ud569 \\(D\\subseteq \\mathbb{C}^n\\)\uc5d0 \ub300\ud558\uc5ec, \\(\\mathbb{T}^n\\)\uc774 \\(D\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ubd84\ubcc4 \uc5f0\uc0b0\uc73c\ub85c \uc791\uc6a9\ud558\uba74 \\(D\\)\ub97c \ub2e4\uc911\uace0\ub9ac<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\ub54c \ub9cc\uc57d \\(D\\)\uac00 \uc601\uc5ed(\uc5f4\ub9b0 \uc5f0\uacb0\uc9d1\ud569)\uc774\uba74 \\(D\\)\ub97c \ub77c\uc778\ud558\ub974\ud2b8 \uc601\uc5ed<\/span>(Reinhardt domain)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\uc601\uc5ed\uc740 \ub77c\uc778\ud558\ub974\uce20 \uc601\uc5ed\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(f : D \\mapsto M\\)\uacfc \uadf8 \uc5ed\ud568\uc218\uac00 \ubaa8\ub450 \ud574\uc11d\uc801\uc774\uace0 \\(D\\)\uac00 \ub77c\uc778\ud558\ub974\uce20 \uc601\uc5ed\uc77c\uc9c0\ub77c\ub3c4 \\(M\\)\uc740 \ub77c\uc778\ud558\ub974\uce20 \uc601\uc5ed\uc774 \uc544\ub2d0 \uc218 \uc788\ub2e4. \uc608\ucee8\ub300 \uc77c\ubcc0\uc218 \ud574\uc11d\uc5d0\uc11c \ub9ac\ub9cc \uc0ac\uc0c1 \uc815\ub9ac\uc5d0 \uc758\ud558\uba74 \ubcf5\uc18c\ud3c9\uba74 \uc804\uccb4\uac00 \uc544\ub2cc \uc784\uc758\uc758 \ub2e8\uc21c\uc5f0\uacb0\uc601\uc5ed \\(D\\)\ub294 \ub77c\uc778\ud558\ub974\uce20 \uc601\uc5ed\uc778 \ub2e8\uc704\uc6d0\ud310\uc758 \ub0b4\ubd80\uc601\uc5ed\uacfc \ub3d9\uce58\uc774\uc9c0\ub9cc \\(D\\)\ub294 \ub77c\uc778\ud558\ub974\uce20 \uc601\uc5ed\uc774 \uc544\ub2cc \uacbd\uc6b0\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1. (\uace0\ub9ac \uc601\uc5ed\uc5d0\uc11c \ud558\ub974\ud1a0\ud06c\uc758 \ud655\uc7a5 \uc815\ub9ac)<\/span><\/p>\n

\\(n \\ge 2\\)\uc774\uace0 \\(0 < r < R \\le \\infty\\)\uc774\uba70 \\(\\lVert \\cdot \\rVert\\)\uac00 \\(\\mathbb{C}^n\\)\uc758 \ub178\ub984\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0\n\\[B^n ( r,\\,R) := B_{R}^{n} (0) \\setminus \\overline{B_r^n (0)}\\]\n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc81c\ud55c\uc0ac\uc0c1\ub300\uc751\ud568\uc218\n\\[\\rho : O(B_R^n (0)) \\,\\to\\, O(B^n (r,\\,R)),\\, F\\,\\mapsto\\, F | _{B^n (r,\\,R)}\\]\n\ub294 \uc704\uc0c1\uc801 \ub300\uc218\uc5d0 \uad00\ud55c \ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\uba85\ubc31\ud788 \\(B^n (r,\\,R)\\)\ub294 \ub77c\uc778\ud558\ub974\uce20 \uc601\uc5ed\uc774\ub2e4. \\(e_1 ,\\) \\(\\cdots ,\\) \\(e_n\\)\uc774 \\(\\mathbb{C}^n\\)\uc758 \ud45c\uc900\uae30\uc800\ub77c\uace0 \ud558\uace0 \\(\\lambda\\in (r,\\,R)\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(B^n (r,\\,R)\\)\ub294 \\(j\\)\ubc88\uc9f8 \uc131\ubd84\uc774 \\(\\lambda\\)\uc774\uace0 \ub098\uba38\uc9c0 \uc131\ubd84\uc740 \\(0\\)\uc778 \uc784\uc758\uc758 \ubca1\ud130 \\(\\lambda e_j\\)\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c4\ub2e4. \ub530\ub77c\uc11c
\n\\[\\begin{align}
\n\\hat{B^n} (r,\\,R) &= \\left\\{ (t_1 z_1 ,\\, \\cdots ,\\,t_n z_n ) \\,\\vert\\, z\\in B(r,\\,R) ,\\, 0 \\le t_j \\le 1,\\, j=1,\\,\\cdots,\\,n \\right\\} \\\\[6pt]
\n&= \\left\\{ (t_1 z_1 ,\\, \\cdots ,\\, t_n z_n ) \\,\\vert\\, z < \\lVert z \\rVert < R ,\\, 0 \\le t_j \\le 1 ,\\, j=1 ,\\, \\cdots ,\\, n \\right\\}\\\\[6pt]\n&= \\left\\{ z\\in \\mathbb{C}^n \\,\\vert\\, 0\\le \\lVert z \\rVert < R \\right\\} \\\\[6pt]\n&= B_R^n (0)\n\\end{align}\\]\n\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c\n\\[\\rho : O (B_R^n (0)) \\,\\to\\, O(B^n ( r,\\,R)) ,\\, F \\,\\mapsto\\, F|_{B(r,\\,R)}\\]\n\ub294 \ub300\uc218\uc5d0 \uad00\ud55c \ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4. \ub354\uc6b1\uc774 \uc704 \ud568\uc218\ub294 \uc704\uc0c1\uc801 \ub300\uc218\uc5d0 \uad00\ud55c \ub3d9\ud615\uc0ac\uc0c1\uc774 \ub41c\ub2e4.\n<\/span><\/p>\n<\/div>\n

\uc704 \uc815\ub9ac\ub85c\ubd80\ud130 \ub2e4\uc74c\uacfc \uac19\uc740 \ub180\ub77c\uc6b4 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

\n

\ub530\ub984\uc815\ub9ac 1. (\ub2e4\ubcc0\uc218 \ubcf5\uc18c\ud568\uc218\uc758 \ud2b9\uc774\uc810 \uc81c\uac70)<\/span><\/p>\n

2\uac1c \uc774\uc0c1\uc758 \ubcc0\uc218\ub97c \uac00\uc9c4 \ud574\uc11d\uc801 \ubcf5\uc18c\ud568\uc218\ub294 \uace0\ub9bd\uc9c4\uc131\ud2b9\uc774\uc810\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\\(f\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569 \\(U \\subseteq \\mathbb{C}^n\\)\uc5d0\uc11c \ud574\uc11d\uc801\uc774\uace0 \\(n \\ge 2\\)\uc774\uba70 \\(a\\in U\\)\ub77c\uace0 \ud558\uc790. \uc88c\ud45c\uc758 \ud3c9\ud589\uc774\ub3d9\uc744 \uc774\uc6a9\ud558\uba74 \\(a=0\\)\uc774\ub77c\uace0 \ud574\ub3c4 \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\ub294\ub2e4. \\(U\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ubbc0\ub85c \\( 0 < r < R\\)\uc778 \\(r\\)\uc640 \\(R\\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[0 \\notin B^n (r,\\,R) \\subseteq U\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(f\\in O(U)\\)\uc774\ubbc0\ub85c \uc81c\ud55c\uc0ac\uc0c1 \\(f | _{B^n (r,\\,R)}\\)\ub294 \ud574\uc11d\uc801\uc774\ub2e4.<\/p>\n

\ub354\uc6b1\uc774 \uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(f\\)\uc758 \uc815\uc758\uc5ed\uc744 \ud655\uc7a5\ud55c \ud574\uc11d\uc801 \ud568\uc218 \\(F \\in O(B_R^n (0))\\)\uc774 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud2b9\uc774\uc810 \\(0\\)\uc740 \uc81c\uac70 \uac00\ub2a5\ud558\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\ub354 \uc77c\ubc18\uc801\uc778 \uc601\uc5ed\uc5d0\uc11c\uc758 \ud558\ub974\ud1a0\ud06c \ud655\uc7a5 \uc815\ub9ac\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\uc815\ub9ac 2. (\ud558\ub974\ud1a0\ud06c\uc758 \ud655\uc7a5 \uc815\ub9ac)<\/span><\/p>\n

\\(X\\)\uac00 \\(\\mathbb{C}^n\\)\uc5d0\uc11c \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \\(n \\ge 2\\)\uc774\uba70 \\(K \\subseteq X\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\uace0 \\(X\\setminus K\\)\uac00 \uc5f0\uacb0\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc81c\ud55c\uc0ac\uc0c1 \ub300\uc751\ud568\uc218
\n\\[\\rho : O(X) \\,\\to\\, O(X\\setminus K)\\]
\n\ub294 \ubcf5\uc18c \ub300\uc218\uc5d0 \uad00\ud55c \ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\uc77c\uce58 \uc815\ub9ac(identity theorem)\uc5d0 \uc758\ud558\uc5ec \\(\\rho\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc784\uc740 \uc790\uba85\ud558\ubbc0\ub85c \\(\\rho\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc784\uc744 \uc99d\uba85\ud558\uba74 \ucda9\ubd84\ud558\ub2e4. \uc5f4\ub9b0\uc9d1\ud569\uc774\uba74\uc11c \\(K \\subseteq C \\subseteq X\\)\ub85c\uc11c \uc0c1\ub300\uc801 \ucef4\ud329\ud2b8\uc778 \uc9d1\ud569 \\(C\\)\ub97c \ud0dd\ud55c\ub2e4. \ub610\ud55c \ucef4\ud329\ud2b8 \ubc1b\uce68\uc744 \uac16\uace0 \\(\\varphi | _C = 1\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub9e4\ub044\ub7ec\uc6b4 \ud568\uc218 \\(\\varphi : X \\,\\to\\,[0,\\,1]\\)\uc744 \ud0dd\ud55c\ub2e4.<\/p>\n

\\(f\\in O(X\\setminus K)\\)\ub77c\uace0 \ud558\uace0 \ub9e4\ub044\ub7ec\uc6b4 \ud568\uc218 \\(h : X \\,\\to\\,\\mathbb{C}\\)\ub97c
\n\\[z \\,\\mapsto\\,
\n\\begin{cases}
\n(1-\\varphi (z)) f(z) & \\quad \\text{if} \\,\\, z\\in X \\setminus K \\\\[6pt]
\n0 & \\quad \\text{if} \\,\\, z\\in K
\n\\end{cases}\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4. \uadf8\ub7ec\uba74
\n\\[h|_{X\\setminus \\operatorname{supp} \\varphi} = f|_{X\\setminus \\operatorname{supp} \\varphi} \\tag{1}\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ud2b9\ud788 \\(\\operatorname{supp}\\varphi\\)\uc758 \ubc14\uae65\uc5d0\uc11c \\(d ‘ ‘ f = d ‘ ‘ h = 0\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \\(d ‘ ‘ h\\)\ub294 \\(X\\) \ubc16\uc5d0\uc11c \\(d ‘ ‘ h \\in \\epsilon^{0,\\,1} (\\mathbb{C}^n )\\)\uc73c\ub85c\uc11c \ud655\uc7a5 \uac00\ub2a5\ud558\ub2e4. \\(d ‘ ‘ h\\)\uc758 \ubc1b\uce68\uc740 \ucef4\ud329\ud2b8 \uc9d1\ud569 \\(\\operatorname{supp} \\varphi\\)\uc758 \ub2eb\ud78c \ubd80\ubd84\uc9d1\ud569\uc774\ubbc0\ub85c \ucef4\ud329\ud2b8\uc774\ub2e4. \ub530\ub77c\uc11c \ucef4\ud329\ud2b8 \ubc1b\uce68\uc744 \uac16\ub294 \ud568\uc218 \\(g\\in \\epsilon (\\mathbb{C}^n )\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(d ‘ ‘ g = d ‘ ‘ h\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(g\\)\ub294 \\(\\mathbb{C}^n \\setminus \\operatorname{supp} \\varphi\\) \uc704\uc5d0\uc11c \ud574\uc11d\uc801\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(\\operatorname{supp} \\varphi\\) \ubc16\uc5d0\uc11c \\(d ‘ ‘ h = 0\\)\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \\(d ‘ ‘ h = d ‘ ‘ g\\)\uc774\ubbc0\ub85c \ud574\uc11d\uc801 \ud568\uc218 \\(F : X \\,\\to\\,\\mathbb{C}\\)\ub97c
\n\\[F := h-g\\]
\n\ub85c\uc11c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\uc774\uc81c \\(F\\)\uac00 \\(f\\)\uc758 \ud574\uc11d\uc801 \ud655\uc7a5\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc790. \\(W\\)\uac00 \\(\\mathbb{C}^n \\setminus \\operatorname{supp} \\varphi\\)\uc758 \uc720\uc77c\ud55c \uc720\uacc4\uac00 \uc544\ub2cc \uc5f0\uacb0\uc131\ubd84\uc774\ub77c\uace0 \ud558\uc790. \\(\\operatorname{supp} g\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ubbc0\ub85c \uc77c\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(g\\)\ub294 \\(W\\)\uc5d0\uc11c \\(0\\)\uc774 \ub41c\ub2e4. \\(X\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub77c\ub294 \uc0ac\uc2e4\uacfc
\n\\[\\partial W \\subseteq \\partial (\\mathbb{C}^n \\setminus \\operatorname{supp} \\varphi )
\n= \\partial \\operatorname{supp}\\varphi \\subseteq X\\]
\n\ub77c\ub294 \uc0ac\uc2e4\ub85c\ubd80\ud130 \\(X\\cap W \\ne \\varnothing\\)\uc744 \uc5bb\ub294\ub2e4. \\(X\\cap W\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub77c\ub294 \uac83\uacfc
\n\\[\\varnothing \\ne X\\cap W \\subseteq X\\cap (\\mathbb{C}^n \\setminus \\operatorname{supp} \\varphi ) = X \\setminus \\operatorname{supp} \\varphi\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc5fc\ub450\uc5d0 \ub450\uace0 \uc99d\uba85\uc744 \uc774\uc5b4\uac00\uc790.
\n\\[F | _{X\\cap W} = h | _{X \\cap W} = f|_{X \\cap W}\\]
\n\uc774\uba70, \\(\\varphi | _K =1\\)\uc774\ubbc0\ub85c \\(K \\subseteq \\operatorname{supp} \\varphi\\)\uc774\uace0, \ub530\ub77c\uc11c \\(X \\setminus \\operatorname{supp} \\varphi \\subseteq X \\setminus K\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc815\ub9ac\uc758 \uc804\uc81c \uc870\uac74\uc5d0 \uc758\ud558\uc5ec \\(X\\setminus K\\)\ub294 \uc601\uc5ed\uc774\uace0 \\(F | _{X\\setminus K}\\)\uc640 \\(f\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc5f4\ub9b0 \ubd80\ubd84\uc9d1\ud569 \\(X \\cap W\\) \uc704\uc5d0\uc11c \uc11c\ub85c \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc77c\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
\n\\[F|_{X\\setminus K} = f\\]
\n\uc774\ubbc0\ub85c \uc6d0\ud558\ub294 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.
\n<\/span>\n<\/p>\n<\/div>\n

\n

\ub530\ub984\uc815\ub9ac 2. (\ub2e4\ubcc0\uc218 \ubcf5\uc18c\ud574\uc11d\uc801 \ud568\uc218\uc758 \uc601\uc9d1\ud569\uc758 \ube44\ucef4\ud329\ud2b8\uc131)<\/span><\/p>\n

\\(D\\)\uac00 \\(\\mathbb{C}^n\\)\uc5d0\uc11c\uc758 \uc601\uc5ed\uc774\uba70 \\(n \\ge 2\\)\uc774\uace0 \\(f\\in O(D)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc601\uc9d1\ud569 \\(N(f)\\)\uc740 \ucef4\ud329\ud2b8\uac00 \uc544\ub2c8\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\\(f\\)\uac00 \\(f=0\\)\uc778 \uc0c1\uc218\ud568\uc218\ub77c\uba74 \uc790\uba85\ud558\uac8c \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(N(f)\\)\uc758 \uc5ec\ucc28\uc6d0\uc740 \\(1\\)\uc774\uace0, \\(D\\setminus N(f)\\)\ub294 \uc5f0\uacb0\uc9d1\ud569\uc774\uba70, \ud568\uc218 \\(1\/f\\)\ub294 \\(D\\setminus N(f)\\)\uc5d0\uc11c \ud574\uc11d\uc801\uc774\ub2e4. \ub9cc\uc57d \\(N(f)\\)\uac00 \ucef4\ud329\ud2b8\ub77c\uba74 \ud558\ub974\ud1a0\ud06c\uc758 \ud655\uc7a5 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \ud574\uc11d\uc801 \ud568\uc218 \\(F\\in O(D)\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[F|_{D\\setminus N(f)} = \\frac{1}{f}\\]
\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ud0a8\ub2e4. \uc591\ubcc0\uc5d0 \\(f\\)\ub97c \uacf1\ud558\uba74 \\(D\\setminus N(f)\\) \uc704\uc5d0\uc11c \\(F \\cdot f = 1\\)\uc744 \uc5bb\ub294\ub2e4. \\(f\\)\ub294 \uc0c1\uc218\ud568\uc218 \\(0\\)\uc774 \uc544\ub2c8\ubbc0\ub85c \\(D\\setminus N(f)\\)\ub294 \\(D\\setminus N(f)\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc5f4\ub9b0 \ubd80\ubd84\uc9d1\ud569\uc774\ub2e4. \ub530\ub77c\uc11c \uc77c\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(D\\) \uc804\uccb4\uc5d0\uc11c \\(F \\cdot f = 1\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ub098 \uc774\uac83\uc740 \\(f|_{N(f)} = 0\\)\uc774\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\ubcf4\uae30 1.<\/span>
\n\ub530\ub984\uc815\ub9ac 2\ub294 \uc2e4\ud574\uc11d\uc801 \ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub294 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \ud568\uc218
\n\\[f : \\mathbb{C}^n \\,\\to\\,\\mathbb{C} ,\\, z\\,\\mapsto\\, \\lVert z \\rVert _2 ^2 -1\\]
\n\uc758 \uc601\uc9d1\ud569\uc740 \\(\\mathbb{C}^n\\)\uc5d0\uc11c\uc758 \ucef4\ud329\ud2b8 \ub2e8\uc704\uad6c\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\ubcf4\uae30 2.<\/span>
\n\ub2e4\ud56d\ubc29\uc815\uc2dd \\(p(z_1 ,\\, \\cdots ,\\, z_n ) = z_1\\)\uc758 \uc601\uc9d1\ud569\uc740 \\(\\left\\{ 0 \\right\\} \\times \\mathbb{C}^{n-1}\\)\uc774\ub2e4. \\(n > 1\\)\uc77c \ub54c \uc774 \uc9d1\ud569\uc740 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\ud558\ub974\ud1a0\ud06c\uc758 \ud655\uc7a5 \uc815\ub9ac\uc758 \uc0c1\uc138\ud55c \uc99d\uba85\uc744 \uc54c\uace0\uc790 \ud558\ub294 \uc0ac\ub78c\uc740 \ub2e4\uc74c \ubb38\uc11c\ub97c \ucc38\uace0\ud558\uae30 \ubc14\ub780\ub2e4:<\/p>\n