{"id":360,"date":"2017-01-18T09:33:14","date_gmt":"2017-01-18T09:33:14","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=360"},"modified":"2017-01-18T09:39:46","modified_gmt":"2017-01-18T09:39:46","slug":"fujihezuoye","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=360","title":{"rendered":"\u590d\u51e0\u4f55\u4f5c\u4e1a"},"content":{"rendered":"

1. \u7b2c\u4e00\u6b21<\/a><\/span>\n\n

  1. \u8bc1\u660e\u5168\u7eaf\u51fd\u6570$f=u+iv$\u7684\u5b9e\u90e8$u$\u4e0e\u865a\u90e8$v$\u662f\u8c03\u548c\u7684.<\/li>
  2. \u5bf9\u5168\u7eaf\u51fd\u6570\u8bc1\u660e\u6781\u5927\u503c\u539f\u7406.<\/li>
  3. \u5047\u8bbe$U\\subset\\mathbb{C}^n$\u662f\u4e00\u4e2a\u5f00\u96c6, \u800c$f:U\\to\\mathbb{C}$\u662f\u5168\u7eaf\u7684. \u8bc1\u660e\u5bf9$n\\geq2$, \u96f6\u70b9\u96c6$Z(f)$\u4e0d\u53ef\u80fd\u662f\u4e00\u4e2a\u5355\u70b9\u96c6. \u7c7b\u4f3c\u5730, \u8bc1\u660e\u5bf9\u5168\u7eaf\u51fd\u6570$f:\\mathbb{C}^n\\to\\mathbb{C}$, $n\\geq2$\u4ee5\u53ca$w=\\mathrm{Im}(f)$, \u5b58\u5728$z\\in f^{-1}(w)$, \u4f7f\u5f97$\\|z\\|>0$.<\/li>
  4. \u4ee4$\\set{f_i}$\u662f\u5f00\u96c6$U\\subset\\mathbb{C}^n$\u4e0a\u5217\u5168\u7eaf\u51fd\u6570, \u5047\u8bbe\u5bf9\u4efb\u4f55$V\\subset\\subset U$, \u6709$f_i$\u5728$V$\u4e0a\u4e00\u81f4\u6536\u655b\u5230$g$. \u8bc1\u660e$g$\u4e5f\u662f\u5168\u7eaf\u7684.<\/li>
  5. \u4ee4$f:U\\to V$\u662f\u4e00\u4e2a\u5168\u7eaf\u6620\u5c04. \u8bc1\u660e\u81ea\u7136\u62c9\u56de\u6620\u7167$f^*:\\mathcal{A}^k(V)\\to\\mathcal{A}^k(U)$\u53c8\u5230\u4e86$\\mathcal{A}^{p,q}(U)$\u5230$\\mathcal{A}^{p,q}(V)$\u7684\u6620\u5c04. \u8fd9\u4e5f\u8868\u660e$f^*\\partial\\alpha=\\partial f^*\\alpha$, $f^*\\bar\\partial\\alpha=\\partial f^*\\alpha$.<\/li>
  6. \u4ee4$B\\subset\\mathbb{C}^n$\u662f\u591a\u5706\u76d8\u4e14$\\alpha\\in\\mathcal{A}^{p,q}$\u662f$d$-\u95ed\u7684, $p,q\\geq1$. \u8bc1\u660e\u5b58\u5728$\\gamma\\in\\mathcal{A}^{p-1,q-1}(B)$\u4f7f\u5f97$\\partial\\bar\\partial\\gamma=\\alpha$.<\/li><\/ol>
    \n2. \u7b2c\u4e8c\u6b21    <\/a><\/span>\n\n
    1. \u4ee4$(M,J)$\u662f\u590d\u6d41\u5f62\u4e14$X$\u662f$M$\u4e0a\u4e00\u4e2a\u5411\u91cf\u573a. \u8bc1\u660e, \u5bf9\u4efb\u4f55\u5411\u91cf\u573a$Y$\u6210\u7acb$[X,JY]=J[X,Y]$\u5f53\u4e14\u4ec5\u5f53$X^{1,0}=\\frac{1}{2}(X-\\sqrt{-1}JX)$\u662f\u5168\u7eaf\u7684.<\/li>
    2. \u4ee4$M$\u662f$2n$\u7ef4\u7684\u5149\u6ed1\u6d41\u5f62, \u4e14\u5177\u6709\u590d\u7ed3\u6784$J$. \u8bc1\u660e$J$\u7684\u7ed5\u7387(torsion)\u5b9a\u4e49\u4e3a
      \n$$
      \n\\tau(X,Y)=[JX,JY]-[X,Y]-J[X,JY]-J[JX,Y].
      \n$$
      \n\u8bc1\u660e
      1. $\\tau$\u662f\u4e00\u4e2a\u5f20\u91cf.<\/li>
      2. $\\tau(X,Y)=-\\tau(Y,X)$, $\\tau(JX,Y)=-J\\tau(Y,X)$.<\/li>
      3. \u5982\u679c$n=1$, \u5219$\\tau=0$, \u4ece\u800c\u7531Newlander-Nirenberg\u5b9a\u7406\u77e5$J$\u662f\u53ef\u79ef\u7684.
        \n
        \u6ce8\u8bb0 1<\/span>.<\/span> \u7531\u4e8e\u5728\u4efb\u4f55\u53ef\u5b9a\u5411\u4e8c\u7ef4\u9ece\u66fc\u6d41\u5f62$(\\Sigma,g)$\u4e0a, \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49$J$\u4e3a\u9006\u65f6\u9488\u65cb\u8f6c$\\pi\/2$, \u56e0\u6b64$\\Sigma$\u662f\u4e00\u4e2a\u9ece\u66fc\u9762.
        \n<\/div><\/li>
      4. \u8bc1\u660e\u4efb\u4f55\u4ece$\\mathbb{P}^n$\u51fa\u53d1\u5230\u590d\u73af\u9762\u7684\u5168\u7eaf\u6620\u5c04, \u82e5$n=1$\u90fd\u662f\u5e38\u503c\u7684. \u82e5$n>1$\u53c8\u5982\u4f55?<\/li>
      5. \u8bc1\u660eP65\u9875\u7684\u95ee\u98982.1.12.<\/li>
      6. \u5047\u8bbe\u7b49\u6e29\u5750\u6807\u7cfb(isothermal coordinate)\u7684\u5b58\u5728\u6027, \u8bc1\u660e\u4efb\u4f55\u53ef\u5b9a\u54112\u7ef4\u9ece\u66fc\u6d41\u5f62\u81ea\u7136\u662f\u590d1\u7ef4\u7684\u590d\u6d41\u5f62(\u5373\u9ece\u66fc\u9762).<\/li>
      7. \u5047\u8bbe$\\Sigma$\u662f\u9ece\u66fc\u9762. \u8bc1\u660e$\\Sigma$\u4e0a\u7684\u4e00\u4e2a\u4e9a\u7eaf\u51fd\u6570\u5b9a\u4e49\u4e86\u4ece$\\Sigma$\u5230$\\mathbb{P}^1$\u7684\u5168\u7eaf\u6620\u7167, \u53cd\u4e4b\u4ea6\u7136.<\/li><\/ol><\/li><\/ol>\n

        3. \u7b2c\u4e09\u6b21<\/a><\/span>\n\n

        1. \u7528Newlander-Nirenberg\u5b9a\u7406\u8bc1\u660e\u95ee\u98982.6.10.<\/li>
        2. \u8bc1\u660e\u95ee\u98983.1.2, 3.1.4, 3.1.6, 3.1.12, 3.2.6, 3.2.16.<\/li>
        3. \u4ee4$\\theta$\u662f\u7d27\u9ece\u66fc\u9762$\\Sigma$\u4e0a\u7684\u4e9a\u7eaf1-\u5f62\u5f0f. \u5bf9\u4efb\u4f55\u7684$p\\in\\Sigma$, \u5b9a\u4e49$\\theta$\u5728$p$\u5904\u7684\u7559\u6570(residue)\u4e3a
          \n\\[
          \n\\mathrm{res}(\\theta;p)=\\frac{1}{2\\pi\\sqrt{-1}}\\int_{\\gamma}\\theta,
          \n\\]
          \n\u5176\u4e2d$\\gamma$\u662f$p$\u5904\u4e00\u4e2a\u9006\u65f6\u9488\u5b9a\u5411\u7684\u95ed\u66f2\u7ebf. \u8bc1\u660e\u5b83\u662f\u826f\u5b9a\u7684\u4e14
          \n\\[
          \n\\sum_{p\\in\\Sigma}\\mathrm{res}(\\theta;p)=0.
          \n\\]<\/li>
        4. \u8003\u5bdf$\\mathbb{C}$\u4e0a\u7684\u4e9a\u7eaf1-\u5f62\u5f0f$\\frac{z^3}{1+z^2}dz$. \u5b83\u662f\u5426\u5b9a\u4e49\u4e86\u9ece\u66fc\u7403$\\hat{\\mathbb{C}}$\u4e0a\u7684\u4e9a\u7eaf1-\u5f62\u5f0f? \u5982\u679c\u662f, \u627e\u51fa\u5176\u6240\u6709\u96f6\u70b9\u4e0e\u6781\u70b9\u4ee5\u53ca\u5b83\u4eec\u5728$\\hat{\\mathbb{C}}$\u4e0a\u7684\u7559\u6570.<\/li>
        5. \u8003\u5bdf\u4ee3\u6570\u66f2\u9762$\\Sigma\\subset\\mathbb{P}^2$:
          \n\\[
          \nz_2^2z_0=4z_1^3+az_1z_0^2+bz_0^3.
          \n\\]
          1. \u5df2\u77e5$\\Sigma$\u662f\u8fde\u901a\u7684, \u5728$a,b$\u6ee1\u8db3\u4ec0\u4e48\u6761\u4ef6\u4e0b, $\\Sigma$\u662f\u8fde\u901a\u7684?<\/li>
          2. \u5047\u8bbe$a,b$\u6ee1\u8db3\u5982\u4e0a\u6761\u4ef6. \u6211\u4eec\u5b9a\u4e49\u4e24\u4e2a\u4e9a\u7eaf\u51fd\u6570$f=z_1\/z_0$\u4ee5\u53ca$g=z_2\/z_0$. \u6c42\u5b83\u4eec\u7684\u96f6\u70b9\u4e0e\u6781\u70b9(\u5305\u62ec\u91cd\u6570).<\/li>
          3. \u8003\u5bdf\u4e9a\u7eaf\u5fae\u5206$\\theta=df\/g$. \u6c42\u5b83\u7684\u96f6\u70b9\u4e0e\u6781\u70b9.<\/li>
          4. \u4f60\u53ef\u4ee5\u63cf\u8ff0$\\Sigma$\u4e0a\u6240\u6709\u4e9a\u7eaf\u5fae\u5206\u6240\u6210\u7684\u7a7a\u95f4\u5417? $\\Sigma$\u7684\u4e8f\u683c\u4e3a\u591a\u5c11?<\/li><\/ol><\/li><\/ol>4. \u6ce8\u8bb0<\/a><\/span>\n\n\u8fd9\u4e3b\u8981\u53c2\u8003\u4e86Huybrechts\u7684\u590d\u51e0\u4f55[1<\/a>].<\/p>\n
            \n\t
            \u53c2\u8003\u6587\u732e<\/div>\n\t
            1. D. Huybrechts<\/span>, Complex geometry<\/a>, Universitext<\/span>, Springer-Verlag, Berlin<\/span>, 2005<\/span>. xii+309<\/span>. An introduction<\/span>. MR2093043<\/a><\/li><\/ol><\/div>","protected":false},"excerpt":{"rendered":"

              1. \u7b2c\u4e00\u6b21 \u8bc1\u660e\u5168\u7eaf\u51fd\u6570$f=u+iv$\u7684\u5b9e\u90e8$u$\u4e0e\u865a\u90e8$v$\u662f\u8c03\u548c\u7684.\u5bf9\u5168\u7eaf\u51fd\u6570\u8bc1\u660e\u6781\u5927\u503c… \u7ee7\u7eed\u9605\u8bfb\u590d\u51e0\u4f55\u4f5c\u4e1a<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[66,65],"class_list":["post-360","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-xiti","tag-fujihe","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/360","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=360"}],"version-history":[{"count":5,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/360\/revisions"}],"predecessor-version":[{"id":365,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/360\/revisions\/365"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=360"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=360"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=360"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}