Cartan’s conjecture for moving hypersurfaces

  • Qiming Yan
  • Guangsheng Yu


Let f be a holomorphic curve in \({\mathbb {P}}^n({{\mathbb {C}}})\) and let \(\mathcal {D}=\{D_1,\ldots ,D_q\}\) be a family of moving hypersurfaces defined by a set of homogeneous polynomials \(\mathcal {Q}=\{Q_1,\ldots ,Q_q\}\). For \(j=1,\ldots ,q\), denote by \(Q_j=\sum \nolimits _{i_0+\cdots +i_n=d_j}a_{j,I}(z)x_0^{i_0}\cdots x_n^{i_n}\), where \(I=(i_0,\ldots ,i_n)\in {\mathbb {Z}}_{\ge 0}^{n+1}\) and \(a_{j,I}(z)\) are entire functions on \({{\mathbb {C}}}\) without common zeros. Let \(\mathcal {K}_{\mathcal {Q}}\) be the smallest subfield of meromorphic function field \(\mathcal {M}\) which contains \({{\mathbb {C}}}\) and all \(\frac{a_{j,I'}(z)}{a_{j,I''}(z)}\) with \(a_{j,I''}(z)\not \equiv 0\), \(1\le j\le q\). In previous known second main theorems for f and \(\mathcal {D}\), f is usually assumed to be algebraically nondegenerate over \(\mathcal {K}_{\mathcal {Q}}\). In this paper, we prove a second main theorem in which f is only assumed to be nonconstant. This result can be regarded as a generalization of Cartan’s conjecture for moving hypersurfaces.


Nevanlinna theory Second main theorem Moving hypersurfaces 

Mathematics Subject Classification

30D35 32H30 


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Authors and Affiliations

  1. 1.Department of MathematicsTongji UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of Shanghai for Science and TechnologyShanghaiPeople’s Republic of China

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