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Cartan’s conjecture for moving hypersurfaces

  • Qiming Yan
  • Guangsheng Yu
Article
  • 47 Downloads

Abstract

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.

Keywords

Nevanlinna theory Second main theorem Moving hypersurfaces 

Mathematics Subject Classification

30D35 32H30 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

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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