A generalized second main theorem for closed subschemes

Abstract

Let $Y_1,\dots ,Y_q$ be closed subschemes which are located in $\ell$-subgeneral position with index $\kappa$ in a complex projective variety $X$ of dimension $n.$ Let $A$ be an ample Cartier divisor on $X.$ We obtain that if a holomorphic curve $f:\mathbb{C}\to X$ is Zariski-dense, then for every $\epsilon >0,$ \begin{eqnarray*} \sum^{q}_{j=1}ε_{Y_{j}}(A)m_{f}(r,Y_{j})\leq_{exc} \left(\frac{(\ell-n+κ)(n+1)}κ+ε\right)T_{f,A}(r). \end{eqnarray*}This generalizes the second main theorems for general position case due to Heier-Levin [AM J. Math. 143(2021), no. 1, 213-226] and subgeneral position case due to He-Ru [J. Number Theory 229(2021), 125-141]. In particular, whenever all the $Y_j$ are reduced to Cartier divisors, we also give a second main theorem with the distributive constant. The corresponding Schmidt's subspace theorem for closed subschemes in Diophantine approximation is also given.

17 pages. This is the final verion which is accepted and will appear in Annales Polonici Mathematici. arXiv admin note: text overlap with arXiv:1910.07966 by other authors

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