高维Nevanlinna理论中第二基本定理及其应用

The research object of this proposal is the second main theorem of Nevanlinna theory of holomorphic maps in several complex variables, we hope to further develop the Nevanlinna theory from the point of view of geometry and number theory, and study the core problems in this field as well as its applications in Diophantine approximation and geometric hyperbolicity. This proposal focuses on the development of new concepts and new methods that have emerged in Nevanlinna theory in recent years. These include new filtration methods, the (birational-)Nevanlinna constant, the new concepts on singularities of divisors and general position, the application of Seshadri constants and so on. The research contents include the second main theorems of algebraic (non-)degenerate holomorphic curves in complex projective spaces or projective varieties intersecting hypersurfaces in general position; the second main theorems of moving hypersurfaces of slowly growth; the applications of second main theorems, which mainly involves the estimates of the upper bound of the greatest common divisor, the second main theorems of higher co-dimensional case, and the geometric hyperbolicity; the correspondence of the above results of Nevanlinna in Diophantine approximation.

本课题的研究对象为多复变全纯映射的Nevanlinna理论的第二基本定理，我们希望从几何和数论的观点进一步发展Nevanlinna理论，研究这一领域的核心问题以及在Diophantine逼近和几何双曲中的应用。主要致力于发展近几年Nevanlinna理论研究中出现的一些新概念和新方法，包含新的filtration方法，（双有理）Nevanlinna常数，关于除子奇性和一般位置的指标的新概念，Seshadri常数的应用等。研究内容包括复射影空间或射影簇中代数（非）退化的全纯曲线与处于一般位置的超曲面交的第二基本定理；研究慢增长的活动超曲面情形的第二基本定理；第二基本定理的应用：主要涉及最大公因子上界估计等处理高余维对象的问题和几何双曲问题；利用Vojta字典研究上述Nevanlinna理论中的结果在Diophantine逼近中的对应。

本项目研究对象为多复变Nevanlinna理论及其与数论中丢番图逼近理论的对应。主要研究成果包含以下几个方面：进一步细化了除子处于次一般位置的条件，提出了相应的“指标”的概念，并建立了关于全纯曲线的更精细的第二基本定理型结果；研究了活动超曲面的Nevanlinna理论第二基本定理，考虑了全纯曲线代数退化和非常值的情形；通过Nevanlinna理论和丢番图逼近理论的联系，给出了对应的Schmidt子空间定理型结果并研究了相关应用；将除子情形的方法应用到高余维对象的研究中，建立了相关第二基本定理型结果，并研究了在几何双曲问题中的应用。

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