Second Main Theorem for Meromorphic Maps into Algebraic Varieties Intersecting Moving Hypersurfaces Targets

We prove a second main theorem type estimate in Nevanlinna theory for holomorphic curves f : Y → X from finite covering spaces Y → ℂ of the complex plane ℂ into complex projective manifolds X of maximal albanese dimension. If X is moreover of general type, then this implies that the special set of X is a proper subset of X. For a projective curve C in such X, our estimate also yields an upper bound of the ratio of the degree of C to the geometric genus of C, provided that C is not contained in a proper exceptional subset in X.

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Algebro-geometric version of Nevanlinna’s lemma on logarithmic derivative and applications

Nagoya Mathematical Journal ◽

10.1017/s0027763000008710 ◽

2004 ◽

Vol 173 ◽

pp. 23-63 ◽

Cited By ~ 5

Author(s):

Katsutoshi Yamanoi

Keyword(s):

Nevanlinna Theory ◽

Affine Space ◽

Projective Variety ◽

Logarithmic Derivative ◽

Second Main Theorem ◽

Smooth Complex ◽

Complex Projective Variety ◽

Meromorphic Maps ◽

Complex Projective ◽

The Second Main Theorem

AbstractIn this paper we shall establish some generalization of Nevanlinna’s Lemma on Logarithmic Derivative to the case of meromorphic maps from a finite analytic covering space over the m-dimensional complex affine space ℂm to a smooth complex projective variety. Then we shall apply this to “the Second Main Theorem” in Nevanlinna theory in several complex variables.

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Degenerated second main theorem for holomorphic curves into algebraic varieties

International Journal of Mathematics ◽

10.1142/s0129167x20500421 ◽

2020 ◽

Vol 31 (06) ◽

pp. 2050042

Author(s):

Lei Shi

Keyword(s):

Holomorphic Curves ◽

Algebraic Varieties ◽

Second Main Theorem ◽

Counting Functions ◽

The Second Main Theorem

In this paper, under the refinement of the subgeneral position, we give an improvement for the Second Main Theorem with truncated counting functions of algebraically non-degenerate holomorphic curves into algebraic varieties [Formula: see text] intersecting divisors in subgeneral position with some index.

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On Meromorphic Maps of Algebraic Varieties

Annals of Mathematics ◽

10.2307/1970675 ◽

1969 ◽

Vol 89 (2) ◽

pp. 391 ◽

Cited By ~ 12

Author(s):

Wei-Liang Chow

Keyword(s):

Algebraic Varieties ◽

Meromorphic Maps

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Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties

Nagoya Mathematical Journal ◽

10.1017/s0027763000019504 ◽

1981 ◽

Vol 83 ◽

pp. 213-233 ◽

Cited By ~ 23

Author(s):

Junjiro Noguchi

Keyword(s):

Meromorphic Functions ◽

Holomorphic Mapping ◽

Holomorphic Curves ◽

Compact Complex Manifold ◽

Algebraic Varieties ◽

Second Main Theorem ◽

Smooth Variety ◽

Complex Projective ◽

Logarithmic Derivatives ◽

The Second Main Theorem

Nevanlinna’s lemma on logarithmic derivatives played an essential role in the proof of the second main theorem for meromorphic functions on the complex plane C (cf., e.g., [17]). In [19, Lemma 2.3] it was generalized for entire holomorphic curves f: C → M in a compact complex manifold M (Lemma 2.3 in [19] is still valid for non-Kähler M). Here we call, in general, a holomorphic mapping from a domain of C or a Riemann surface into M a holomorphic curve in M, and sometimes use it in the sense of its image if no confusion occurs. Applying the above generalized lemma on logarithmic derivatives to holomorphic curves f: C → V in a complex projective algebraic smooth variety V and making use of Ochiai [22, Theorem A], we had an inequality of the second main theorem type for f and divisors on V (see [19, Main Theorem] and [20]). Other generalizations of Nevanlinna’s lemma on logarithmic derivatives were obtained by Nevanlinna [16], Griffiths-King [10, § 9] and Vitter [23].

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A defect relation for meromorphic maps on generalized p-parabolic manifolds intersecting hypersurfaces in complex projective algebraic varieties

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091512000284 ◽

2013 ◽

Vol 56 (2) ◽

pp. 551-574 ◽

Cited By ~ 1

Author(s):

Qi Han

Keyword(s):

Algebraic Varieties ◽

Defect Relation ◽

Meromorphic Maps ◽

Complex Projective

AbstractWe establish a defect relation for algebraically non-degenerate meromorphic maps over generalized p-parabolic manifolds that intersect hypersurfaces in smooth projective algebraic varieties, extending certain results of H. Cartan, L. Ahlfors, W. Stoll, M. Ru, P. M. Wong and Philip P. W. Wong and others.

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Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

10.23943/princeton/9780691160504.001.0001 ◽

2017 ◽

Author(s):

Claire Voisin

Keyword(s):

Recent Work ◽

Chow Groups ◽

Ring Structure ◽

Complete Intersections ◽

Algebraic Varieties ◽

Chow Ring ◽

Hodge Structures ◽

Chow Rings ◽

Geometric Methods ◽

The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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