Nevanlinna Invariant
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In mathematics, the Nevanlinna invariant of an ample divisor ''D'' on a normal
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...
''X'' is a real number connected with the rate of growth of the number of rational points on the variety with respect to the embedding defined by the divisor. The concept is named after
Rolf Nevanlinna Rolf Herman Nevanlinna (né Neovius; 22 October 1895 – 28 May 1980) was a Finnish mathematician who made significant contributions to complex analysis. Background Nevanlinna was born Rolf Herman Neovius, becoming Nevanlinna in 1906 when his fa ...
.


Formal definition

Formally, α(''D'') is the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
of the rational numbers ''r'' such that K_X + r D is in the closed real cone of effective divisors in the
Néron–Severi group In algebraic geometry, the Néron–Severi group of a variety is the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. Its rank is called the Picard number. It is nam ...
of ''X''. If α is negative, then ''X'' is pseudo-canonical. It is expected that α(''D'') is always a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
.


Connection with height zeta function

The Nevanlinna invariant has similar formal properties to the abscissa of convergence of the
height zeta function In mathematics, the height zeta function of an algebraic variety or more generally a subset of a variety encodes the distribution of points of given height. Definition If ''S'' is a set with height function ''H'', such that there are only finitely ...
and it is conjectured that they are essentially the same. More precisely, Batyrev–Manin conjectured the following. Let ''X'' be a projective variety over a number field ''K'' with ample divisor ''D'' giving rise to an embedding and height function ''H'', and let ''U'' denote a Xariski open subset of ''X''. Let α = α(''D'') be the Nevanlinna invariant of ''D'' and β the abscissa of convergence of ''Z''(''U'', ''H''; ''s''). Then for every ε > 0 there is a ''U'' such that β < α + ε: in the opposite direction, if α > 0 then α = β for all sufficiently large fields ''K'' and sufficiently small ''U''.


References

* * {{cite book , first=Serge , last=Lang , authorlink=Serge Lang , title=Survey of Diophantine Geometry , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...
, year=1997 , isbn=3-540-61223-8 , zbl=0869.11051 Diophantine geometry Geometry of divisors