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 is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...
''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; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
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 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 (for example, The set of all ...
.


Connection with height zeta function

The Nevanlinna invariant has similar formal properties to the abscissa of convergence of the height zeta function 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 in ...
, year=1997 , isbn=3-540-61223-8 , zbl=0869.11051 Diophantine geometry Geometry of divisors