Nevanlinna theory

Nevanlinna theoryMathworldPlanetmath deals with quantitative aspects of entire holomorphic mappings into complex varieties. Let ω=ij,kωjkdzjdz¯k be a smooth (1,1)-form on a complex manifold X. Suppose that ω is positive definitePlanetmathPlanetmath, i.e. the Hermitian matrixMathworldPlanetmath (ωjk(z)) is positive definite at every point. Thus ω can be viewed as a Hermitian metric on the tangent bundle TX.

If f:X is an entire curve, the growth indicatrix of f is the function




It is clear that tf,ω(ϱ) is nothing more than the area with respect to ω of the image f(D(0,ϱ)) of the disk in the complex line centered at 0 and of ray ϱ. Now let L be an ample holomorphic line bundle over X (compact, connected). Then L carries an Hermitian metric h with positive curvature ω=Θh(L) and so, in this case, one can suppose that our ω is in fact the curvature of this line bundle. Furthermore, it is clear that if one is merely interested in the order of growth O(Tf,ω(r)) when r goes to infinityMathworldPlanetmath, then this order is independent of the choice of ω.

Let us consider an hypersurface H={σ=0}X defined by a global section of L: one would like to “measure”the intersectionsMathworldPlanetmathPlanetmath of the entire curve f:X with H. For this purpose one looks at the holomorphic function σf:L and introduces the enumerating of zeros function


where nf,σ(ϱ)= number of zeros of σf on D(0,ϱ) counted with multiplicities. Finally one introduces the function mf,σ, called proximity function, defined by


This function is non negative, once one has normalized σ with a constant in such a way that σh1. Morally, mf,σ(r) is bigger and bigger when f often goes near H={σ=0} on the circle of ray r.

The first fundamental theorem of Nevanlinna states the following:

Let (L,h) be an Hermitian line bundle with curvature form ω=Θh(L)>0. For all sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath σ of L and all curve f:CX such that f(C) is not entirely contained in the hypersurface H={σ=0}, one has


In particular, the order of growth of the left hand side when r does not depend on the choice of σ, but only on the growth indicatrix of f.

Classically, one introduces the defect of f with respect to H={σ=0}, defined by

δσ(f)=lim infrmf,σ(r)Tf,ω(r)[0,1].

In particular, the defect is equal to 1 if σf is never zero, and equal to 0 if the enumerating of zeros function Nf,σ(r) grows as much as possible. One of the most important results of Nevanlinna theory concerns the entire curves which map into the Riemann sphere 1 and states that the sum of defects a1δa(f) is at most equal to 2. One of the essential steps for the proof of this statement is an estimate of the proximity function of the logarithmic derivativeMathworldPlanetmath of a meromorphic function.

More precisely the following “logarithmic derivative lemma” holds:

Let f:CP1 be a meromorphic function and Dplogf the p-th logarithmic derivative of f. Then, for all ε>0, there exists a set E of finite Lebesgue measure in R+ such that


An important consequence of the logarithmic derivative lemma is what is called the second fundamental theorem of Nevanlinna from which it follows immediately the estimate for the sum of the defects introduced above:

Let f:CP1 be a meromorphic function. Define the ramification divisor Rf of f as the sum ej[wj] where the wj’s are the points where f is zero and the ej’s are the multiplicities of zero of f at wj (where f(wj)= one looks at 1/f instead of f). Then, for all finite setMathworldPlanetmath {aj}P1, there exists a subset ER+ of finite Lebesgue measure such that


where NRf(r)=r0rnRf(ϱ)𝑑ϱ/ϱ is the enumerating function of the ramification divisor.


J.-P. Demailly, Variétés projectives hyperboliques et équations différentielles algébriques. (French) [Hyperbolic projective varieties and algebraic differential equations] Journée en l’Honneur de Henri Cartan, 3–17, SMF Journ. Annu., 1997, Soc. Math. France, Paris, 1997.

Title Nevanlinna theory
Canonical name NevanlinnaTheory
Date of creation 2013-03-22 15:36:32
Last modified on 2013-03-22 15:36:32
Owner Simone (5904)
Last modified by Simone (5904)
Numerical id 14
Author Simone (5904)
Entry type Topic
Classification msc 32A22
Classification msc 30D35
Related topic ErnstLindelof