Nevanlinna theory deals with quantitative aspects of entire holomorphic mappings into complex varieties. Let be a smooth -form on a complex manifold . Suppose that is positive definite, i.e. the Hermitian matrix is positive definite at every point. Thus can be viewed as a Hermitian metric on the tangent bundle .
If is an entire curve, the growth indicatrix of is the function
It is clear that is nothing more than the area with respect to of the image of the disk in the complex line centered at and of ray . Now let be an ample holomorphic line bundle over (compact, connected). Then carries an Hermitian metric with positive curvature and so, in this case, one can suppose that our is in fact the curvature of this line bundle. Furthermore, it is clear that if one is merely interested in the order of growth when goes to infinity, then this order is independent of the choice of .
Let us consider an hypersurface defined by a global section of : one would like to “measure”the intersections of the entire curve with . For this purpose one looks at the holomorphic function and introduces the enumerating of zeros function
where number of zeros of on counted with multiplicities. Finally one introduces the function , called proximity function, defined by
This function is non negative, once one has normalized with a constant in such a way that . Morally, is bigger and bigger when often goes near on the circle of ray .
The first fundamental theorem of Nevanlinna states the following:
In particular, the order of growth of the left hand side when does not depend on the choice of , but only on the growth indicatrix of .
Classically, one introduces the defect of with respect to , defined by
In particular, the defect is equal to if is never zero, and equal to if the enumerating of zeros function grows as much as possible. One of the most important results of Nevanlinna theory concerns the entire curves which map into the Riemann sphere and states that the sum of defects is at most equal to . One of the essential steps for the proof of this statement is an estimate of the proximity function of the logarithmic derivative of a meromorphic function.
More precisely the following “logarithmic derivative lemma” holds:
Let be a meromorphic function and the -th logarithmic derivative of . Then, for all , there exists a set of finite Lebesgue measure in 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 be a meromorphic function. Define the ramification divisor of as the sum where the ’s are the points where is zero and the ’s are the multiplicities of zero of at (where one looks at instead of ). Then, for all finite set , there exists a subset of finite Lebesgue measure such that
where 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.
|Date of creation||2013-03-22 15:36:32|
|Last modified on||2013-03-22 15:36:32|
|Last modified by||Simone (5904)|