# Nevanlinna theory

Nevanlinna theory deals with quantitative aspects of entire holomorphic mappings into complex varieties. Let $\omega=i\sum_{j,k}\omega_{jk}\,dz_{j}\wedge d\overline{z}_{k}$ be a smooth $(1,1)$-form on a complex manifold $X$. Suppose that $\omega$ is positive definite, i.e. the Hermitian matrix $(\omega_{jk}(z))$ is positive definite at every point. Thus $\omega$ can be viewed as a Hermitian metric on the tangent bundle $T_{X}$.

If $f\colon\mathbb{C}\to X$ is an entire curve, the growth indicatrix of $f$ is the function

 $T_{f,\omega}(r)=\int_{r_{0}}^{r}t_{f,\omega}(\rho)\,\frac{d\varrho}{\varrho},$

where

 $t_{f,\omega}(\varrho)=\int_{D(0,\varrho)}f^{*}\omega.$

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

Let us consider an hypersurface $H=\{\sigma=0\}\subset X$ defined by a global section of $L$: one would like to “measure”the intersections of the entire curve $f\colon\mathbb{C}\to X$ with $H$. For this purpose one looks at the holomorphic function $\sigma\circ f\colon\mathbb{C}\to L$ and introduces the enumerating of zeros function

 $N_{f,\sigma}(r)=\int_{r_{0}}^{r}n_{f,\sigma}(\varrho)\,\frac{d\varrho}{\varrho},$

where $n_{f,\sigma}(\varrho)=$ number of zeros of $\sigma\circ f$ on $D(0,\varrho)$ counted with multiplicities. Finally one introduces the function $m_{f,\sigma}$, called proximity function, defined by

 $m_{f,\sigma}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}\log\frac{1}{\|(\sigma\circ f)(re% ^{i\vartheta})\|_{h}}\,d\vartheta.$

This function is non negative, once one has normalized $\sigma$ with a constant in such a way that $\|\sigma\|_{h}\leq 1$. Morally, $m_{f,\sigma}(r)$ is bigger and bigger when $f$ often goes near $H=\{\sigma=0\}$ 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 $\omega=\Theta_{h}(L)>0$. For all section $\sigma$ of $L$ and all curve $f\colon\mathbb{C}\to X$ such that $f(\mathbb{C})$ is not entirely contained in the hypersurface $H=\{\sigma=0\}$, one has

 $m_{f,\sigma}(r)+N_{f,\sigma}(r)=T_{f,\omega}(r)+O(1).$

In particular, the order of growth of the left hand side when $r\to\infty$ does not depend on the choice of $\sigma$, but only on the growth indicatrix of $f$.

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

 $\delta_{\sigma}(f)=\liminf_{r\to\infty}\frac{m_{f,\sigma}(r)}{T_{f,\omega}(r)}% \in[0,1].$

In particular, the defect is equal to $1$ if $\sigma\circ f$ is never zero, and equal to $0$ if the enumerating of zeros function $N_{f,\sigma}(r)$ grows as much as possible. One of the most important results of Nevanlinna theory concerns the entire curves which map into the Riemann sphere $\mathbb{P}^{1}$ and states that the sum of defects $\sum_{a\in\mathbb{P}^{1}}\delta_{a}(f)$ 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 derivative of a meromorphic function.

More precisely the following “logarithmic derivative lemma” holds:

Let $f\colon\mathbb{C}\to\mathbb{P}^{1}$ be a meromorphic function and $D^{p}\log f$ the $p$-th logarithmic derivative of $f$. Then, for all $\varepsilon>0$, there exists a set $E$ of finite Lebesgue measure in $\mathbb{R}_{+}$ such that

 $m_{D^{p}\log f,\infty}(r)\leq(p+1+\varepsilon)(\log r+\log_{+}T_{f,\omega}(r))% +O(1),\quad\forall r\in\mathbb{R}_{+}\setminus E.$

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 $f\colon\mathbb{C}\to\mathbb{P}^{1}$ be a meromorphic function. Define the ramification divisor $R_{f}$ of $f$ as the sum $\sum e_{j}[w_{j}]$ where the $w_{j}$’s are the points where $f^{\prime}$ is zero and the $e_{j}$’s are the multiplicities of zero of $f^{\prime}$ at $w_{j}$ (where $f(w_{j})=\infty$ one looks at $1/f$ instead of $f$). Then, for all finite set $\{a_{j}\}\subset\mathbb{P}^{1}$, there exists a subset $E\subset\mathbb{R}_{+}$ of finite Lebesgue measure such that

 $N_{R_{f}}(r)+\sum_{j}m_{f,a_{j}}(r)\leq 2T_{f,\omega}(r)+O(\log r+\log_{+}T_{f% ,\omega}(r)),\quad\forall r\in\mathbb{R}_{+}\setminus E,$

where $N_{R_{f}}(r)=\int_{r_{0}}^{r}n_{R_{f}}(\varrho)\,d\varrho/\varrho$ is the enumerating function of the ramification divisor.

## References

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 NevanlinnaTheory 2013-03-22 15:36:32 2013-03-22 15:36:32 Simone (5904) Simone (5904) 14 Simone (5904) Topic msc 32A22 msc 30D35 ErnstLindelof