# Enhancing Polygonal Building Segmentation via Oriented Corners

Mohammad Moein Sheikholeslami1, Muhammad Kamran1, Andreas Wichmann2, and Gunho
Sohn1
1Department of Earth and Space Science and Engineering, York University,
Canada
2Institute for Applied Photogrammetry and Geoinformatics, Jade University,
Germany
{mmoein, mkamran9<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding
author.

###### Abstract

The growing demand for high-resolution maps across various applications has
underscored the necessity of accurately segmenting building vectors from
overhead imagery. However, current deep neural networks often produce raster
data outputs, leading to the need for extensive post-processing that
compromises the fidelity, regularity, and simplicity of building
representations. In response, this paper introduces a novel deep convolutional
neural network named OriCornerNet, which directly extracts delineated building
polygons from input images. Specifically, our approach involves a deep model
that predicts building footprint masks, corners, and orientation vectors that
indicate directions toward adjacent corners. These predictions are then used
to reconstruct an initial polygon, followed by iterative refinement using a
graph convolutional network that leverages semantic and geometric features.
Our method inherently generates simplified polygons by initializing the
refinement process with predicted corners. Also, including geometric
information from oriented corners contributes to producing more regular and
accurate results. Performance evaluations conducted on SpaceNet Vegas and
CrowdAI-small datasets demonstrate the competitive efficacy of our approach
compared to the state-of-the-art in building segmentation from overhead
imagery.

##  1 Introduction

The integration of high-resolution satellite imagery and unmanned aerial
vehicle (UAV) data has had a profound impact on sectors like urban planning
and monitoring of the built environment. Despite these advancements, the
creation of accurate maps remains a resource-intensive endeavor, particularly
in the realm of building extraction. This process traditionally comprises two
key phases: firstly, the segmentation of building footprints from imagery, and
secondly, the conversion of these footprints into vector formats compatible
with geographic information systems (GIS). Challenges such as shadows, tree
obstructions, and errors in raster quantization often lead to imperfect masks
during segmentation, resulting in suboptimal polygons during the vectorization
process (see Figure 1).

Figure 1: Comparison between the common approach for extracting buildings from
imagery and OriCornerNet. The common approach suffers from redundant vertices
and irregular polygons.

Within the domain of deep learning methods for polygonal building
segmentation, two primary categories have emerged: 1) two-step methodologies
and 2) direct polygonal segmentation approaches [3, 6]. Two-step methods, akin
to conventional techniques, involve initial raster segmentation followed by
post-processing steps, including vectorization and potential simplification.
They frequently employ auxiliary representations such as frame fields [4],
directional indicators [13], or attraction fields [15] to aid in the
vectorization process. However, these methods are constrained by their
reliance on offline post-processing, which can fail under certain conditions.
In contrast, direct polygonal segmentation strategies directly predict
building polygons from input images, sidestepping the challenges associated
with two-step methodologies. Nevertheless, these methods can be intricate to
train, computationally demanding, and may encounter difficulties such as
irregular predictions or missing corners [7, 17, 19].

Figure 2: The architecture of OriCornerNet.

This paper presents a novel deep neural network named OriCornerNet, designed
to enhance the performance of a baseline direct polygonal segmentation model
called R-PolyGCN [17]. OriCornerNet achieves this by integrating the detected
oriented corners as an auxiliary representation. This innovative approach,
similar to direct segmentation methods, eliminates the need for post-
processing steps like vectorization. Also, it utilizes the auxiliary
representation to guide the generation of building polygons like two-step
methodologies. In this context, ’orientation’ refers to vectors associated
with each corner that point towards adjacent corners. Our method uses
auxiliary representation to generate regular polygons, identifying occluded
corners and capturing the architectural nuances effectively.

##  2 Related Work

In this section, we will briefly review polygonal segmentation methods that
utilize auxiliary information to guide them through the vectorization process.
These methods are mostly two-step approaches. For example, Girard _et al_. [4]
predict a raster segmentation along with frame fields and use the latter to
vectorize the predicted raster segmentation. Similarly, Shu _et al_. [13]
predict corners and direction vectors using a deep network and then connect
the corners by a boundary tracing algorithm. Another approach adopted by [18]
formulates the polygons as a combination of vertex and direction maps
predicted by deep networks. Then, a polygon generation algorithm is used to
generate the final output. Recently, HiSup [15] proposed to use the attraction
field as a mid-level supervision in mask and corner segmentation tasks,
leading to a state-of-the-art performance in polygonal building segmentation.

The major disadvantage of the methods mentioned earlier is requiring rule-
based post-processing, which is susceptible to failure. Additionally, two-step
methods experience significant issues with raster quantization error, as noted
in [4]. This emphasizes the need for direct methods to effectively use
auxiliary information.

##  3 Methodology

For our model, we selected R-PolyGCN [17] as the baseline, which comprises
three main stages: an instance segmentation model, graph initialization, and
graph convolutional networks (GCNs). Then, we modified each stage to improve
polygonal segmentation performance using oriented corners, as shown in Figure
2 and described below.

###  3.1 Instance Segmentation Baseline

Following R-PolyGCN [17], we use Mask R-CNN [5] as the baseline for semantic
mask segmentation. In addition, we add corner and orientation heads. Given an
input image $I\in\mathbb{R}^{800\times 800\times 3}$, semantic features
$S\in\mathbb{R}^{28\times 28\times 256}$ are extracted for each RoI and inputs
to the heads, then the corner head, fed by the $S$, predicts a heatmap
$H\in\mathbb{R}^{28\times 28}$ and xy offset maps
$\Delta\in\mathbb{R}^{28\times 28\times 2}$. To be concise, we only bring the
losses for corner and orientation heads:

$L_{\text{heat.}}=-\frac{1}{N}\sum_{i=1}^{N}(w.y_{i}.\log
x_{i}+(1-y_{i})\log(1-x_{i}))$ (1)

and

$\begin{split}&L_{\text{offset}}=\frac{1}{N}\sum_{i=1}^{N}L_{i},\\\
&L_{i}=\begin{cases}0.5(x_{i}-y_{i})^{2},&\text{if }|x_{i}-y_{i}|<1\\\
|x_{i}-y_{i}|-0.5,&\text{otherwise}\end{cases}\end{split}$ (2)

where $x_{i}$, $y_{i}$, $w_{p}$, and $N$ are, respectively, prediction,
target, weight of the positive class, and number of pixels in each task. For
orientation representation on each edge pixel, we use two unit vectors
$O_{cw}=(\cos{\theta_{cw}},\sin{\theta_{cw}})$ and
$O_{ccw}=(\cos{\theta_{ccw}},\sin{\theta_{ccw}})$ pointing to the next corner
in the polygon in respectively clockwise and counter-clockwise directions.
Thus, four channels are regressed and supervised on ground truth edge pixels
using the below loss:

$\begin{split}&L_{\text{orient.}}=\frac{1}{N_{e}}\sum_{i=1}^{N_{e}}L_{i},\\\
&L_{i}=\begin{cases}0.5(x_{i}-y_{i})^{2},&\text{if }|x_{i}-y_{i}|<1\\\
|x_{i}-y_{i}|-0.5,&\text{otherwise}\end{cases}\end{split}$ (3)

where $x_{i}$, $y_{i}$, and $N_{e}$ are prediction, target, and the number of
edge pixels.

###  3.2 Initialization Module

Many methods in literature [17, 10, 9, 12] use a fixed number of predefined
corners, e.g., located on the perimeter of a circle, for creating the initial
graph. However, we use the predicted corners and connect them based on their
distances from the mask contour. Also, we propose to combine semantic corners
(mask’s contour points) with the detected ones to enhance the graph
initialization accuracy. In this sense, predicted corners further than
$\delta_{cor2cont}$ from the contour are removed. Moreover, the semantic
corners further than $\delta_{sem2graph}$ from the graph constructed using
only detected corners are added to the corners. Lastly, predicted orientations
and corner heatmaps, i.e. geometric features $G\in\mathbb{R}^{28\times
28\times 5}$, are concatenated with RoI semantic features $S$ to construct
$F=concat\\{S,G\\}\in\mathbb{R}^{28\times 28\times 261}$.

###  3.3 GCNs

This part includes three GCN modules as described in [9]. Getting the initial
graph, GCN modules repeatedly sample the feature vectors for each corner from
$F$ and predict their position offsets. As the number of corners in prediction
and target may vary, a bi-projection [2] loss is used:

$L_{\text{poly}}=\frac{1}{N}(\sum_{(p,\bar{p})\in S_{ic}}\lVert
p-\bar{p}\rVert_{2}+\sum_{(q,\bar{q})\in S_{ac}}\lVert q-\bar{q}\rVert_{2})$
(4)

where $S_{ic}$ represents the set of initially matched corners based on their
distance, and $S_{ac}$ denotes the set of additional corners matched to their
projection on the nearest edge. Additionally, $N$ stands for the number of
corners in the polygon with a higher count. We also incorporate two
regularization losses: 1) orientation consistency loss and 2) orthogonality
loss. Orientation consistency loss penalizes the error between predicted
orientation vectors in each edge:

$L_{\text{ori-cons}}(P)=\frac{1}{n}\sum_{i=1}^{n}(1+\langle
O_{ccw}^{i},O_{cw}^{i+1}\rangle)$ (5)

where $n$ is the number of vertices in polygon $P$ and $\langle.,.\rangle$ is
the inner product of the vectors. The idea is that the orientation vectors of
two consecutive corners should point to each other. On the other hand,
orthogonality loss encourages all the inner angles to be orthogonal, as is
frequently seen in buildings:

$L_{\text{ortho.}}(P)=\frac{1}{n}\sum_{i=1}^{n}\min_{\omega_{peak}}{\mathopen{|}\omega_{i}-\omega_{peak}\mathclose{|}}$
(6)

where $n$ is the number of inner angles in polygon $P$, $\omega$ is the inner
angle and
$\omega_{peak}\in\\{0^{\circ},90^{\circ},180^{\circ},270^{\circ}\\}$. The
final loss is a weighted sum of all the losses.

##  4 Experiments

###  4.1 Datasets and Settings

The proposed network’s performance is evaluated using the SpaceNet Vegas
dataset [14] to demonstrate its superiority over R-PolyGCN, its baseline, and
several competing methods. The dataset contains 3,851 images, with a size of
650$\times$650 pixels and a pixel size of 0.3 meter. The dataset is randomly
divided into training, validation, and test sets. We trained the model using a
single NVIDIA GeForce RTX 3090 GPU with a batch size of 1 for 50 epochs.

Additionally, the CrowdAI-small dataset [11] is utilized for both ablation
studies and comparison with state-of-the-art methods. This dataset consists of
10,186 images sized at 300$\times$300 pixels, with a pixel size of 0.3 meter.
Training on the CrowdAI-small dataset is performed for 40 epochs on a single
NVIDIA GeForce RTX 6000 GPU with a batch size of 1. Images from both datasets
are resized to 800$\times$800 pixels for input to the network. Also,
$\delta_{cor2cont}=5$ and $\delta_{sem2graph}=5$ pixels for both datasets.

###  4.2 Evaluation Metrics

Two groups of metrics have been utilized in the performance assessment of the
proposed method: 1) Raster-based MS-COCO metrics [8], such as mean average
precision (AP) and recall (AR) over different IoU thresholds, and 2) Vector-
based metrics, including PoLiS [1] and C-IoU [15]. PoLiS calculates the
average distance between the matched vertices in polygons $P$ and $Q$:

$\displaystyle PoLiS(P,Q)=\frac{1}{2N_{P}}\sum_{p_{j}\in P}\min_{q\in Q}\lVert
p_{j}-q\rVert+$ (7) $\displaystyle\frac{1}{2N_{Q}}\sum_{q_{k}\in Q}\min_{p\in
P}\lVert q_{k}-b\rVert,$

where $N_{P}$ and $N_{Q}$ are, respectively, the number of vertices in the
predicted and ground truth polygons. Furthermore, C-IoU [15] measures the
polygon complexity as well as its IoU:

$\begin{split}&{CIoU}(P,Q)=IoU(P,Q).(1-D(N_{P},N_{Q})),\end{split}$ (8)

where $D(N_{P},N_{Q})=|N_{P}-N_{Q}|/(N_{P}+N_{Q})$ is the relative difference
between the number of vertices in the polygons.

###  4.3 Preliminary Results and Discussion

To assess the effectiveness of utilizing oriented corners, we conducted a
comparison between the proposed method and its baseline, R-PolyGCN [17].
Raster-based metrics, specifically AP and AR, exhibited a significant
improvement of 16% and 14%, respectively, as shown in Table 1. Furthermore,
vector-based metrics indicated that our method achieved more precise corners
without compromising polygon complexity, highlighting the effectiveness of
corner-based initialization and the multi-task learning approach for
orientation (see Figure 3). Also, the proposed method is compared with two
two-step (Mask R-CNN [5] and FrameField [4]) and two direct (PolyMapper [7]
and BiSVP [16]) polygonal segmentation methods. OriCornerNet proved to
outperform them by 4% in AP and by 11% in terms of AR.

Table 1: Quantitative results of OriCornerNet and rival methods on SpaceNet Vegas dataset. The results for the rival methods except R-PolyGCN are from [16]. Lower PoLiS values are better. Method | AP | AR | PoLiS $\downarrow$ | C-IoU
---|---|---|---|---
Mask R-CNN [5] | 0.47 | 0.55 | - | -
FrameField [4] | 0.54 | 0.59 | - | -
PolyMapper [7] | 0.52 | 0.58 | - | -
BISVP [16] | 0.53 | 0.59 | - | -
R-PolyGCN [17] | 0.43 | 0.57 | 3.20 | 0.55
Ours | 0.58 | 0.70 | 2.29 | 0.80

(a) Prediction

(b) Ground truth

Figure 3: Oriented corners on Vegas dataset. OriCornerNet can grasp building
regular shapes and detect obscured corners.

An ablation study was conducted on the CrowdAI-small dataset to quantitatively
evaluate the contribution of each component in the proposed method. According
to Table 2, the study revealed that combining semantic corners with the
detected corners could significantly improve performance in terms of both
raster-based and vector-based metrics. Additionally, the orientation
consistency loss increased the AP by 0.5% and AR by 0.2%. Ultimately, the full
method, which includes the above-mentioned components plus the orthogonality
loss, delivered the best performance in terms of all metrics. Further,
compared to HiSup, the state-of-the-art method, our method demonstrated
competitive results in terms of AP and considerably better results in terms of
AR and vector-based metrics. Figure 4 provides qualitative results,
demonstrating the proposed method’s more robust performance in predicting
regular and simpler polygons.

Table 2: Ablation study of OriCornerNet and comparison to the state-of-the-art
method on CrowdAI-small dataset. The lower values of PoLiS are better.

Method | AP(%) | AR(%) | PoLiS $\downarrow$ | C-IoU
---|---|---|---|---
Baseline | 52.3 | 66.6 | 1.80 | 78.40
\+ Semantic corners | 54.6 | 68.1 | 1.74 | 79.02
\+ Orientation consistency loss | 55.1 | 68.3 | 1.76 | 79.15
\+ Orthogonality loss | 56.1 | 69.5 | 1.72 | 79.48
HiSup [15] | 56.9 | 60.8 | 1.99 | 76.09

(a) HiSup

(b) Ours

(c) Ground truth

Figure 4: Qualitative results of Hisup, the proposed method, and ground truth
on CrowdAI-small dataset.

##  5 Conclusion

In this work, we have proposed a new method for generating building polygons
using oriented corners as an auxiliary representation in an end-to-end direct
polygonal segmentation network. The network utilizes both semantic and
geometric features to reconstruct the building polygons. It also refines them
via GCNs by applying some geometric regularization terms. The experimental
results on two datasets have demonstrated the effectiveness of our proposed
method in extracting delineated polygons. In the future, we plan to focus on
handling buildings with holes and improving the simplification using
orientation information.

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# Analytic Ax-Schanuel Theorem for semi-abelian varieties
and Nevanlinna theory

J. Noguchi Research supported in part by Grant-in-Aid for Scientific Research
(C) 19K03511, Japan Society for the Promotion of Science.

(Graduate School of Mathematical Sciences
The University of Tokyo, Tokyo, Japan
e-mail<EMAIL_ADDRESS>
###### Abstract

The purpose of this paper is to explore Nevanlinna theory of the entire curve
$\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f:=(\exp_{A}f,f):{\mathbf{C}}\to
A\times{\mathop{\mathrm{Lie}}}(A)$ associated with an entire curve
$f:{\mathbf{C}}\to{\mathop{\mathrm{Lie}}}(A)$, where
$\exp_{A}:{\mathop{\mathrm{Lie}}}(A)\to A$ is an exponential map of a semi-
abelian variety $A$. Firstly we give a Nevanlinna theoretic proof to the
analytic Ax-Schanuel Theorem for semi-abelian varieties, which was proved by
J. Ax 1972 in the case of formal power series (Ax-Schanuel Theorem). We assume
some non-degeneracy condition for $f$ such that the elements of the vector-
valued function $f(z)-f(0)\in{\mathop{\mathrm{Lie}}}(A)\cong{\mathbf{C}}^{n}$
are ${\mathbf{Q}}$-linearly independent in the case of
$A=({\mathbf{C}}^{*})^{n}$. Then by making use of the Log Bloch-Ochiai Theorem
and a key estimate which we show, we prove that ${\mathrm{tr.}\deg}\hskip
0.3pt_{\mathbf{C}}\,\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f\geq n+1.$

Our next aim is to establish a 2nd Main Theorem for $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f$ and its $k$-jet lifts with truncated
counting functions at level one.

Keywords: Nevanlinna theory; analytic Ax-Schanuel; semi-abelian Schanuel; Log
Bloch-Ochiai; value distribution theory.

AMS MSC2020: 32H30, 11J89, 11D61.

## 1\. Introduction

Let $A$ be a semi-abelian variety of dimension $n\,(>0)$ with Lie algebra
${\mathop{\mathrm{Lie}}}(A)$ and an exponential map
$\exp_{A}:{\mathop{\mathrm{Lie}}}(A)\to A$. Let
$f:{\mathbf{C}}\to{\mathop{\mathrm{Lie}}}(A)$ be an entire curve (i.e., a
holomorphic map from ${\mathbf{C}}$ into ${\mathop{\mathrm{Lie}}}(A)$), and
set

(1.1) $\exp_{A}f:=\exp_{A}\circ f:{\mathbf{C}}\to A,\quad\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f:=(\exp_{A}f,f):{\mathbf{C}}\to
A\times{\mathop{\mathrm{Lie}}}(A).$

We denote by $\overline{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f({\mathbf{C}})}^{\mathrm{Zar}}$ the Zariski
closure of $\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f({\mathbf{C}})$ in
$A\times{\mathop{\mathrm{Lie}}}(A)$. Put

${\mathrm{tr.}\deg}\hskip 0.3pt_{\mathbf{C}}\,\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f:=\dim_{\mathbf{C}}\overline{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f({\mathbf{C}})}^{\mathrm{Zar}}.$

We say that $f$ is “$A$-(resp. non)degenerate” if

######  1.2.

  * there is a (resp. no) connected algebraic proper subgroup $G\subsetneq A$ such that $G+\exp_{A}f(0)\supset\exp_{A}f({\mathbf{C}})$.

(1) It is our first aim to give a Nevanlinna theoretic proof to the analytic
version of the Ax-Schanuel Theorem for semi-abelian varieties.

######  Theorem 1.3.

If an entire curve $f:z\in{\mathbf{C}}\to{\mathop{\mathrm{Lie}}}(A)$ with a
semi-abelian variety $A$ is $A$-nondegenerate, then ${\mathrm{tr.}\deg}\hskip
0.3pt_{\mathbf{C}}\,\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f\geq n+1.$

The present study is motivated by S. Lang [11], and J. Ax [1], [2], where Ax
proved the above statement for formally analytic maps
$f:{\mathbf{C}}\to{\mathop{\mathrm{Lie}}}(A)$, i.e., $f$ represented by a
vector of formal power series (Ax-Schanuel Theorem); he dealt with the case of
formal power series of several variables. The proof of Ax [1], [2] is based on
Kolchin’s theory of differential algebra. Note that for
$f:{\mathbf{C}}\to{\mathbf{C}}^{n}$ and
$\exp_{({\mathbf{C}}^{*})^{n}}:{\mathbf{C}}^{n}\ni(w_{j})\mapsto(e^{w_{j}})\in({\mathbf{C}}^{*})^{n}$,
$f$ is $({\mathbf{C}}^{*})^{n}$-nondegenerate if and only if
$f_{j}(z)-f_{j}(0)~{}(1\leq j\leq n)$ are ${\mathbf{Q}}$-linearly independent.

Because of the nature of the statement, it is natural and interesting to look
for an analytic proof in the analytic setting. The present proof relies on the
Log Bloch-Ochiai Theorem [13], [14], and Nevanlinna theory of entire curves
into semi-abelian varieties (cf. Noguchi-Winkelmann [20] in general).

The above Ax-Schanuel Theorem was obtained as an analogue of the following
attractive conjecture:

######  1.4.

Schanuel’s Conjecture (cf. Lang [11] p. 30). Let
$\alpha=(\alpha_{j})\in{\mathbf{C}}^{n}$ be a vector such that
$\alpha_{j}~{}(1\leq j\leq n)$ are ${\mathbf{Q}}$-linearly independent, and
set $\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\alpha=((e^{\alpha_{j}}),\alpha)\in({\mathbf{C}}^{*})^{n}\times{\mathbf{C}}^{n}$.
Then, ${\mathrm{tr.}\deg}\hskip 0.3pt_{\mathbf{Q}}\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\alpha\geq n.$

This conjecture is known only for $n=1$ (Gel’fond-Schneider ([11], [28]);
Hilbert’s 7th Problem), and even in $n=2$ it implies the algebraic
independence of $e$ and $\pi$; cf., e.g., Waldschmidt [28] §1, Jones-Wilkie
[10] for more.

(2) It is our second aim to establish a 2nd Main Theorem for $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f:{\mathbf{C}}\to
A\times{\mathop{\mathrm{Lie}}}(A)$ and its jet lifts. Let $J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f):{\mathbf{C}}\to
J_{k}(A\times{\mathop{\mathrm{Lie}}}(A))$ be the $k$-jet lift of $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f$ and let $X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)$ be the Zariski closure of the image
$J_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)({\mathbf{C}})$ in
$J_{k}(A\times{\mathop{\mathrm{Lie}}}(A))$. Because of the flat structure of
$J_{k}(A)\cong A\times J_{k,A}$ we may write

(1.5) $\displaystyle X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)\subset
A\times{\widehat{J}_{k,A}}\subset
J_{k}(A\times{\mathop{\mathrm{Lie}}}(A)),\quad{\widehat{J}_{k,A}}:={\mathop{\mathrm{Lie}}}(A)\times
J_{k,A}\cong{\mathbf{C}}^{n}\times{\mathbf{C}}^{nk},$

where $J_{k,A}$ is the jet part of $J_{k}(A)$ (see §4.4 for details). Let
$\hskip 3.0pt{\overline{\hskip-3.0pt\widehat{J}}_{k,A}}$ be a projective
compactification of ${\widehat{J}_{k,A}}$; e.g., $\hskip
3.0pt{\overline{\hskip-3.0pt\widehat{J}}_{k,A}}={\mathbf{P}^{n}(\mathbf{C})}\times{\mathbf{P}}^{nk}({\mathbf{C}})$.

The main result is as follows (cf. §§2, 4 for notation):

######  Theorem 1.6 (2nd Main Theorem).

Let $A$ be a semi-abelian variety. Let $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f:{\mathbf{C}}\to
A\times{\mathop{\mathrm{Lie}}}(A)$ be the entire curve associated with an
$A$-nondegenerate entire curve $f:{\mathbf{C}}\to{\mathop{\mathrm{Lie}}}(A)$.
Then we have:

  1. (i)

Let $Z$ be a reduced algebraic subset of $X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)~{}(\subset A\times{\widehat{J}_{k,A}})$
$(k\geqq 0)$. Then there exists an equivariant projective smooth
compactification $\bar{A}$ of $A$ with the closure $\bar{X}_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)$ (resp. $\bar{Z}$) of $X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)$ (resp. $Z$) in $\bar{A}\times\hskip
3.0pt{\overline{\hskip-3.0pt\widehat{J}}_{k,A}}$ such that

(1.7) $T_{J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r,\omega_{\bar{Z}})\leq
N_{1}(r,J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}Z)+S_{\varepsilon,\exp_{A}f}(r).$

  2. (ii)

Moreover, if ${\mathrm{codim}}_{X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}Z\geqq 2$, then

(1.8) $T_{J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r,\omega_{\bar{Z}})=S_{\varepsilon,\exp_{A}f}(r).$

  3. (iii)

($k=0$) In particular, if $D$ is a reduced effective algebraic divisor on
$A\times{\mathop{\mathrm{Lie}}}(A)$ such that $D\not\supset X_{0}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)$, then there is an equivariant projective
smooth compactification $\bar{A}$ of $A$ such that

(1.9) $T_{\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r,L(\bar{D}))\leq
N_{1}(r,(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}D)+S_{\varepsilon,\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r),$

where $\bar{D}\subset\bar{A}\times\overline{{\mathop{\mathrm{Lie}}}(A)}$ with
$\overline{{\mathop{\mathrm{Lie}}}(A)}:=\hskip
3.0pt{\overline{\hskip-3.0pt\widehat{J}}_{0,A}}$.

As an application we obtain the following (cf. 4.1 (ii), 4.2 (6) in §4 below,
and Corvaja-Noguchi [5] for entire curves into semi-abelian varieties):

######  Theorem 1.10.

Let $\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f:{\mathbf{C}}\to A$ be as in
Theorem 1.6 and let $D$ be an $A$-big divisor on $X_{0}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)$. Then there is an irreducible component
$E$ of $D$ such that $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f({\mathbf{C}})\cap E$ is Zariski dense in
$E$; in particular, the cardinality $|\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f({\mathbf{C}})\cap D|=\infty$.

Here, $D$ being $A$-big means roughly that $D$ contains a big divisor in
$A$-factor (see Definition 4.19).

The present paper is organized so that in §2 we prepare the notation from
Nevanlinna theory and prove a key estimate lemma (see Lemma 2.5). We give a
proof of Theorem 1.3 in §3 and those of Theorems 1.6 and 1.10 in §4. In §5 we
discuss some examples. In §6 we remark the generalizations from ${\mathbf{C}}$
to affine algebraic curves and other domains ($1$-dimensional).

######  Remark 1.11.

We would like to refer to some other results related to the present subject
and remarks.

  1. (i)

J. Tsimerman [27] gave another proof to the above Ax-Schanuel Theorem [1] for
$A=({\mathbf{C}}^{*})^{n}$ by means of the “o-minimal structure” theory, and
similarly Y. Peterzil and S. Starchenko [24] for semi-abelian varieties.

  2. (ii)

In Noguchi [17], there is a direct application of the Log Bloch-Ochiai Theorem
to the proof of Raynaud’s Theorem (Manin-Mumford Conjecture) through the aid
of the “o-minimal structure” theory.

  3. (iii)

In the analytic theory we can think of the “values”; it might be an advantage
of the analytic theory, compared with the formally analytic theory.

Acknowledgment. The author is grateful to Professors P. Corvaja and U. Zannier
for interesting discussions on topics related to Diophantine approximation.

## 2\. Order functions and a key lemma

### 2.1. Order functions

In general for order functions, cf. Hayman [9] Chap’s. 1, 2, Noguchi-Ochiai
[18] Chap. V, and Noguchi-Winkelmann [20] Chap. 2.

Let $X$ be a compact complex space with reduced structure sheaf
${\mathcal{O}}_{X}$, and let $\mathscr{I}\subset{\mathcal{O}}_{X}$ be a
coherent ideal sheaf. Let $f:{\mathbf{C}}\to X$ be an entire curve. We are
going to define an order function $T_{f}(r,\omega_{\mathscr{I}})$ of $f$ with
respect to $\mathscr{I}$ (see [20] §2.4 for details: Note that the projective
algebraic condition on $X$ is unnecessary because of $1$-dimensionality of the
domain ${\mathbf{C}}$).

The pull-back $f^{*}\mathscr{I}$ is an effective divisor on ${\mathbf{C}}$,
unless $f({\mathbf{C}})\subset\mathrm{Supp}\,\mathscr{I}~{}(:=\\{x\in
X:\mathscr{I}_{x}\not={\mathcal{O}}(X)_{x}\\})$, which we assume. Denoting by
${\mathrm{ord}}_{z}f^{*}\mathscr{I}$ the order of $f^{*}\mathscr{I}$ at
$z\in{\mathbf{C}}$, we define the counting functions of $f^{*}\mathscr{I}$
truncated at level $l\in\mathbf{N}\cup\\{\infty\\}$ by

$n_{l}(t,f^{*}\mathscr{I})=\sum_{|z|<t}\min\\{{\mathrm{ord}}_{z}f^{*}\mathscr{I},l\\},\quad
N_{l}(t,f^{*}\mathscr{I})=\int_{1}^{r}\frac{n_{l}(t,f^{*}\mathscr{I})}{t}dt,\quad
r>1.$

Let $\phi_{\mathscr{I}}(x)~{}(x\in X)$ be the proximity potential of
$\mathscr{I}$ (see [20] §2.4). Then we have the pull-back
$f^{*}\phi_{\mathscr{I}}(z)$ such that it is of $C^{\infty}$-class outside the
singular set $f^{-1}(\mathrm{Supp}\,\mathscr{I})$ and at a singular point
$a\in f^{-1}(\mathrm{Supp}\,\mathscr{I})$ it is written locally as

$f^{*}\phi_{\mathscr{I}}(z)=\lambda_{a}\log|z-a|+\hbox{$C^{\infty}$-term},$

where $\lambda_{a}\in\mathbf{N}$ (positive integers). We set the proximity
function of $f$ for $\mathscr{I}$ by

$m_{f}(r,\mathscr{I})=\frac{1}{2\pi}\int_{|z|=r}-f^{*}\phi_{\mathscr{I}}(re^{i\theta})d\theta.$

With $\omega_{\mathscr{I}}:=\frac{1}{\pi
i}{\partial}\bar{\partial}\phi_{\mathscr{I}}$ we define the order function of
$f$ with respect to $\omega_{\mathscr{I}}$ by

$T_{f}(r,\omega_{\mathscr{I}})=\int_{1}^{r}\frac{dt}{t}\int_{|z|<t}f^{*}\omega_{\mathscr{I}}\quad(r>1).$

We have a so-called First Main Theorem for $f$ and $\mathscr{I}$ (see [20]
Theorem 2.4.9):

(2.1)
$T_{f}(r,\omega_{\mathscr{I}})=N_{\infty}(r,f^{*}\mathscr{I})+m_{f}(r,\mathscr{I})-m_{f}(1,\mathscr{I}).$

In the case where $\mathscr{I}=\mathscr{I}{\langle}Y{\rangle}$ is the ideal
sheaf of an analytic subset $Y$ possibly non-reduced of $X$, we write

$f^{*}\mathscr{I}{\langle}Y{\rangle}=f^{*}Y,\quad\omega_{Y}=\omega_{\mathscr{I}{\langle}Y{\rangle}},\quad
T_{f}(r,\omega_{Y})=T_{f}(r,\omega_{\mathscr{I}{\langle}Y{\rangle}}).$

If $X$ is smooth and $Y$ is a divisor $D$ on $X$. Then we have the line bundle
$L(D)\to X$ associated with $D$, and the first Chern class $c_{1}(D)$. Then,
$\omega_{D}\in c_{1}(D)$, and it is common to write

$T_{f}(r,\omega_{D})=T_{f}(r,L(D))=T_{f}(r,c_{1}(D))$

for the order functions.

Assume that $X$ is projective algebraic. Let $D$ and $D^{\prime}$ be big
divisors on $X$ such that
$f({\mathbf{C}})\not\subset\mathrm{Supp}\,D\cup\mathrm{Supp}\,D^{\prime}$.
Then we have

$T_{f}(r,L(D))=O\left(T_{f}(r,L(D^{\prime}))\right),\quad
T_{f}(r,L(D^{\prime}))=O\left(T_{f}(r,L(D))\right).$

Therefore in an estimate such as $O\left(T_{f}(r,L(D))\right)$ the choice of
of $D$ or $L(D)$ does not matter; in such a case we simply write $T_{f}(r)$
for $T_{f}(r,L(D))$ with respect to some ample or big line bundle $L(D)$ over
$X$ as far as $f({\mathbf{C}})\not\subset\mathrm{Supp}\,D$; however, once it
is chosen, it is fixed.

Note that the followings are equivalent for $f:{\mathbf{C}}\to X$:

######  2.2.

  1. (i)

$f$ is rational (not transcendental);

  2. (ii)

$\underset{r\to\infty}{\underline{\lim}}\;\frac{T_{f}(r)}{\log r}<\infty$.

  3. (iii)

$\underset{r\to\infty}{\overline{\lim}}\;\frac{T_{f}(r)}{\log r}<\infty$;
i.e., $T_{f}(r)=O(\log r)$.

The order $\rho_{f}$ of $f:{\mathbf{C}}\to X$ is defined by

$0\leq\rho_{f}=\underset{r\to\infty}{\overline{\lim}}\;\frac{\log
T_{f}(r)}{\log r}\leq\infty.$

If $\rho_{f}<\infty$, $f$ is said to be of finite order and

$T_{f}(r)=O(r^{\rho_{f}+\varepsilon}),\qquad^{\forall}\varepsilon>0.$

### 2.2. Key lemma

Let $A$ be a semi-abelian variety. Here we fix an isomorphism
${\mathop{\mathrm{Lie}}}(A)\cong{\mathbf{C}}^{n}$ with coordinates
$(z_{1},\ldots,z_{n})$. Let $f=(f_{1},\ldots,f_{n}):z\in{\mathbf{C}}\to
f(z)\in{\mathbf{C}}^{n}(\cong{\mathop{\mathrm{Lie}}}(A))$ be an entire curve.
In the present paper we assume that $f$ is non-constant. We denote by
$T(r,f_{j})$ Nevanlinna’s order function (see [9], [20] Chap. 1), and set

$T_{f}(r)=\max_{1\leq j\leq n}T(r,f_{j}).$

Let $\bar{A}$ be an equivariant smooth projective compactification of $A$ with
boundary divisor ${\partial}A=\bar{A}\setminus A$ of simple normal crossing
type (cf. [20] Chap. 5), and let $L\to{\bar{A}}$ be an ample line bundle. We
set the order function of $\exp_{A}f:{\mathbf{C}}\to A$ by

$T_{\exp_{A}f}(r)=T_{\exp_{A}f}(r,L).$

For $\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f=(\exp_{A}f,f):{\mathbf{C}}\to
A\times{\mathbf{C}}^{n}$ we set

$T_{\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r):=T_{\exp_{A}f}(r)+T_{f}(r).$

By [20] Theorem 6.1.9 and 2.2 above we see

######  Proposition 2.3.

Let the notation be as above. The followings are equivalent:

  1. (i)

$f:{\mathbf{C}}\to{\mathbf{C}}^{n}$ is rational (i.e., $f_{j}$ are
polynomials).

  2. (ii)

$\exp_{A}f$ is of finite order.

  3. (iii)

$\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f$ is of finite order.

As usual in Nevanlinna theory we use the symbol “$S_{\exp_{A}f}(r)$” for a
small term such that

(2.4) $S_{\exp_{A}f}(r)=O(\log^{+}T_{\exp_{A}f}(r))+O(\log r)+O(1)~{}~{}||,$

where “$||$” stands for the validity of the estimate except for $r$ in
exceptional intervals with finite total length, and for $\exp_{A}f$ of finite
order there are no such exceptional intervals. We use the notation through the
paper.

The following is the key lemma for the estimates in the arguments henceforth.

######  Lemma 2.5.

With the notation above, we have:

  1. (i)

$T_{f}(r)=S_{\exp_{A}f}(r)$.

  2. (ii)

$T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r)=T_{\exp_{A}f}(r)+S_{\exp_{A}f}(r)$.

Proof.  It suffices to prove (i). When the order of $\exp_{A}f$ is finite,
then $f$ is rational, and so $T_{f}(r)=O(\log r)$ without exceptional
intervals.

In general, we take a representation of the semi-abelian variety $A$

$0\to({\mathbf{C}}^{*})^{p}\to A\to A_{0}\to 0,$

where $A_{0}$ is an abelian variety. Then $A$ has a structure of locally flat
$({\mathbf{C}}^{*})^{p}$-principal bundle with transition transformation by
$(S^{1})^{p}:=\\{\zeta\in{\mathbf{C}}:|\zeta|=1\\}^{p}$-multiplication (cf.
[20] §6.1). After a change of indices of the coordinates $(z_{j})$ and a
linear transform of $(z_{j})$ we have the following expression of the order
function $T_{\exp_{A}f}(r)$ (cf. ibid.):

(2.6) $\displaystyle f(z)$
$\displaystyle=(f_{1}(z),\ldots,f_{p}(z),f_{p+1},\ldots,f_{n}(z)),$
$\displaystyle T_{1}(r)$
$\displaystyle:=\sum_{j=1}^{p}T\left(r,e^{f_{j}}\right),$ $\displaystyle
T_{2}(r)$
$\displaystyle:=\frac{1}{4\pi}\int_{|z|=r}\sum_{j+1}^{n}|f_{j}(z)|^{2}\,d\theta-\frac{1}{4\pi}\int_{|z|=1}\sum_{j+1}^{n}|f_{j}(z)|^{2}\,d\theta,$
$\displaystyle T_{\exp_{A}f}(r)$ $\displaystyle=T_{1}(r)+T_{2}(r).$

For $f_{j}~{}(p+1\leq j\leq n)$ we have

$\displaystyle T(r,f_{j})$
$\displaystyle=\frac{1}{2\pi}\int_{|z|=r}\log^{+}|f_{j}|\,d\theta\leq\frac{1}{4\pi}\int_{|z|=r}\log(1+|f_{j}|^{2})\,d\theta$
$\displaystyle=\frac{1}{2}\log\left(1+\frac{1}{2\pi}\int_{|z|=r}|f_{j}|^{2}\,d\theta\right)\leq\frac{1}{2}\log^{+}T_{2}(r)+O(1)$
$\displaystyle=O(\log^{+}T_{\exp_{A}f}(r))+O(1).$

For $f_{j}~{}(1\leq j\leq p)$ we have

$T\left(r,e^{f_{j}}\right)=\frac{1}{2\pi}\int_{|z|=r}\log^{+}\left|e^{f_{j}(z)}\right|\,d\theta=\frac{1}{2\pi}\int_{|z|=r}{\Re\,}^{+}f_{j}(z)\,d\theta,$

where ${\Re\,}^{+}f_{j}:=\max\\{{\Re\,}f_{j},0\\}$ with the real part
${\Re\,}f_{j}$. With the imaginary part ${\Im\,}f_{j}(0)$ and the complex
Poisson kernel we write

$f_{j}(z)=\frac{1}{2\pi}\int_{|\zeta|=R}\frac{\zeta+z}{\zeta-z}{\Re\,}f_{j}(\zeta)\,d\theta+{\Im\,}f_{j}(0).$

For $|z|=r<R$ we get

$|f_{j}(z)|\leq\frac{R+r}{R-r}\cdot\frac{1}{2\pi}\int_{|\zeta|=R}{\Re\,}^{+}f_{j}(\zeta)\,d\theta+|{\Im\,}f_{j}(0)|.$

Then we have

$\displaystyle T(r,f_{j})$
$\displaystyle\leq\frac{1}{2\pi}\int_{|\zeta|=r}\log(1+|f_{j}(z)|)\,d\theta\leq\log\left(1+\frac{1}{2\pi}\int_{|z|=r}|f_{j}(z)|\,d\theta\right)$
$\displaystyle\leq\log\left(1+\frac{R+r}{R-r}\cdot\frac{1}{2\pi}\int_{|\zeta|=R}{\Re\,}^{+}f_{j}(\zeta)\,d\theta+O(1)\right)$
$\displaystyle\leq\log\left(1+\frac{R+r}{R-r}\cdot T_{1}(R)+O(1)\right)$
$\displaystyle\leq\log^{+}\left(\frac{R+r}{R-r}\cdot T_{1}(R)\right)+O(1)$

Now we take $R=r+1/T_{1}(r)$, so that

$\displaystyle T(r,f_{j})\leq\log^{+}\left((2r+1)T_{1}(r)\cdot
T_{1}\left(r+\frac{1}{T_{1}(r)}\right)\right)+O(1)$

Borel’s Lemma (cf. Hayman [9] Lemma 2.4) implies

$T_{1}\left(r+\frac{1}{T_{1}(r)}\right)\leq 2T_{1}(r)\quad||.$

Therefore it follows that

$\displaystyle T(r,f_{j})$ $\displaystyle=O(\log^{+}T_{1}(r))+O(\log r)+O(1)$
$\displaystyle=O(\log^{+}T_{\exp_{A}f}(r))+O(\log r)+O(1)\quad||.$

The proof is completed. ∎

## 3\. Proof of Theorem 1.3

By [14] we see that the Zariski closure
$\overline{\exp_{A}f({\mathbf{C}})}^{\mathrm{Zar}}$ in $A$ is a translate of a
connected algebraic subgroup (a semi-abelian subvariety) of $A$. It follows
from $A$-nondegeneracy 1.2 that
$\overline{\exp_{A}f({\mathbf{C}})}^{\mathrm{Zar}}=A$, so that

(3.1) ${\mathrm{tr.}\deg}\hskip 0.3pt_{\mathbf{C}}\,\exp_{A}f=n.$

Let ${\mathbf{C}}(A)$ (resp. ${\mathbf{C}}(f)$) be the rational function field
of $A$ (resp. the field generated by $f_{j}~{}(1\leq j\leq n)$ over
${\mathbf{C}}$). We denote by ${\mathrm{tr.}\deg}\hskip
0.3pt_{{\mathbf{C}}(f)}(\exp_{A}f)^{*}{\mathbf{C}}(A)$ the transcendence
degree of the pull-backed field $(\exp_{A}f)^{*}{\mathbf{C}}(A)$ over
${\mathbf{C}}(f)$. We prove:

######  Lemma 3.2.

With the notation above we have

(3.3) ${\mathrm{tr.}\deg}\hskip
0.3pt_{{\mathbf{C}}(f)}\,(\exp_{A}f)^{*}{\mathbf{C}}(A)\geq 1.$

Proof.  We take a transcendence basis $\\{\phi_{j}\\}_{j=1}^{n}$ of
${\mathbf{C}}(A)$ over ${\mathbf{C}}$ such that
$\hat{\phi}_{j}:=\phi_{j}\circ\exp_{A}f$ are defined as non-constant
meromorphic functions, and

$\hat{\phi}:=\left(\hat{\phi}_{1},\ldots,\hat{\phi}_{n}\right).$

Assume contrarily that (3.3) is false; i.e.,

(3.4) ${\mathrm{tr.}\deg}\hskip
0.3pt_{{\mathbf{C}}(f)}\hat{\phi}=0\quad(\hbox{regarded as
}\hat{\phi}=\\{\hat{\phi}_{j}\\}_{j=1}^{n}).$

Then all $\hat{\phi}_{j}$ are algebraic over ${\mathbf{C}}(f)$. There are non-
zero polynomials $P_{j}(t)$ in one variable with coefficients in
${\mathbf{C}}(f)$ such that

(3.5) $P_{j}(\hat{\phi}_{j})=0,\quad 1\leq j\leq n.$

By [20] Lemma 2.5.15 we have

$T(r,\hat{\phi}_{j})=O(T_{f}(r))+O(1).$

With setting $\widehat{T}(r):=\max_{1\leq j\leq n}T(r,\hat{\phi}_{j})$ we thus
obtain

$\widehat{T}(r)=O(T_{f}(r))+O(1).$

On the other hand, it follows from [20] Theorem 2.5.18 that

$T_{\exp_{A}f}(r)=O(\widehat{T}(r))+O(1).$

Therefore we see that

$T_{\exp_{A}f}(r)=O(T_{f}(r))+O(1).$

But this contradicts Lemma 2.5. ∎

Continuation of the proof of Theorem 1.3: By (3.1), ${\mathrm{tr.}\deg}\hskip
0.3pt_{{\mathbf{C}}}\\{f,\hat{\phi}\\}\geq n$. For proof by contradiction we
assume that

${\mathrm{tr.}\deg}\hskip 0.3pt_{\mathbf{C}}\,\\{f,\hat{\phi}\\}=n.$

Then all $f_{j}$ are algebraic over ${\mathbf{C}}(\hat{\phi})$, so that there
are non-trivial algebraic relations,

(3.6)
$P_{j}(f_{j},\hat{\phi})=P_{j}(f_{j},\hat{\phi}_{1},\ldots,\hat{\phi}_{n})=0,\quad
1\leq j\leq n.$

If ${\mathrm{tr.}\deg}\hskip 0.3pt_{\mathbf{C}}\\{f_{j}\\}_{j=1}^{n}=n$, the
assumption implies ${\mathrm{tr.}\deg}\hskip
0.3pt_{{\mathbf{C}}(f)}\\{f,\hat{\phi}\\}=0$; this does not take place by
Lemma 3.2. Therefore ${\mathrm{tr.}\deg}\hskip
0.3pt_{\mathbf{C}}\\{f_{j}\\}_{j=1}^{n}<n$, and hence there is a non-trivial
algebraic relation over ${\mathbf{C}}$:

(3.7) $Q(f_{1},\ldots,f_{n})=0.$

If, to say, $f_{1}$ is contained in (3.7), we take the resultant of (3.7) and
(3.6) ($j=1$) with respect to $f_{1}$, which yields a non-trivial algebraic
relation

$Q_{1}(f_{2},\ldots,f_{n},\hat{\phi})=0.$

After repeating this process at most $n$-times we eliminate
$f_{1},\ldots,f_{n}$ in (3.7) to obtain a non-trivial algebraic relation

$\hat{Q}\left(\hat{\phi}_{1},\ldots,\hat{\phi}_{n}\right)=0\,;$

this again contradicts the algebraic independence of
$\hat{\phi}_{1},\ldots,\hat{\phi}_{n}$.

The proof of Theorem 1.3 is completed. ∎

######  Remark 3.8.

  1. (i)

It is noticed that the logarithmic function $\log(1+t)\in{\mathbf{C}}[[t]]$
can be dealt with in the Ax-Schanuel Theorem as a formal power series, but
cannot in our Theorem 1.3. To deal with the case of a finite unit disk as a
domain instead of ${\mathbf{C}}$ we need some growth condition for
$T_{\exp_{A}f}(r)$ (cf. §6.3).

  2. (ii)

Let $\exp_{A}:{\mathbf{C}}^{n}={\mathop{\mathrm{Lie}}}(A)\to A$ be as above.
We have a semi-lattice
$\Lambda={\mathrm{Ker}\,}\exp_{A}\subset{\mathbf{C}}^{n}$ (the periods of
$A$). Then an entire curve $f:{\mathbf{C}}\to{\mathbf{C}}^{n}$ is $A$-(resp.
non)degenerate if and only if there is a (resp. no) complex vector subspace
$E\subsetneq{\mathbf{C}}^{n}$ such that $E\supset(f({\mathbf{C}})-f(0))$ and
$E/(E\cap\Lambda)$ is a semi-abelian variety. Therefore if $\Lambda$ is
concerned, it would be better to say $f$ being $\Lambda$-(non)degenerate.

## 4\. Nevanlinna theory of entire curves $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f$

### 4.1. Back ground

In the same monograph [11] (1966) as Schanuel’s Conjecture 1.4 was mentioned,
S. Lang raised an interesting question (p. 32):

######  4.1.

  1. (i)

Let $\varphi:{\mathbf{C}}\to A$ be a $1$-parameter subgroup of an abelian
variety $A$ (say Zariski dense), and let $D$ be a hyperplane section of $A$.
Then, is $\varphi({\mathbf{C}})\cap D\not=\emptyset$?

  2. (ii)

And unless $\psi$ is algebraic, is the cardinality $|\varphi({\mathbf{C}})\cap
D|=\infty$?

It has developed roughly as follows (a non-complete list):

######  4.2.

  1. (1)

J. Ax [2] (1972) gave an affirmative answer to 4.1 (i) above.

  2. (2)

P.A. Griffiths [8] (Problem F, 1972) generalized problem 4.1 (i) for entire
curves into $A$ (so-called Lang’s Conjecture).

  3. (3)

At the Taniguchi Symposium “Geometric Function Theory in Katata 1978 organized
by S. Murakami (chair), the author formulated a 2nd main theorem for entire
curves $f:{\mathbf{C}}\to A$ and a divisor $D$ on $A$ as a conjecture, which
implies (2) above and 4.1 (i) as well; see Noguchi–Ochiai [18] (p. 248,
1984/’90).

  4. (4)

Siu and Yeung [26] (1996) solved Lang’s Conjecture (2) above for entire curves
into abelian varieties, and Noguchi [16] (1998) generalized it for entire
curves into semi-abelian varieties with another proof, which unifies the
result for abelian varieties and the classical E. Borel’s results for
$({\mathbf{C}}^{*})^{n}$.

  5. (5)

Noguchi, Winkelmann and Yamanoi [21] (2000), [22] (2002) proved (3) above, the
2nd Main Theorem for entire curves into semi-abelian varieties, and finally in
[23] proved it with counting functions truncated at level one.

  6. (6)

P. Corvaja and J. Noguchi [5] (2012) solved affirmatively 4.1 (ii) for entire
curves into semi-abelian varieties, $f:{\mathbf{C}}\to A$ by making use of the
2nd Main Theorem of (5) above. It is noticed that 4.1 (ii) had been open even
for $1$-parameter subgroups of abelian varieties.

It is natural and interesting to ask questions similar to the above for

$\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f:z\in{\mathbf{C}}\to(\exp_{A}f(z),f(z))\in
A\times{\mathop{\mathrm{Lie}}}(A)$

in view of the analytic Ax-Schanuel Theorem 1.3 and Schanuel’s Conjecture 1.4.

### 4.2. 2nd Main Theorem

We denote by $S_{\varepsilon,\exp_{A}f}(r)~{}(\geq 0)$ a small term such that
for every $\varepsilon>0$

$S_{\varepsilon,\exp_{A}f}(r)\leq\varepsilon T_{\exp_{A}f}(r)+O(\log
r)~{}~{}||_{\varepsilon}$

(cf. (2.4)).

The main aim is to prove Theorem 1.6 for $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f:{\mathbf{C}}\to
A\times{\mathop{\mathrm{Lie}}}(A)$, which is very analogous to [20] Theorem
6.5.1 (cf. [23]) for $\exp_{A}f:{\mathbf{C}}\to A$. The proof of Theorem 1.6
is in fact an adaptation of the arguments in [20] Chap. 6 (cf. [22], [23]) by
making use of Lemmata 2.5, 4.11 for the projection $\widehat{I}_{k}$ of (4.10)
below. Henceforward we will sketch the key points.

The way to obtain the compactifications of $\bar{A}$ and $X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)$ in Theorem 1.6 is not written precisely in
[20] Theorem 6.5.1, but it follows from the arguments of the proof there.

### 4.3. Reduction

Let $f:{\mathbf{C}}\to{\mathop{\mathrm{Lie}}}(A)$ be an entire curve. By the
Log Bloch–Ochiai Theorem ([20] Theorem 6.2.1, [13], [14])
$\overline{\exp_{A}f({\mathbf{C}})}^{\mathrm{Zar}}$ is a translate of a
subgroup $B$ of $A$. By a translate we may assume that
$\overline{\exp_{A}f({\mathbf{C}})}^{\mathrm{Zar}}=B$. Then
$f({\mathbf{C}})\subset{\mathop{\mathrm{Lie}}}(B)~{}(\subset{\mathop{\mathrm{Lie}}}(A))$,
and

$\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f:{\mathbf{C}}\longrightarrow
B\times{\mathop{\mathrm{Lie}}}(B)\subset A\times{\mathop{\mathrm{Lie}}}(A).$

Now, $f:{\mathbf{C}}\to{\mathop{\mathrm{Lie}}}(B)$ is $B$-nondegenerate.
Therefore without loss of generality we may assume that $B=A$, i.e, $f$ is
$A$-nondegenerate.

### 4.4. Jet bundles

We keep the same notation as in the previous subsections. Let
$f:{\mathbf{C}}\to{\mathbf{C}}^{n}$ be an $A$-nondegenerate entire curve. We
would like to study the value distribution of $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f:{\mathbf{C}}\to A\times{\mathbf{C}}^{n}$.

Let $J_{k}(A)\to A$ (resp.
$J_{k}({\mathop{\mathrm{Lie}}}(A))\to{\mathop{\mathrm{Lie}}}(A)$) be the
$k$-th jet bundle over $A$ (resp. ${\mathop{\mathrm{Lie}}}(A)$). Because of
the flat structure of the logarithmic tangent (and cotangent, as well) bundle
over $A$ (cf. [20] §4.6.3), we have the trivializations:

(4.5) $\displaystyle\begin{array}[]{cc}J_{k}(A)\cong A\times
J_{k,A},&J_{k,A}\cong{\mathbf{C}}^{nk},\\\
J_{k}({\mathop{\mathrm{Lie}}}(A))\cong{\mathop{\mathrm{Lie}}}(A)\times
J_{k,{\mathop{\mathrm{Lie}}}(A)},&J_{k,{\mathop{\mathrm{Lie}}}(A)}\cong{\mathbf{C}}^{nk},\end{array}$

where $J_{k,A}$ and $J_{k,{\mathop{\mathrm{Lie}}}(A)}$ are the so-called jet-
parts of $A$ and ${\mathop{\mathrm{Lie}}}(A)$, respectively. Through the
exponential map $\exp_{A}:{\mathop{\mathrm{Lie}}}(A)\to A$ we have the natural
isomorphism $J_{k,A}\cong J_{k,{\mathop{\mathrm{Lie}}}(A)}$, which are
identified. Therefore we have

(4.6) $J_{k}(A\times{\mathop{\mathrm{Lie}}}(A))\cong
A\times{\mathop{\mathrm{Lie}}}(A)\times J_{k,A}\times J_{k,A}.$

Let $\Delta_{k}\subset J_{k,A}\times J_{k,A}$ be the diagonal, and let
$J_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f):{\mathbf{C}}\to
J_{k}(A\times{\mathop{\mathrm{Lie}}}(A))$ be the $k$-jet lift of $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f$. Then we see that

(4.7) $J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f):z\in{\mathbf{C}}\to(\exp_{A}f(z),f(z),J_{k,f}(z),J_{k,f}(z))\in
A\times{\mathop{\mathrm{Lie}}}(A)\times\Delta_{k},$

where $J_{k,f}$ is the jet part of
$J_{k}(f)=(f,J_{k,f})\in{\mathop{\mathrm{Lie}}}(A)\times J_{k,A}$. For the
sake of simplicity we identify $\Delta_{k}=J_{k,A}$ and write

(4.8) $J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f):z\in{\mathbf{C}}\to(\exp_{A}f(z),f(z),J_{k,f}(z))\in
A\times{\mathop{\mathrm{Lie}}}(A)\times J_{k,A}(\subset
J_{k}(A\times{\mathop{\mathrm{Lie}}}(A))).$

We put

(4.9) $\displaystyle{\widehat{J}_{k,A}}$
$\displaystyle={\mathop{\mathrm{Lie}}}(A)\times
J_{k,A}~{}\cong{\mathbf{C}}^{n}\times{\mathbf{C}}^{nk}\quad(\hbox{\em extended
jet part}),$ $\displaystyle X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)$
$\displaystyle=\overline{J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)({\mathbf{C}})}^{\mathrm{Zar}}\subset
A\times{\widehat{J}_{k,A}}.$

We define the extended jet projection by

(4.10) $\widehat{I}_{k}:X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)~{}(\subset
A\times{\widehat{J}_{k,A}})\longrightarrow{\widehat{J}_{k,A}},$

which will play the role of the jet projection $I_{k}$ (cf. [20] p. 151) for
entire curves into semi-abelian varieties in [20] Chap’s. 4–6.

Let $L_{\bar{A}}\to\bar{A}$ be a big line bundle over a projective
compactification $\bar{A}$ of $A$. We take a compactification $\hskip
3.0pt{\overline{\hskip-3.0pt\widehat{J}}_{k,A}}$ of ${\widehat{J}_{k,A}}$,
e.g.,

$\hskip
3.0pt{\overline{\hskip-3.0pt\widehat{J}}_{k,A}}=\overline{{\mathbf{C}}^{n}\times{\mathbf{C}}^{nk}}={\mathbf{P}^{n}(\mathbf{C})}\times{\mathbf{P}}^{nk}({\mathbf{C}}),$

and the ample line bundle $H=O_{\mathbf{P}^{n}(\mathbf{C})}(1)\otimes
O_{{\mathbf{P}}^{nk}({\mathbf{C}})}(1)\to\hskip
3.0pt{\overline{\hskip-3.0pt\widehat{J}}_{k,A}}$, with which we define

$T_{\widehat{I}_{k}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r)=T_{\widehat{I}_{k}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r,H).$

######  Lemma 4.11.

For $\widehat{I}_{k}$ we have

(4.12) $T_{\widehat{I}_{k}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r)=S_{\exp_{A}f}(r).$

Proof.  Since

$\widehat{I}_{k}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f):z\in{\mathbf{C}}\to(f(z),J_{k,\exp_{A}f}(z))\in{\mathop{\mathrm{Lie}}}(A)\times
J_{k,A},$

it follows from Lemma on logarithmic derivative for $\exp_{A}f$ ([13], [20]
§4.7) that

$T_{J_{k,\exp_{A}f}}(r)=S_{\exp_{A}f}(r).$

This combined with Lemma 2.5 implies (4.12). ∎

### 4.5. $A$-action

We consider an $A$-action on $A\times{\mathop{\mathrm{Lie}}}(A)\times
J_{k,A}\subset J_{k}(A\times{\mathop{\mathrm{Lie}}}(A))$ by

$(a,(x,v,w))\in A\times\left(A\times{\mathop{\mathrm{Lie}}}(A)\times
J_{k,A}\right)\to(a+x,v,w)\in A\times{\mathop{\mathrm{Lie}}}(A)\times
J_{k,A}.$

We denote the stabilizer subgroup of $X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)$ by

$\mathrm{St}(X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f))=\mathrm{St}_{A}(X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f))=\\{a\in A:a+X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)=X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)\\},$

and by $\mathrm{St}(X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f))^{0}$ the
identity component.

######  Lemma 4.13.

With the notation above, $\mathrm{St}(X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f))^{0}\not=\\{0\\}$.

Proof.  If otherwise, $\mathrm{St}(X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f))^{0}=\\{0\\}$. We consider the $l$-jet
space $J_{l}(X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f))$ of
$X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)$ (“jet of jet”) with induced
projection

$d^{l}\widehat{I}_{k}:J_{l}(X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f))~{}(\subset
J_{l}(A\times{\mathop{\mathrm{Lie}}}(A)\times J_{k,A}))\to
J_{l}({\mathop{\mathrm{Lie}}}(A)\times J_{k,A}).$

By [20] Lemma 6.2.4, there is a large number $l\in\mathbf{N}$ such that the
differential $d(d^{l}\widehat{I}_{k})$ is non-degenerate at general points of
$J_{l}(X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f))$. Therefore we have

$\displaystyle T_{\exp_{A}f}(r)$ $\displaystyle\leq T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r)=O\left(T_{J_{l}(J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f))}(r)\right)=O\left(T_{d^{l}\widehat{I}_{k}(J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f))}(r)\right).$

On the other hand, $T_{d^{l}\widehat{I}_{k}(J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f))}(r)=S_{\varepsilon,\exp_{A}f}(r)$ by Lemma
4.11; it is a contradiction. ∎

######  Proposition 4.14 (cf. [20] Theorem 6.2.6).

Let $B=\mathrm{St}(X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f))^{0}$ and
set the quotient map

(4.15) $q_{B}:X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)\to X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)/B~{}(\subset(A/B)\times{\mathop{\mathrm{Lie}}}(A)\times
J_{k,A}~{}).$

Then $T_{q_{B}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r)=S_{\exp_{A}f}(r)$.

Proof.  The semi-abelian variety $A/B$ acts on
$(A/B)\times{\mathop{\mathrm{Lie}}}(A)\times J_{k,A}$ by the translations of
the first factor and the identity for the other factors. Then
$\mathrm{St}_{A/B}(X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)/B)^{0}=\\{0\\}$. As in the proof of Lemma
4.13, with a large $l$ the projection

$\rho_{l}:J_{l}(X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)/B)\to
J_{l}({\mathop{\mathrm{Lie}}}(A)\times J_{k,A})$

has a non-degenerate differential $d\rho_{l}$ at general points (see [20]
Lemma 6.2.4). Therefore we have

$T_{d^{l}q_{B}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r)=O\left(T_{\rho_{l}\circ d^{l}q_{B}\circ
J_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r)\right),$

where $d^{l}q_{B}:J_{l}(X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f))\to
J_{l}(X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)/B)$ is the induced
morphism from $q_{B}$. It follows from $\rho_{l}\circ
d^{l}q_{B}=d^{l}\widehat{I}_{k}$ and Lemma 4.11 that

$T_{d^{l}q_{B}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r)=S_{\exp_{A}f}(r).$

On the other hand we have by Lemma 2.5

$T_{d^{l}q_{B}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r)=T_{q_{B}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r)+S_{\exp_{A}f}(r).$

Thus we deduce that $T_{q_{B}\circ J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r)=S_{\exp_{A}f}(r)$. ∎

### 4.6. Proof of Theorem 1.6

Let the notation be as in Theorem 1.6. Through the arguments of the proof of
[20] Theorem 6.5.6 with replacing the jet projection $I_{k}$ there by the
extended jet projection $\widehat{I}_{k}$ (cf. (4.10)) and $q_{k}^{B}$ by
$q_{B}$ (cf. (4.15)), we deduce that there are a number $l_{0}\in\mathbf{N}$
and a compactification $\bar{A}$ of $A$ such that

(4.16) $T_{J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r,\omega_{\bar{Z}})=N_{l_{0}}(r,J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}Z)+S_{\exp_{A}f}(r),$

where $\bar{X}_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)$ (resp.
$\bar{Z}$) is the closure of $X_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)$
(resp. $Z$) in $\bar{A}\times\hskip
3.0pt{\overline{\hskip-3.0pt\widehat{J}}_{k,A}}$.

Next, we show (ii). It follows from (4.16) that

$T_{J_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r,\omega_{\bar{Z}})\leq
l_{0}N_{1}(r,J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}Z)+S_{\exp_{A}f}(r).$

Now, making use of the assumption ${\mathrm{codim}}_{X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}Z\geqq 2$ together with Lemmata 2.5 and
4.11, we deduce from the arguments of [20] §6.5.3, adapted for $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f:{\mathbf{C}}\to
A\times{\mathop{\mathrm{Lie}}}(A)$ that

(4.17) $N_{1}(r,J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}Z)=S_{\varepsilon,\exp_{A}f}(r).$

Thus, (1.8) is deduced, and (ii) is finished.

Now, we go back to the proof of (i). It follows from the First Main Theorem
2.1 that

(4.18) $N_{1}(r,J_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}Z)\leq
N_{\infty}(r,J_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}Z)\leq
T_{J_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r,\bar{Z}).$

Note that (i) is finished in the case of ${\mathrm{codim}}_{X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}Z\geq 2$. Thus we consider the case where
$Z$ is an effective reduced divisor $D$ on $X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)$. Let $D=\sum D_{i}$ be the irreducible
decomposition. We deduce from (4.16) that

$\displaystyle T_{J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r,\omega_{\bar{D}})$ $\displaystyle\leq
N_{l_{0}}(r,J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}D)+S_{\exp_{A}f}(r)$ $\displaystyle\leq
N_{1}(r,J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}D)+l_{0}\sum_{i<j}N_{1}(r,J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}(D_{i}\cap D_{j}))$
$\displaystyle\quad+l_{0}\sum_{i}N_{1}(r,J_{k+1}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}J_{1}(D_{i}))+S_{\exp_{A}f}(r)$

(cf. [20] (6.5.51)). Since ${\mathrm{codim}}_{X_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}D_{i}\cap D_{j}\geq 2$, (4.17) implies

$N_{1}(r,J_{k}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}(D_{i}\cap
D_{j}))=S_{\varepsilon,\exp_{A}f}(r).$

We have $J_{k+1}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f):{\mathbf{C}}\to
X_{k+1}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)\subset
J_{k+1}(A\times{\mathop{\mathrm{Lie}}}(A))$ and $B=\mathrm{St}(X_{k+1}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f))^{0}$. For each $D_{i}$ we have two cases:
(1) $B\subset\mathrm{St}(D_{i})^{0}$ and (2)
$B\not\subset\mathrm{St}(D_{i})^{0}$. In the first case (1) we have by using
$q_{B}$ that

$N_{1}(r,J_{k+1}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}J_{1}(D))=S_{\exp_{A}f}(r)$

(see [20] §6.5.4 (b)). In the second case (2), we have by [20] Lemma 6.5.50

${\mathrm{codim}}_{X_{k+1}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(X_{k+1}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)\cap J_{1}(D_{i}))\geq 2.$

Then it is deduced from (4.17) with $k+1$ that

$N_{1}(r,J_{k+1}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}J_{1}(D))=S_{\varepsilon,\exp_{A}f}(r).$

Thus, (1.7) follows.

(iii) The case of $k=0$ is a special case of (i); the proof of Theorem 1.6 is
completed. ∎

We consider the fundamental case where $k=0$ and $Z$ is a reduced divisor $D$
on $X_{0}(f)$. Let $p_{1}:X_{0}(f)\,(\subset
A\times{\mathop{\mathrm{Lie}}}(A))\to A$ be the projection to the first factor
$A$.

######  Definition 4.19.

We say that $D$ is $A$-big if for a big divisor $E$ on $A$ (i.e., the closure
$\bar{E}$ in a compactification $\bar{A}$ of $A$ is big) the complete linear
system $|mD-p_{1}^{*}E|$ with large $m\in\mathbf{N}$ contains an effective
divisor on $X_{0}(f)$.

If $D$ is $A$-big, then

(4.20) $\displaystyle T_{\exp_{A}f}(r)$ $\displaystyle=O(T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r,\omega_{\bar{D}})),$ $\displaystyle
T_{\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r)$ $\displaystyle=O(T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r,\omega_{\bar{D}}))\quad||.$

######  Corollary 4.21.

Let $f:{\mathbf{C}}\to{\mathop{\mathrm{Lie}}}(A)$ be an $A$-nondegenerate
entire curve and let $D$ be a reduced divisor on $X_{0}(f)$. If
${\mathrm{ord}}_{z}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}D\geq 2$ for
all $z\in\mathrm{Supp}\,(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}D$
except for finitely points of $\mathrm{Supp}\,(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}D$, then $D$ is not $A$-big.

Proof.  If $D$ is $A$-big, it follows from the 2nd Main Theorem 1.6 and (4.20)
that

$\displaystyle T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r,\omega_{\bar{D}})$ $\displaystyle\leq
N_{1}(r,(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}D)+S_{\varepsilon,\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}$
$\displaystyle\leq\frac{1}{2}N_{\infty}(r,(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}D)+S_{\varepsilon,\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}$ $\displaystyle\leq\frac{1}{2}T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r,\omega_{\bar{D}})+\varepsilon T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r,\omega_{\bar{D}})+O(\log
r)~{}~{}||_{\varepsilon}.$

This implies a contradiction ‘$1\leq\frac{1}{2}$’. ∎

######  Remark 4.22.

This corollary is motivated through a discussion with Corvaja and Zannier on
their related or analogous results in rational function fields of Corvaja and
Zannier [7] (see [15] too).

### 4.7. Proof of Theorem 1.10

With the notation of the theorem we set

$Z=\overline{\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f({\mathbf{C}})\cap
D}^{\mathrm{Zar}}\subset D.$

If ${\mathrm{codim}}_{X_{0}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)}Z\geq
2$, Theorem 1.6 (ii) would imply

$N_{1}(r,(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}D)=N_{1}(r,(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}Z)=S_{\varepsilon,\exp_{A}f}(r).$

It follows from Theorem 1.6 (i) and (4.20) that for every $\varepsilon>0$

$T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r,\omega_{\bar{D}})\leq\varepsilon
T_{\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r,\omega_{\bar{D}})+O(\log
r)~{}~{}||_{\varepsilon}:$

This is a contradiction.

Therefore, $Z$ has an irreducible component of codimension one in
$X_{0}(\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f)$, which is an irreducible
component of $D$. ∎

## 5\. Examples

To discuss examples it is convenient to write
${\mathop{\mathrm{Lie}}}(A)\times A$ for $A\times{\mathop{\mathrm{Lie}}}(A)$,
so that in this section we use the notation

$\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f=(f,\exp_{A}f):{\mathbf{C}}\longrightarrow{\mathop{\mathrm{Lie}}}(A)\times
A;$

there will be no confusion.

(a) The optimality of (3.1): Let $A=({\mathbf{C}}^{*})^{n}$ and let
$\alpha_{j},1\leq j\leq n$, be complex numbers, linearly independent over
${\mathbf{Q}}$. Then the entire curve
$\phi(z)=(\alpha_{1}z,\ldots,\alpha_{n}z)~{}(\in{\mathbf{C}}^{n}={\mathop{\mathrm{Lie}}}(A))$
is $A$-nondegenerate with the natural exponential map
$\exp:{\mathbf{C}}^{n}\ni(z_{j})\mapsto(e^{z_{j}})\in A$, and so

${\mathrm{tr.}\deg}\hskip 0.3pt_{\mathbf{C}}\\{z,e^{\alpha_{1}z},\ldots
e^{\alpha_{n}z}\\}=n+1.$

Let ${\mathbf{P}^{n}(\mathbf{C})}\supset{\mathbf{C}}^{n}$ and
$\bar{A}:={\mathbf{P}^{1}(\mathbf{C})}^{n}\supset A$ be the compactifications
and let $T_{\exp\phi}(r)$ and $T_{\hskip 1.0pt{\widehat{\mathrm{ex}}}\phi}(r)$
denote the order functions with respect to the products of point bundles. We
write $\hskip 1.0pt{\widehat{\mathrm{ex}}}\,\phi=(\phi,\exp\phi)$. Then,

$T_{\hskip 1.0pt{\widehat{\mathrm{ex}}}\,\phi}(r)=T_{\exp\phi}(r)+O(\log
r)=\frac{\sum_{j=1}^{n}|\alpha_{j}|}{\pi}r+O(\log r).$

Let $P(z_{1},\ldots,z_{n},w_{1},\ldots,w_{n})$ be a polynomial of degree
$d_{j}$ in $w_{j}~{}(1\leq j\leq n)$. We assume the condition:

######  5.1.

  1. (i)

The divisor on ${\mathbf{P}^{n}(\mathbf{C})}\times\bar{A}$ defined by the zero
of $P$ is reduced and equal to the closure $\bar{D}_{P}$ of the divisor
$D_{P}$ defined by $\\{P=0\\}\cap({\mathop{\mathrm{Lie}}}(A)\times A)$.

  2. (ii)

$D_{P}$ is $A$-big (see Definition 4.19).

We need this condition; otherwise, to say, if $P=z_{1}$, then
$D_{P}=\\{z_{1}=0\\}$ and $D_{P}$ is not $A$-big; there is only one root $z=0$
in $P(\hskip 1.0pt{\widehat{\mathrm{ex}}}\,\phi(z))=0$. We have

$T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\phi}(r,L(\bar{D}_{P}))=\frac{\sum_{j=1}^{n}d_{j}|\alpha_{j}|}{\pi}r+O(\log
r).$

It is yet, in general, hard to find a root of $P(\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\phi(z))=0$, but by Theorem 1.6 (iii)

(5.2) $\displaystyle T_{\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\phi}(r,L(\bar{D}_{P}))=N_{\infty}(r,(\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\phi)^{*}D_{P})+O(\log r)=N_{1}(r,(\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\phi)^{*}D_{P})+S_{\varepsilon,\exp\phi}(r).$

The above estimate of $N_{\infty}(r,(\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\phi)^{*}D_{P})$ is classical due to Borel-
Nevanlinna, but that of $N_{1}(r,(\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\phi)^{*}D_{P})$ is new; moreover from Theorem
1.10 we obtain

(5.3) $\overline{\hskip 1.0pt{\widehat{\mathrm{ex}}}\phi({\mathbf{C}})\cap
D_{P}}^{\mathrm{Zar}}=D_{P}.$

By Corollary 4.21 there is no entire function $g(z)$ such that

(5.4) $P(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f(z))=(g(z))^{m}\quad(m\in\mathbf{N}\,\geq
2).$

In view of the transcendence problem of $\pi$ and $e$ the above example in the
case of $n=2$ and complex vector $\varpi_{0}=(1,2\pi i)\in{\mathbf{C}}^{2}$ is
of a special interest. We consider the induced $1$-parameter subgroup

(5.5) $\displaystyle\phi_{0}(z)$
$\displaystyle=z\varpi_{0}\in{\mathbf{C}}^{2},\qquad z\in{\mathbf{C}},$
$\displaystyle\hskip 1.0pt{\widehat{\mathrm{ex}}}\,\phi_{0}(z)$
$\displaystyle=(z,2\pi iz,e^{z},e^{2\pi
iz})=(z_{1},z_{2},w_{1},w_{2})\in{\mathbf{C}}^{2}\times({\mathbf{C}}^{*})^{2}.$

Let $P(z_{1},z_{2},w_{1},w_{2})$ be a polynomial with integral coefficients
satisfying condition 5.1. If $P(\hskip
1.0pt{\widehat{\mathrm{ex}}}\,\phi_{0}(\zeta))=0$, then

$\zeta,~{}2\pi i\zeta,~{}e^{\zeta},~{}e^{2\pi i\zeta}$

are algebraically dependent, and there are infinitely many such points, for
which (5.3) and (5.2) hold.

(c) Let $\exp_{(j)}(z)~{}(j=1,2,\ldots)$ denote the $j$-times iteration of the
exponential function $e^{z}$. We set

$f_{1}(z)=z,~{}f_{j}(z)=\exp_{(j-1)}(z),~{}2\leq j\leq n.$

Then we have

${\mathrm{tr.}\deg}\hskip
0.3pt_{\mathbf{C}}\\{f_{1},\ldots,f_{n},e^{f_{1}},\ldots,e^{f_{n}}\\}={\mathrm{tr.}\deg}\hskip
0.3pt_{\mathbf{C}}\\{f_{1},\ldots,f_{n},f_{n+1}\\}=n+1.$

In this case, $\exp f$ is of infinite order and $T_{\exp f}(r)$ has a growth
such that $T_{\exp f}(r)\sim\exp_{(n)}(r)$.

(d) (Cf. Brownawell-Kubota [4]) Let
$\exp_{A}:{\mathop{\mathrm{Lie}}}(A)\cong{\mathbf{C}}^{n}\to A$ be an
exponential map of a semi-abelian variety $A$. In general if $f_{j}~{}(1\leq
j\leq n)$ are entire functions, linearly independent over ${\mathbf{C}}$, then
$f=(f_{j}):{\mathbf{C}}\to{\mathbf{C}}^{n}$ is $A$-nondegenerate. In
particular, let $n=l+m$ and let $\wp_{j}(w)~{}(1\leq j\leq m)$ be
Weierstrass’s pe-functions. Then

${\mathrm{tr.}\deg}\hskip
0.3pt_{\mathbf{C}}\\{f_{1},\ldots,f_{l+m},e^{f_{1}},\ldots,e^{f_{l}},\wp_{1}(f_{l+1}),\ldots,\wp_{m}(f_{l+m})\\}\geq
l+m+1.$

(e) Let $f_{1}(z)=z,f_{2}(z)=z$. Then they are not linearly independent over
${\mathbf{C}}$. Let $E_{j}~{}(j=1,2)$ be elliptic curves which are not
isogenous to each other. Let $A=E_{1}\times E_{2}$ and
$\exp_{A}:{\mathbf{C}}^{2}\to A$ be an exponential map. Then
$f=(f_{1},f_{2}):{\mathbf{C}}\to{\mathbf{C}}^{2}$ is $A$-nondegenerate, and so

${\mathrm{tr.}\deg}\hskip 0.3pt_{\mathbf{C}}\\{z,\wp_{1}(z),\wp_{2}(z)\\}=3.$

This should be known classically.

Let
$\overline{{\mathop{\mathrm{Lie}}}(E_{1})\times{\mathop{\mathrm{Lie}}}(E_{2})}\times
A={\mathbf{P}^{2}(\mathbf{C})}\times E_{1}\times E_{2}$ be the
compactification with product line bundle $L$ (resp. $L_{0}$) of the
hyperplane bundle and the point-bundles over
${\mathbf{P}^{2}(\mathbf{C})}\times E_{1}\times E_{2}$ (resp. $E_{1}\times
E_{2}$). Then, $L$ is ample and the order functions satisfy

$T_{\hskip 1.0pt{\widehat{\mathrm{ex}}}f}(r,L)=T_{\exp f}(r,L_{0})+2\log
r+O(1)=\frac{\pi
r^{2}}{2}\left(\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}+o(1)\right),$

where $\lambda_{j}$ is the surface area of the fundamental domain of
$\wp_{j}~{}(j=1,2)$. Let $P(z_{1},z_{2},w_{1},w_{2})$ be an irreducible
polynomial, involving $w_{1}$ and $w_{2}$, and of degree $d_{1}$ (resp.
$d_{2}$) with respect to $w_{1}$ (resp. $w_{2}$). We consider
$w_{j}=\wp_{j}~{}(j=1,2)$ a rational function of $E_{j}$, and denote by
$D_{P}$ the divisor defined by the zeros of $P$ on $Lie(A)\times A$. Let
$\Xi_{P}$ denote the zero divisor on ${\mathbf{C}}$ defined by
$P(z,z,\wp_{1}(z),\wp_{2}(z))=0$. Then we have

$N_{\infty}(r,\Xi_{P})=N_{1}(r,\Xi_{P})+S_{\varepsilon,\exp f}(r)={\pi
r^{2}}\left(\frac{d_{1}}{\lambda_{1}}+\frac{d_{2}}{\lambda_{2}}+o(1)\right)+S_{\exp
f}(r).$

It also follows from Theorem 1.10 that $\overline{\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f({\mathbf{C}})\cap D_{P}}^{\mathrm{Zar}}$
contains an irreducible component of $D_{P}$. By Corollary 4.21 we see that
there is no meromorphic function $g(z)$ on ${\mathbf{C}}$ satisfying

(5.6) $P(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f(z))=g(z)^{m}\quad(m\in\mathbf{N},\,\geq 2).$

(f) Set $f_{1}=z,f_{2}=z^{2},f_{3}=z$. Then these are not linearly independent
over ${\mathbf{C}}$. Let $\exp_{E}:{\mathbf{C}}\to E$ be an exponential map of
an elliptic curve $E$ with Weierstrass’ $\wp(w)$. Set
$A=({\mathbf{C}}^{*})^{2}\times E$ with $\exp_{A}:{\mathbf{C}}^{3}\to A$. Then
$f=(f_{j}):{\mathbf{C}}\to{\mathbf{C}}^{3}$ is $A$-nondegenerate, and so

${\mathrm{tr.}\deg}\hskip 0.3pt_{\mathbf{C}}\\{z,e^{z},e^{z^{2}},\wp(z)\\}=4.$

The order function of $\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f$ has a growth,
$T_{\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f}(r)\sim r^{2}$.

## 6\. Remarks to affine algebraic curves and other domains

### 6.1. Affine algebraic curves

In this section $A$ denotes a semi-abelian variety.

Let $R$ be a complex affine algebraic curve and let
$f:R\to{\mathop{\mathrm{Lie}}}(A)$ be a holomorphic curve. We may consider
$\exp_{A}f:R\to A$ and $\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f:R\to
A\times{\mathop{\mathrm{Lie}}}(A)$.

In general for a holomorphic curve $g:R\to A$ the arguments up to obtaining an
estimate such as (4.16) work (cf. [20] Chap. 6), but further to advance to the
estimates of the 2nd Main Theorem 1.6 with counting functions truncated at
level one, we need to lift $g$ to $\tilde{g}:R\to{\mathop{\mathrm{Lie}}}(A)$,
which does not exists in general, since $R$ is not simply connected. But in
the present case we begin with a holomorphic curve
$f:R\to{\mathop{\mathrm{Lie}}}(A)$, which is a lift of $\exp_{A}f:R\to A$.
Therefore we can advance the arguments further there.

Let $\bar{R}$ be the compactification of $R$ by adding a finite number of
points. To study $\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f$ we may localize
the problem about an infinite point $a\in\bar{R}\setminus R$; we take a disk
neighborhood $\Delta$ of $a$ in $\bar{R}$. Then
$\Delta^{*}=\Delta\setminus\\{a\\}$ is a punctured disk, and the analysis of
transcendental properties of $\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f$ is
reduced to that of the restriction $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f|_{\Delta^{*}}$ (see the next).

### 6.2. Punctured disk

Let $\Delta^{*}$ be a punctured disk. As mentioned above, although
$\Delta^{*}$ is not simply connected, it does not cause a difficulty here,
since we give in first a holomorphic curve

(6.1)
$f:z\in\Delta^{*}\longrightarrow(f_{1}(z),f_{2}(z),\ldots,f_{n}(z))\in{\mathbf{C}}^{n}\cong{\mathop{\mathrm{Lie}}}(A).$

For a notational convenience we put the puncture at infinity and introduce a
coordinate $z$ such that

$\Delta^{*}=\\{|z|>1\\},\qquad\Delta=\Delta^{*}\cup\\{\infty\\}.$

Let $F(z)$ be one of $f_{j}(z)$. Then $F(z)$ is expanded to a Laurent series

(6.2) $F(z)=\sum_{\nu>0}c_{\nu}z^{\nu}+\sum_{\nu\leq
0}c_{\nu}z^{\nu}=F_{\mathrm{m}}(z)+F_{\mathrm{b}}(z).$

Fix $r_{0}>1$. Then $F_{\mathrm{b}}(z)$ and their derivatives are bounded in
$\\{|z|\geq r_{0}\\}$, and $F_{\mathrm{m}}(z)$ is the main part of the
expansion:

(6.3)
$\frac{d^{k}}{dz^{k}}F(z)=\frac{d^{k}}{dz^{k}}F_{\mathrm{m}}(z)+O(1),\qquad|z|\geq
r_{0},~{}k\geq 0.$

Note that $F_{\mathrm{m}}(z)$ is holomorphic in ${\mathbf{C}}$. Applying the
key Lemma 2.5 for $F_{\mathrm{m}}(z)$, we deduce the key Lemma 2.5 for $f$ ,
$\exp_{A}f$ and $r>r_{0}$. We can then deduce the 2nd Main Theorem 1.6 for
$\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f:\Delta^{*}\to
A\times{\mathop{\mathrm{Lie}}}(A)$ and $J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)$ as well for $r>r_{0}$.

### 6.3. Finite disk

We consider the case of a disk of ${\mathbf{C}}$ with finite radius, to say,
the unit disk $\Delta$. Let $f:\Delta\to{\mathop{\mathrm{Lie}}}(A)$ be a
holomorphic curve. In this hyperbolic case, to make the proofs of key Lemma
2.5 and Lemma on logarithmic derivatives to work at least for a sequence
$r_{\nu}\nearrow 1~{}(\nu\to\infty)$, we need a technical condition on the
growth of the order function $T_{\exp_{A}f}(r)$ such that

(6.4) $\underset{r\to
1}{\overline{\lim}}\;\frac{T_{\exp_{A}f}(r)}{\log\frac{1}{1-r}}=\infty$

(cf. Nevanlinna [12] Chap. VI, Hayman [9] §2.3). Under this condition the 2nd
Main Theorem 1.6 for $\hskip 1.0pt{\widehat{\mathrm{ex}}}_{A}f:\Delta\to
A\times{\mathop{\mathrm{Lie}}}(A)$ is deduced.

### 6.4. Open Riemann surfaces

Let $R$ be an open Riemann surface. The generalizations of Nevanlinna theory
for meromorphic functions on $R$, holomorphic maps from $R$ to another Riemann
surface and holomorphic curves from $R$ into ${\mathbf{P}^{n}(\mathbf{C})}$
are classical (cf., e.g., Sario-Noshiro [25], Wu [29]). There one uses a
finite (hyperbolic case) or infinite (parabolic case) exhaustion function
$\tau:R\to[0,r_{0})$ with $r_{0}\leq\infty$ such that $\tau$ is harmonic
outside a compact subset of $R$.

Similarly it is formally possible to extend the 2nd Main Theorem 1.6 for
holomorphic curves $f:R\to{\mathop{\mathrm{Lie}}}(A)$ and $\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f:R\to A\times{\mathop{\mathrm{Lie}}}(A)$.
There we use the differential ${\partial}\tau$, holomorphic where $\tau$ is
harmonic. For a logarithmic $1$-form $\omega$ on $A$ we take the ratio
$F=(\exp_{A}f)^{*}\omega/{\partial}\tau$, which may have poles at zeros of
${\partial}\tau$. The counting functions of those zeros is the counting
function $E_{R}(r)$ of the Euler numbers of $\\{\tau<r\\}$, which appears in
the estimates (4.16) and (1.7) under a growth assumption such as (6.4) with
$r_{0}=1$ in hyperbolic case (i.e., $r_{0}<\infty$):

(6.5) $\displaystyle T_{J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r,\omega_{\bar{Z}})=N_{l_{0}}(r,J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}Z)+C(k,l_{0})E_{R}(r)$
$\displaystyle\hskip 120.0pt+O(\log^{+}T_{\exp_{A}f}(r))+O(\log r)~{}~{}||,$
(6.6) $\displaystyle(1-\varepsilon)T_{J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)}(r,\omega_{\bar{Z},J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)})\leq N_{1}(r,J_{k}(\hskip
1.0pt{\widehat{\mathrm{ex}}}_{A}f)^{*}Z)+C^{\prime}(\varepsilon,k)E_{R}(r)+O(\log
r)~{}~{}||_{\varepsilon}.$

Here $C(k,l_{0})$ and $C^{\prime}(\varepsilon,k)$ are positive constants such
that $C(k,l_{0}),C^{\prime}(\varepsilon,k)\nearrow\infty$ as
$k,l_{0}\nearrow\infty$ and $\varepsilon\searrow 0$; there are no estimates
for $C(k,l_{0}),C^{\prime}(\varepsilon,k)$. Therefore, in order to obtain a
meaningful consequence we need a technical condition (besides (6.4) in
hyperbolic case) such that

$\underset{r\to r_{0}}{\overline{\lim}}\;\frac{E_{R}(r)}{T_{\exp_{A}f}(r)}=0.$

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Junjiro Noguchi

Graduate School of Mathematical Sciences

The University of Tokyo

Komaba, Meguro-ku, Tokyo 153-8914, Japan

e-mail<EMAIL_ADDRESS>