#  Boundary asymptotics of the relative Bergman kernel metric for curves

Robert Xin Dong

###### Abstract

We study the behaviors of the relative Bergman kernel metrics on holomorphic
families of degenerating hyperelliptic Riemann surfaces and their Jacobian
varieties. Near a node or cusp, we obtain precise asymptotic formulas with
explicit coefficients. In general the Bergman kernels on a given cuspidal
family do not always converge to that on the regular part of the limiting
surface, which is different from the nodal case. It turns out that information
on both the singularity and complex structure contributes to various
asymptotic behaviors of the Bergman kernel. Our method involves the classical
Taylor expansion for Abelian differentials and period matrices.

††footnotetext:  Keywords. Bergman kernel metric, cusp, degeneration, Hodge
theory, hyperelliptic Riemann surface, Jacobian variety, node, Ohsawa-
Takegoshi $L^{2}$ extension, period matrix, positivity of direct image
Mathematics Subject Classification. Primary 32A25; Secondary 32G20, 32G15,
14H45, 32Q28, 14D06

## Introduction

On an $n$-dimensional complex manifold $X$, the Bergman kernel is a
reproducing kernel of the space of square integrable holomorphic top forms.
For the canonical bundle $\mathcal{K}$, which is the $n$th exterior power of
the cotangent bundle on $X$, the Bergman kernel is defined as

$\kappa:=\sum e_{j}\otimes\overline{e_{j}},$ (0.1)

where $\\{e_{j}\\}$ is a complete orthonormal basis of
$H_{L^{2}}^{0}(X,\mathcal{K})$. Thus, the Bergman kernel is a canonical
volume-form determined only by the complex structure. As the complex structure
deforms, the log-plurisubharmonic variation results of the Bergman kernels on
pseudoconvex domains were obtained by Maitani and Yamaguchi [49] and later by
Berndtsson [3, 4]. More generally, further important developments on general
Stein manifolds and complex projective algebraic manifolds [4, 70, 9, 48, 59,
68, 20] state certain positivity properties of the direct images of the
relative canonical bundles, and turn out to have intimate connections with the
space of Kähler metrics [5, 21, 22, 69], the invariance of plurigenera [47,
63, 64, 57, 10], and the sharp Ohsawa-Takegoshi theorem [56, 51, 62, 11, 36,
8, 13, 37, 38, 18]. Conversely, the Ohsawa-Takegoshi theorem with optimal
constant was applied in [36, 13] to prove the variation results of the Bergman
kernels.

Consider the family $\mathfrak{X}\xlongrightarrow{\pi}\Delta,$ where
$\mathfrak{X}$ is a complex $(n+1)$-dimensional Stein manifold fibred over the
unit disc $\Delta\subset\mathbb{C}$, and $\pi$ is a surjective holomorphic map
with connected fibers. Moreover, $\pi$ is a holomorphic submersion over
$\Delta^{*}:=\Delta\setminus\\{0\\}$. For $\lambda\in\Delta^{*}$, the fibres
$X_{\lambda}=\pi^{-1}(\lambda)$ are $n$-dimensional Stein manifolds, whose
Bergman kernel are denoted by $\kappa_{X_{\lambda}}$. Let $(z,\lambda)$ be the
coordinate of $X_{\lambda}\times\Delta^{*}$, which is the local trivialization
of the fibration, and write $\kappa_{X_{\lambda}}:=k_{\lambda}(z)dz\otimes
d\bar{z}$ locally. Then, the above mentioned positivity results imply that
$\psi:=\log k_{\lambda}(z)$ is subharmonic with respect to
$\lambda\in\Delta^{*}$, i.e.,

$\frac{\partial^{2}\psi}{\partial{\lambda}\partial{\bar{\lambda}}}\,\geq 0,$
(0.2)

if the Bergman kernel is not identically zero. Here as $\lambda$ varies,
$e^{\psi}$ induces on $\mathcal{K}_{\mathfrak{X}/\Delta}$ the so-called
relative Bergman kernel metric, which is represented by different local
functions, given different local trivializations (see [9]). When the fibration
$\pi$ has a singular fiber $X_{0}$, our goal is to quantitatively characterize
at degeneration the asymptotic of $\psi$, as $\lambda\to 0$.

In fact, the study of the asymptotic analysis of integrals is a very classical
subject and has been pursued by many experts in the field of Hodge theory [42,
43, 75, 15, 16, 71], especially the nilpotent orbit theorem [35, 60] in the
variations of Hodge structures. In [34], the (pluri)subharmonicity in the base
direction of the above $\psi$ was described as “a pleasant surprise” by
Griffiths. It is known classically that the earlier works of Griffiths [33,
34], Fujita [32], as well as other important results in algebraic geometry
including [44, 46] played a decisive role in understanding the behavior of
$\psi$. For more background on the $L^{2}$-analysis in several complex
variables, and its interaction with Hodge theory, see [7, 17, 52, 53, 58, 54,
55].

In the theory of Riemann surfaces and their moduli space, the degenerations of
analytic differentials have also been extensively studied via a general method
called the pinching coordinate (see [31, 73, 50] and the references therein).
The spaces of degenerating Riemannn surfaces correspond to paths in the moduli
space leading to the boundary points, and are obtained from compact surfaces
by shrinking finitely many closed loops to points, called nodes. Near nodal
singularities of general curves, Wentworth [72] obtained precise estimates for
the Arakelov metric with applications to arithmetic geometry and string theory
(see also [19, 41]); Habermann and Jost [39, 40] studied the behavior of the
Bergman kernels and their induced $L^{2}$ metrics on Teichmüller spaces, with
a strong motivation in minimal surface theory. Specifically, the result in
[39] shows that the Bergman kernels on a degenerating family of compact
Riemann surfaces converge to that on the normalization of the limiting nodal
curve, with the second term having asymptotic growth $(-\log|\lambda|)^{-1}$,
as $\lambda\to 0$.

On the other hand, there exist worse singularities such as cusps, with which
the pinching coordinate method cannot deal. Boucksom and Jonsson [12] studied
the asymptotics of volume forms on degenerating compact manifolds and
established a measure-theoretic version of the Kontsevich-Soibelman
conjecture, which deals with the limiting behavior of a family of Calabi-Yau
manifolds approaching a “cusp” in the moduli space boundary. In [23, 24, 25,
26], the author obtained quantitative results for the Bergman kernels on
Legendre and other degenerating families of elliptic curves, by using elliptic
functions as well as a method based on the Taylor series expansion of Abelian
differentials and period matrices (see [14]).

In this paper, for the fibers $X_{\lambda}$ being algebraic curves or their
Jacobian varieties, we determine the precise asymptotic behaviors of $\psi$ at
degeneration, using the classical Taylor expansion method. Our first
motivation is to write down explicitly the asymptotic coefficients, which
involve the complex structure information and reflect the geometry of the base
varieties and their singularities, for various families of hyperelliptic
curves degenerating to singular ones with nodes or cusps. Our second
motivation is to investigate whether similar convergence results hold true for
the Bergman kernel on cuspidal degenerating families of curves, in comparison
with [39, 40, 72, 12, 61]. It turns out that this is not always the case as
seen in Theorem 1.2. However, in Theorem 1.3 we find that the Bergman kernels
on some cuspidal families of curves indeed converge at degeneration. As a last
motivation, we apply the results on higher genus curves to the study of their
Jacobians, and generalize the results in [23, 24, 26] toward higher
dimensions.

The organization of this paper is as follows. In Section 1, we state our main
results on the degeneration of the Bergman kernel. In Section 2, we study the
Bergman kernel on the normalization of general algebraic curves. In Sections
3–4, we work on nodal curves. In Sections 5–6 and 7–8, we work on cuspidal
curves of types I and II, respectively. Our results on the Jacobian varieties
are proved in Section 9.

##  1 Main results

Throughout this paper, we identify the term Riemann surface with smooth
algebraic curve, and consider the complex analytic families of hyperelliptic
curves
$X_{\lambda}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=h_{\lambda}(x)P(x)\\}$ of
genus $g\geq 2$, parametrized by $\lambda\in\Delta^{*}$. Here, $P$ denotes a
polynomial of degree $2g-2$ with complex roots $a_{j}$ such that
$1<|a_{1}|<|a_{2}|<...<|a_{2g-2}|$. For each $\lambda\in\Delta^{*}$,
$h_{\lambda}$ is a degree $3$ polynomial with distinct roots of absolute
values less than or equal to $1$. It is well known that on the smooth curve
$X_{\lambda}$ there exist globally defined Abelian differentials

$\omega_{i}:=\frac{x^{i-1}dx}{y},\,i=1,...,g.$ (1.1)

By its classical construction, the hyperelliptic curve $X_{\lambda}$ can be
realised as a 2-sheeted ramified covering of the Riemann sphere
$\mathbb{P}^{1}$ (see [14]). Take the homology basis $\delta_{i}$ and
$\gamma_{j}$ of $H_{1}(X_{\lambda},\mathbb{Z})$ such that their intersection
numbers are $\delta_{i}\cdot\delta_{j}=0,\gamma_{i}\cdot\gamma_{j}=0$, and
$\delta_{i}\cdot\gamma_{j}=\delta_{i,j}=-\gamma_{j}\cdot\delta_{i}$ (here
$\delta_{i,j}$ is the Kronecker $\delta$). Then, for the above $\omega_{i}\in
H^{0}(X_{\lambda},\mathcal{K})$, we set

$A_{i,j}:=\int_{\delta_{j}}\omega_{i},\quad
B_{i,j}:=\int_{\delta_{j}}\gamma_{i}.$ (1.2)

By the Hodge-Riemann bilinear relation, the matrix $(A_{ij})_{1\leq i,j\leq
g}$ is invertible and define the normalized period matrix $Z:=A^{-1}B$. Then,
$Z$ is symmetric and has a positive definite imaginary part, i.e.,
$\operatorname{Im}Z>0.$ For simplicity, with respect to the chosen homology
basis, we call the matrices $A,B$ and $Z$, the $A$-period matrix, $B$-period
matrix, and normalized period matrix, respectively.

#### Nodal cases

In affine coordinates, consider a family of hyperelliptic curves

$X_{\lambda}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x(x-{\lambda})(x-1)P(x)\\}.$
(1.3)

As $\lambda\to 0$, $X_{\lambda}$ degenerates to a singular curve $X_{0}$ with
a non-separating node. The normalization of $X_{0}$ is a smooth curve
$Y:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=(x-1)P(x)\\}$, whose $A$-period matrix
and normalized period matrix are denoted by $A_{0}$ and $Z_{0}$, respectively.
By the $L^{2}$ removable singularity theorem, $Y$ and the regular part of
$X_{0}$ have the same Bergman kernel, denoted by $\kappa_{0}$. In the local
coordinate $z:=\sqrt{x}$ near $(0,0)$, we write
$\kappa_{X_{\lambda}}=k_{\lambda}(z)dz\otimes d\bar{z}$ and
$\kappa_{0}=k_{0}(z)dz\otimes d\bar{z}$. See (4.1) for the precise formulae of
$\kappa_{0}$.

######  Theorem 1.1 (Nodal cases).

For $X_{\lambda}$ defined by (1.3), as $\lambda\rightarrow 0$, it holds that

  * $(i)$

$\kappa_{X_{\lambda}}\to\kappa_{0}$;

  * $(ii)$

for $0\neq|z|$ small,

$\psi-\log{k}_{0}(z)\sim\frac{\pi}{-\log|\lambda|}\frac{{1-2\operatorname{Re}\sum_{i=1}^{g-1}\left((\operatorname{Im}Z_{0})^{-1}\operatorname{Im}(A_{0}^{-1}\star)\right)_{i}{z}^{2i}}{}}{\sum_{i,j=1}^{g-1}((\operatorname{Im}Z_{0})^{-1})_{i,j}(z^{i}\overline{z}^{j})^{2}},$

where $\star$ is a column vector with $g-1$ rows whose entries all equal
${-2i\sqrt{P(0)}^{-1}}{}$.

Part $(i)$ is essentially due to Habermann and Jost [39], who in fact have
used the the pinching coordinate method to study general (possibly non-
hypereliptic) algebraic curves degenerating to singular ones with separating
or non-separating nodes. To write down the explicit asymptotic coefficients in
Part $(ii)$, we rely on the study of genus two curves in our Theorem 3.1.

#### Cuspidal cases

Since in [31, 73, 50, 72, 39, 40, 41] only the nodal degeneration was
considered, we continue investigating whether similar convergence results hold
true for the Bergman kernel on cuspidal degenerating families of curves, and
find that in general this is not the case. For the singular curve
$X_{0}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x^{3}P(x)\\}$ with an ordinary
cusp at $(0,0)$, its normalization is the smooth curve
$Y:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=xP(x)\\}$, whose $A$-period matrix and
normalized period matrix are denoted by $A_{0}$ and $Z_{0}$, respectively.
Then, $Y$ and the regular part of $X_{0}$ have the same Bergman kernel,
denoted by $\kappa_{0}$. In the following two cases, we explore different
types of families of hyperelliptic curves $X_{\lambda}$ that give rise to
$X_{0}$, as $\lambda\rightarrow 0$. In each case, similarly, in the local
coordinate $z:=\sqrt{x}$ near $(0,0)$, we write
$\kappa_{X_{\lambda}}=k_{\lambda}(z)dz\otimes d\bar{z}$ and
$\kappa_{0}=k_{0}(z)dz\otimes d\bar{z}$. See (6.2) for the precise formulae of
$\kappa_{0}$.

######  Theorem 1.2 (Cuspidal cases, I).

For the hyperelliptic curves

$X_{\lambda}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x(x^{2}-{\lambda})P(x)\\},$
(1.4)

as $\lambda\rightarrow 0$, for $0\neq|z|$ small, it holds that

  * $(i)$

$\quad{k}_{\lambda}(z)\to{k}_{0}(z)+4|z^{4}P(z^{2})|^{-1},\quad
i.e.,\,\kappa_{X_{\lambda}}\not\to\kappa_{0};$

$\noindent{}{{(ii)}}\hskip
42.67912pt\psi-\log\left({k}_{0}(z)+\frac{4}{|z^{4}P(z^{2})|}\right)\sim\frac{-2\operatorname{Re}\sum_{i=1}^{g-1}\left((\operatorname{Im}Z_{0})^{-1}\operatorname{Im}(A_{0}^{-1}(\diamondsuit-\sqrt{-1}*))\right)_{i}{z}^{2i}}{1+\sum_{i,j=1}^{g-1}((\operatorname{Im}Z_{0})^{-1})_{i,j}(z^{i}\overline{z}^{j})^{2}},$

where $\diamondsuit$ and $*$ are column vectors***See their definitions in
Lemmata 6.2 and 6.1 with $g-1$ rows involving $\lambda$.

######  Theorem 1.3 (Cuspidal cases, II).

For the hyperelliptic curves

$X_{\lambda}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x(x-{\lambda})(x-{\lambda^{2}})P(x)\\},$
(1.5)

as $\lambda\rightarrow 0$, it holds that

  * $(i)$

$\kappa_{X_{\lambda}}\to\kappa_{0}$;

  * $(ii)$

for $0\neq|z|$ small,

$\psi-\log{k}_{0}(z)\sim\frac{4\pi}{{k}_{0}(z)|z^{4}P(z^{2})|}\cdot\frac{1}{-\log|\lambda|}.$

Remarks.

  1. (a)

In Theorem 1.2, $(ii)$, the right hand side in the expansion of $\psi$ is
harmonic in $\lambda$ of order $\operatorname{O}(\lambda^{1/4})$, which is
different from Theorems 1.1 and 1.3 where growth order $(-\log|\lambda|)^{-1}$
appears. The harmonicity in the Bergman kernel variation detects the
triviality of the holomorphic fibration, with applications to the Suita
conjecture (see [66, 8, 28, 29, 6]).

  2. (b)

The proofs of Theorems 1.2 and 1.3 rely on the proofs of Theorems 5.1 and 7.1,
respectively, for genus-two curves with more explicit asymptotic coefficients
when $P=(x-a)(x-b)$. In particular, Theorem 7.1 says that the right hand side
in Theorem 1.3, $(ii)$, reduces to

$\frac{\pi\cdot\operatorname{Im}\tau}{-\log|\lambda|\cdot|z|^{4}},$

where $\tau$ is the period (scalar) of the elliptic curve
$\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x(x-a)(x-b)\\}.$

  3. (c)

In Theorems 1.1 and 1.3, the coefficients of the first and second terms in the
expansions of $\psi$ depend only on the information away from the singularity.

For a holomorphic family of Riemann surfaces $X_{\lambda}$ parametrized by
$\lambda\in\Delta^{*}$, each fiberwise Bergman kernel is locally written as
$\kappa_{X_{\lambda}}:=k_{\lambda}(z)dz\otimes d\bar{z}$, in some coordinate
$z$ for some function $k_{\lambda}$. By (0.2), $\psi:=\log k_{\lambda}(z)$ is
subharmonic with respect to $\lambda\in\Delta^{*}$. Moreover, the boundedness
of $\psi$ near $\lambda=0$ in Theorems 1.1, 1.2 and 1.3 yields the following
corollary.

######  Corollary 1.4.

For either family of hyperelliptic curves defined by (1.3), (1.4), or (1.5),
as $\lambda\rightarrow 0$, $\psi:=\log k_{\lambda}(z)$ extends to a
subharmonic function in $\lambda\in\Delta$, where the local coordinate is
$z:=\sqrt{x}\neq 0$. Moreover, $\psi$ has Lelong number zero at $\lambda=0$.

The boundary behavior of canonically defined plurisubharmonic functions often
contains significant information, in general. In our studies of curves, the
subharmonic function $\psi$ tends to be small, not big, near the singular
fibers (cf. [9]).

#### Jacobian varieties

For a curve $X_{\lambda}$ of genus $g\geq 2$, its Jacobian variety
$\text{Jac}(X_{\lambda})$ is a $g$-dimensional complex torus whose Bergman
kernel by the definition (0.1) is written as $\mu_{\lambda}dw\otimes
d\bar{w}$, under the coordinate $w$ induced from $\mathbb{C}^{g}$. For either
family of hyperelliptic curves defined by (1.3), (1.4), or (1.5), we determine
the asymptotic behaviors of the Bergman kernels on the naturally associated
family of Jacobians $\text{Jac}(X_{\lambda})$ as follows.

######  Theorem 1.5 (Jacobian varieties).

As $\lambda\rightarrow 0$, it holds that

$\log\mu_{\lambda}=\begin{cases}-\log({-\log|\lambda|})+\log\frac{\pi}{\det(\operatorname{Im}Z_{0})}+\operatorname{O}\left((\log|\lambda|)^{-1}\right),&\text{for
}X_{\lambda}\text{ in }\eqref{X-node}\,\text{or}\,\eqref{X-cuspII};\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt\par-\log\det(\operatorname{Im}Z_{0})+\operatorname{O}(\lambda^{{1}/{2}}),&\text{for
}X_{\lambda}\text{ in }\eqref{X-cuspI}.\par\end{cases}$

It is worth mentioning that as the curves $X_{\lambda}$ in (1.4) degenerate to
a cuspidal one, their Jacobian varieties $\text{Jac}(X_{\lambda})$ remain
being a manifold (i.e., non-degenerate), as $\lambda\to 0$. However, for two
other families of curves $X_{\lambda}$ in (1.3) and (1.5), their Jacobian
varieties $\text{Jac}(X_{\lambda})$ indeed degenerate. Theorem 1.5 generalizes
the results in [23, 24, 26] on elliptic curves toward higher dimensional
Abelian varieties. One may further compare our results with recent works [30,
45, 65, 67, 74]. An announcement of this paper for curves appeared in [27].

##  2 Bergman kernel on curves with possible singularities

The Bergman kernel, defined in (0.1), is independent of the choices of the
basis for the Hilbert space of $L^{2}$ holomorphic top-forms. It is well known
that the Bergman kernel $\kappa_{X_{\lambda}}$ on the curve $X_{\lambda}$ can
be written as

$\kappa_{X_{\lambda}}:=\sum_{i,j=1}^{g}((\operatorname{Im}Z)^{-1})_{i,j}\,\omega_{i}\otimes\overline{\omega_{j}},$

where $\omega_{i}$ is given in (1.1) and $Z$ is the normalized Riemann period
matrix defined by (1.2). Under certain local coordinate $z$ to be specified as
below, we write $\kappa_{X_{\lambda}}=k_{\lambda}(z)dz\otimes d\bar{z}$. In
order to explore the asymptotic behavior of $\psi:=\log k_{\lambda}(z)$ as
$\lambda\to 0$ when the curve degenerates, we will choose particular local
coordinates near each singularity so as to make the computations less
involved.

#### Local coordinates near a node

A nodal Riemann surface $X$ is a connected complex space such that every point
$p\in X$ has a neighborhood isomorphic to either a disk in $\mathbb{C}$ or
$\\{(x,y)\in\mathbb{C}^{2}\,|\,|x|,|y|<1\,,xy=0\\}$. In the latter, $p$ is a
node. For the family of hyperelliptic curves in (1.3), as $\lambda\to 0$,
$X_{\lambda}$ degenerates to a singular curve $X_{0}$ with a non-separating
node. On $X_{0}$, near the node $(0,0)$, the local coordinate can be taken as
$z=\sqrt{x}$, so it holds that $x=z^{2}$ and $dx=2zdz$. Then, in this
$z$-coordinate,

$\omega_{i}=\frac{2z^{2(i-1)}dz}{\sqrt{(z^{2}-\lambda)(z^{2}-1)P(z^{2})}},\,i=1,...,g,$

and the Bergman kernel $\kappa_{X_{\lambda}}$ can be written as

$\sum_{i,j=1}^{g}((\operatorname{Im}Z)^{-1})_{i,j}(z^{i}\overline{z}^{j})^{2}\,\frac{4dz\otimes
d\bar{z}}{|z^{4}(z^{2}-\lambda)(z^{2}-1)P(z^{2})|}=:{k}_{\lambda}(z)dz\otimes
d\bar{z}.$ (2.1)

Particularly, if $P=(x-a)(x-b)$, then (2.1) reduces to

$4\frac{((\operatorname{Im}Z)^{-1})_{1,1}+((\operatorname{Im}Z)^{-1})_{1,2}\overline{z}^{2}+((\operatorname{Im}Z)^{-1})_{2,1}{z}^{2}+((\operatorname{Im}Z)^{-1})_{2,2}|z|^{4}}{|(z^{2}-1)(z^{2}-a)(z^{2}-b)(z^{2}-\lambda)|}dz\otimes
d\bar{z}.$ (2.2)

The pinching coordinate could be used for general (possibly non-hyperelliptic)
algebraic curves degenerating to a nodal curve, which represents a boundary
point of the Deligne-Mumford compactification of moduli space (see [31, 73,
50, 72]). The non-hyperellipticity of $X_{\lambda}$ is equivalent to the non-
vanishing of the Gaussian curvature of the Bergman kernel
$\kappa_{X_{\lambda}}$. However, in this paper we do not use this effective
pinching coordinate method, which works for both non-separating and separating
nodal cases, since we need to deal with the cuspidal curves as well.

#### Local coordinates near a cusp, Case I

For the family of hyperelliptic curves in (1.4), as $\lambda\to 0$,
$X_{\lambda}$ degenerates to a singular curve $X_{0}$ with a cusp. On $X_{0}$,
near $(0,0)$, the local coordinate can be taken as $z=\sqrt{x}$. Similarly,

$\kappa_{X_{\lambda}}=\sum_{i,j=1}^{g}((\operatorname{Im}Z)^{-1})_{i,j}(z^{i}\overline{z}^{j})^{2}\,\frac{4dz\otimes
d\bar{z}}{|z^{4}(z^{4}-\lambda)\cdot P(z^{2})|}=:{k}_{\lambda}(z)dz\otimes
d\bar{z}.$ (2.3)

Particularly, if $P=(x-a)(x-b)$, then (2.3) reduces to

$4\frac{((\operatorname{Im}Z)^{-1})_{1,1}+((\operatorname{Im}Z)^{-1})_{1,2}\overline{z}^{2}+((\operatorname{Im}Z)^{-1})_{2,1}{z}^{2}+((\operatorname{Im}Z)^{-1})_{2,2}|z|^{4}}{|(z^{2}-a)(z^{2}-b)(z^{4}-\lambda)|}dz\otimes
d\bar{z}.$ (2.4)

#### Local coordinates near a cusp, Case II

For the family of hyperelliptic curves in (1.5), as $\lambda\to 0$,
$X_{\lambda}$ degenerates to a singular curve $X_{0}$ with a cusp. On $X_{0}$,
near $(0,0)$, the local coordinate can be taken as $z=\sqrt{x}$. Similarly,

$\kappa_{X_{\lambda}}=\sum_{i,j=1}^{g}((\operatorname{Im}Z)^{-1})_{i,j}(z^{i}\overline{z}^{j})^{2}\,\frac{4dz\otimes
d\bar{z}}{|z^{4}(z^{2}-\lambda)(z^{2}-\lambda^{2})\cdot
P(z^{2})|}=:{k}_{\lambda}(z)dz\otimes d\bar{z}.$ (2.5)

Particularly, if $P=(x-a)(x-b)$, then (2.5) reduces to

$4\frac{((\operatorname{Im}Z)^{-1})_{1,1}+((\operatorname{Im}Z)^{-1})_{1,2}\overline{z}^{2}+((\operatorname{Im}Z)^{-1})_{2,1}{z}^{2}+((\operatorname{Im}Z)^{-1})_{2,2}|z|^{4}}{|(z^{2}-a)(z^{2}-b)(z^{2}-\lambda)(z^{2}-\lambda^{2})|}dz\otimes
d\bar{z}.$ (2.6)

#### Normalization of singular curves

For the nodal singular curve $X_{0}$ in (1.3), we consider its normalization
$Y:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=(x-1)P(x)\\}$, which is a curve of
genus $g-1$. The regular part of $X_{0}$ is non-compact, and can be seen as
either $X_{0}$ or $Y$ less the node. Let $Z_{0}$ be the period matrix of $Y$.
By the $L^{2}$ removable singularity theorem, $Y$ and the regular part of
$X_{0}$ have the same Bergman kernel which can be written under the local
coordinate $z=\sqrt{x}$ as

$\kappa_{0}=\sum_{i,j=1}^{g-1}((\operatorname{Im}Z_{0})^{-1})_{i,j}(z^{i}\bar{z}^{j})^{2}\,\frac{4dz\otimes
d\bar{z}}{|z^{2}(z^{2}-1)P(z^{2})|},$ (2.7)

where

$\omega_{i}=\frac{x^{(i-1)}dx}{\sqrt{(x-1)P(x)}}=\frac{2z^{2i-1}dz}{\sqrt{(z^{2}-1)P(z^{2})}},\,i=1,...,g-1.$

For the cuspidal singular curve
$X_{0}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x^{3}P(x)\\}$, its normalization
is $Y:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=xP(x)\\}$. Similarly, the Bergman
kernel on $Y$ can be written under the coordinate $z=\sqrt{x}$ as

$\kappa_{0}=\sum_{i,j=1}^{g-1}((\operatorname{Im}Z_{0})^{-1})_{i,j}(z^{i}\bar{z}^{j})^{2}\,\frac{4dz\otimes
d\bar{z}}{|z^{4}P(z^{2})|}.$ (2.8)

##  3 Non-separating node: genus-two curves

In this section, we consider a family of genus two curves

$X_{\lambda}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x(x-{\lambda})(x-1)(x-a)(x-b)\\},$
(3.1)

where $a,b,\lambda$ are distinct complex numbers satisfying
$0<|\lambda|<1<|a|<|b|$. As $\lambda\to 0$, $X_{\lambda}$ degenerates to a
singular curve $X_{0}$ with a non-separating node. The normalization of
$X_{0}$ is an elliptic curve
$\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=(x-1)(x-a)(x-b)\\},$ whose period is $c$
given in (3.6) and let

$c_{1}:=\int_{1}^{a}\frac{\sqrt{ab}dx}{\sqrt{(x-1)(x-a)(x-b)}}$

be a constant depending on $a,b$. The Bergman kernels on $X_{\lambda}$ and on
the normalization of $X_{0}$ are denoted by $\kappa_{X_{\lambda}}$ and
$\kappa_{0}$, respectively. In the local coordinate $z:=\sqrt{x}$ near
$(0,0)$, we write $\kappa_{X_{\lambda}}=k_{\lambda}(z)dz\otimes d\bar{z}$, and
$\kappa_{0}=k_{0}(z)dz\otimes d\bar{z}$. Then, our result on the asymptotic
behaviour of the Bergman kernel $\kappa_{X_{\lambda}}$ with precise
coefficients is stated as follows.

######  Theorem 3.1.

For $X_{\lambda}$ defined by (3.1), as $\lambda\rightarrow 0$, it holds that

$\kappa_{X_{\lambda}}\to\kappa_{0}$;

for small $|z|\neq 0$ ,

$\log{k}_{\lambda}(z)-\log{k}_{0}(z)\sim\frac{\pi}{-\log|\lambda|}{\frac{\operatorname{Im}c+({z}^{2}+\overline{z}^{2})\operatorname{Re}\\{c_{1}^{-1}\\}}{|z|^{4}}}.$

In fact, the results in Section 4 on general hyperelliptic curves largely rely
on our proofs in this section. To prove Theorem 3.1, we need the following two
lemmata by analyzing the asymptotics of the $A$-period matrix and $B$-period
matrix on $X_{\lambda}$.

######  Lemma 3.2.

Under the same assumptions as in Theorem 3.1, as $\lambda\rightarrow 0$, it
holds that

$A=\left(\begin{array}[]{cc}\frac{-2\pi}{\sqrt{ab}}&c_{2}\\\
0&\frac{-2c_{1}}{\sqrt{ab}}\end{array}\right)+\operatorname{O}(\lambda),$

where $c_{2}:=\int_{\delta_{2}}\frac{dx}{x\sqrt{(x-1)(x-a)(x-b)}}$ and
$\lim\limits_{\lambda\to 0}\frac{\operatorname{O}(\lambda)}{\lambda}$ is a
finite matrix.

###### Proof of Lemma 3.2.

We estimate the four entries of $A$ one by one. Firstly,
$A_{1,1}=\int_{\delta_{1}}\omega_{1}$, where $\delta_{1}$ only contains $0$
and $\lambda$. Change the variables by setting $t={\lambda}^{-1},s={x}^{-1}$,
and get the dual cycle $\tilde{\delta}_{1}$ which contains $\\{\infty,t\\}$ so
that $-\tilde{\delta}_{1}$ contains $\\{0,1,{a}^{-1},{b}^{-1}\\}$, for
$|s|\in(1,|t|)$. As $\lambda\rightarrow 0$,

$\displaystyle A_{11}$
$\displaystyle\xlongequal[t={\lambda}^{-1}]{s={x}^{-1}}\int_{\tilde{\delta}_{1}}\frac{-s^{-2}ds}{\sqrt{s^{-1}(s^{-1}-t^{-1})(s^{-1}-1)(s^{-1}-a)(s^{-1}-b)}}$
$\displaystyle=-\int_{-\tilde{\delta}_{1}}\frac{-\sqrt{s}\sqrt{t}ds}{\sqrt{ab(s-1)(s-a^{-1})(s-b^{-1})(s-t)}}$
$\displaystyle=\int_{-\tilde{\delta}_{1}}\frac{\sqrt{s}ds}{\sqrt{-ab(s-1)(s-a^{-1})(s-b^{-1})}}\left(1+\frac{s}{2t}+\operatorname{O}\left(\frac{s^{2}}{{t}^{2}}\right)\right)$
$\displaystyle=\int_{-\tilde{\delta}_{1}}\frac{ds}{-s\sqrt{-ab}}\left(1+\frac{s}{2t}+\operatorname{O}\left(\frac{s^{2}}{{t}^{2}}\right)\right)\left(1+\operatorname{O}({{s^{-1}}})\right)\left(1+\operatorname{O}({{s^{-1}}})\right)\left(1+\operatorname{O}({{s^{-1}}})\right)$
$\displaystyle=\int_{-\tilde{\delta}_{1}}\frac{ds}{-s\sqrt{-ab}}\left(1+\frac{s}{2t}+\operatorname{O}(\frac{s^{2}}{{t}^{2}})\right)\left(1+\operatorname{O}({{s}^{-1}})\right)$
$\displaystyle=\frac{1}{-\sqrt{-ab}}\int_{-\tilde{\delta}_{1}}\left(1+\operatorname{O}(t^{-1})\right)\frac{ds}{s}=\frac{2\pi}{-\sqrt{ab}}\left(1+\operatorname{O}({\lambda})\right).$

Notice that we have used the Maclaurin expansion

$\frac{1}{\sqrt{s-a}}=\begin{cases}{\sqrt{-a}^{-1}}\left(1+\operatorname{O}\left({s}{{a^{-1}}}\right)\right),&|s|<|a|;\\\
\sqrt{s}^{-1}\left(1+\operatorname{O}({{s^{-1}a}})\right),&|s|>|a|.\end{cases}$
(3.2)

Secondly, look at $A_{21}=\int_{\delta_{1}}\omega_{2}$ and similarly it holds
that

$\displaystyle A_{21}$
$\displaystyle=\int_{-\tilde{\delta}_{1}}\frac{ds}{-s^{2}\sqrt{-ab}}\left(1+\frac{s}{2t}+\operatorname{O}\left(\frac{s^{2}}{{t}^{2}}\right)\right)\left(1+\operatorname{O}(s^{-1})\right)$
$\displaystyle=\frac{1}{-\sqrt{-ab}}\int_{-\tilde{\delta}_{1}}\left(\frac{1}{2t}+\operatorname{O}(t^{-2})\right)\frac{ds}{s}=\frac{2\pi}{-\sqrt{ab}}\left(\frac{\lambda}{2}+\operatorname{O}(\lambda^{2})\right).$

Thirdly, let $\delta_{2}$ contain $\\{0,\lambda,1,a\\}$. Then, as $\lambda\to
0$, it holds that

$\displaystyle A_{12}=$
$\displaystyle\int_{\delta_{2}}\frac{dx}{x\sqrt{(x-1)(x-a)(x-b)}}\left(1+\frac{\lambda}{2x}+\operatorname{O}\left(\frac{\lambda^{2}}{x^{2}}\right)\right).$
$\displaystyle=$
$\displaystyle\int_{\delta_{2}}\frac{dx}{x\sqrt{(x-1)(x-a)(x-b)}}\left(1+\operatorname{O}\left(\lambda\right)\right).$

Lastly,

$A_{22}=\int_{\delta_{2}}\frac{dx}{\sqrt{(x-1)(x-a)(x-b)}}\left(1+\operatorname{O}(\lambda)\right)=-2\int_{1}^{a}\frac{dx}{\sqrt{(x-1)(x-a)(x-b)}}\left(1+\operatorname{O}(\lambda)\right).$

∎

######  Lemma 3.3.

Under the same assumptions as in Theorem 3.1, as $\lambda\rightarrow 0$, it
holds that

$B\sim\left(\begin{array}[]{cc}\frac{-2\log\lambda}{\sqrt{-ab}}&d_{1}\\\
\frac{-2}{\sqrt{-ab}}&d_{2}\end{array}\right),$

where $d_{1}:=-2\int_{a}^{b}\frac{dx}{x\sqrt{(x-1)(x-a)(x-b)}}$ and
$d_{2}:=-2\int_{a}^{b}\frac{dx}{\sqrt{(x-1)(x-a)(x-b)}}.$

###### Proof of Lemma 3.3 .

As $t\to\infty$, we make use of the following computations (cf. [14]).

$\int_{1}^{t}\frac{ds}{s\sqrt{s-t}}=\left.\frac{-2}{\sqrt{t}}\sqrt{-1}\log\left(\sqrt{\frac{t}{s}}+\sqrt{\frac{t}{s}-1}\right)\right|_{1}^{t}\sim\frac{\sqrt{-1}}{\sqrt{t}}\log
t.$ (3.3)

$\int_{1}^{t}\frac{ds}{s^{2}\sqrt{s-t}}=\left.\frac{\sqrt{s-t}}{ts}\right|_{1}^{t}+\frac{1}{2t}\int_{1}^{t}\frac{ds}{s\sqrt{s-t}}=\frac{-\sqrt{1-t}}{t}+\frac{\sqrt{-1}}{2t\sqrt{t}}\log
t\sim-\frac{\sqrt{-1}}{\sqrt{t}}.$ (3.4)

In particular, (3.4) yields the boundedness of

$\int_{1}^{t}\frac{\sqrt{t}}{s\sqrt{s-t}}\operatorname{O}(s^{-1})ds.$

More generally, for any $\alpha\geq 1$, as $t\to\infty$, it holds that

$\int_{1}^{t}\frac{ds}{s^{\alpha+1}\sqrt{s-t}}=\left.\frac{\sqrt{s-t}}{\alpha
ts^{\alpha}}\right|_{1}^{t}+\frac{2\alpha-1}{2\alpha
t}\int_{1}^{t}\frac{ds}{s^{\alpha}\sqrt{s-t}}\sim-\frac{\sqrt{-1}}{\alpha\sqrt{t}}.$
(3.5)

Similar to the proof of Lemma 3.2, the four entries of $B$ are estimated one
by one. Firstly, by (3.3) and (3.4), for $|s|\in(1,|t|)$,

$\displaystyle B_{11}=$
$\displaystyle-2\int_{\lambda}^{1}\frac{dx}{\sqrt{x(x-\lambda)(x-1)(x-a)(x-b)}}$
$\displaystyle=$
$\displaystyle-2\int_{t}^{1}\frac{-\sqrt{s}\sqrt{t}ds}{\sqrt{ab(s-1)(s-a^{-1})(s-b^{-1})(s-t)}}$
$\displaystyle=$
$\displaystyle-2\int_{1}^{t}\frac{\sqrt{s}\sqrt{t}ds}{\sqrt{(s-t)ab}}\frac{1}{s\sqrt{s}}\left(1+\operatorname{O}({{s^{-1}}})\right)\left(1+\operatorname{O}({{s^{-1}}})\right)\left(1+\operatorname{O}({{s^{-1}}})\right)$
$\displaystyle=$
$\displaystyle-\frac{2\sqrt{t}}{\sqrt{ab}}\int_{1}^{t}\frac{ds}{s\sqrt{s-t}}\left(1+\operatorname{O}({{s^{-1}}})\right)\sim\frac{-2\log\lambda}{\sqrt{-ab}},$

as $\lambda\to 0$. Secondly, by (3.4),

$B_{21}=-\frac{2\sqrt{t}}{\sqrt{ab}}\int_{1}^{t}\frac{ds}{s^{2}\sqrt{s-t}}\left(1+\operatorname{O}({{s^{-1}}})\right)\sim\frac{-2}{\sqrt{-ab}}.$

Finally, similar to $A_{12}$,

$B_{12}\sim-2\int_{a}^{b}\frac{dx}{x\sqrt{(x-1)(x-a)(x-b)}},$

and

$B_{22}\sim-2\int_{a}^{b}\frac{dx}{\sqrt{(x-1)(x-a)(x-b)}}.$

∎

Combining Lemmata 3.2 and 3.3 with (2.2), we get the asymptotics of the
Bergman kernels.

###### Proof of Theorem 3.1.

Notice that the period is defined as

$c:=\frac{\int_{a}^{b}\frac{dx}{\sqrt{(x-1)(x-a)(x-b)}}}{\int_{1}^{a}\frac{dx}{\sqrt{(x-1)(x-a)(x-b)}}}=\tau\left(\frac{1-b}{1-a}\right),$
(3.6)

where $\tau(\cdot)$ is the inverse of the elliptic modular lambda function. On
the normalization of the nodal curve $X_{0}$, by (2.7), in the local
coordinate $z=\sqrt{x}$, the Bergman kernel is exactly
$\kappa_{0}=k_{0}(z)dz\otimes d\bar{z}$, where

$k_{0}(z)=\frac{4|z|^{2}}{(\operatorname{Im}c)|(z^{2}-1)(z^{2}-a)(z^{2}-b)|}.$
(3.7)

By Lemma 3.2, as $\lambda\to 0$,

$A^{-1}\sim\left(\begin{array}[]{cc}\frac{-2\pi}{\sqrt{ab}}&c_{2}\\\
\operatorname{O}(\lambda)&\frac{-2c_{1}}{\sqrt{ab}}\end{array}\right)^{-1}=\left(\begin{array}[]{cc}\frac{-\sqrt{ab}}{2\pi}&\frac{\sqrt{ab}c_{2}}{{2\pi
d_{2}}}\\\
\operatorname{O}(\lambda)&\frac{\sqrt{ab}}{-2c_{1}}\end{array}\right).$

Let $Z$ denote the period matrix of $X_{\lambda}$. Then, it follows that

$Z=A^{-1}B\sim\left(\begin{array}[]{cc}\frac{-\sqrt{-1}}{\pi}\log\lambda&\frac{abc_{2}d_{2}+\sqrt{ab}{2c_{1}d_{1}}}{{-4\pi
c_{1}}}\\\
{-\sqrt{-1}}{c_{1}}^{-1}&c\end{array}\right),\quad\operatorname{Im}Z\sim\left(\begin{array}[]{cc}-\frac{\log|\lambda|}{\pi}&-\operatorname{Re}\\{c_{1}^{-1}\\}\\\
-\operatorname{Re}\\{c_{1}^{-1}\\}&\operatorname{Im}c\end{array}\right).$

So, as $\lambda\rightarrow 0$, it holds that

$(\operatorname{Im}Z)^{-1}\sim\left(\begin{array}[]{cc}\frac{\pi}{-\log|\lambda|}&\frac{-\operatorname{Re}\\{c_{1}^{-1}\\}\pi}{(\log|\lambda|)\operatorname{Im}c}\\\
\frac{-\operatorname{Re}\\{c_{1}^{-1}\\}\pi}{(\log|\lambda|)\operatorname{Im}c}&({\operatorname{Im}c})^{-1}\end{array}\right).$

By (2.2) and (3.7), $\kappa_{X_{\lambda}}\to\kappa_{0}.$ Moreover, in the
local coordinate $z=\sqrt{x}$ near $(0,0)$,

${k}_{\lambda}(z)-{k}_{0}(z)\sim\frac{4\pi}{|(z^{2}-1)(z^{2}-a)(z^{2}-b)(z^{2}-\lambda)|}\frac{1+\frac{\operatorname{Re}\\{c_{1}^{-1}\\}}{\operatorname{Im}c}({z}^{2}+\overline{z}^{2})}{-\log|\lambda|},$

which yields the conclusion.

∎

Since the moduli space of genus-two curves is 3 dimensional, one may consider
the more general family
$X_{\lambda,a,b}:=\\{y^{2}=x(x-1)(x-\lambda)(x-a)(x-b))\\}$, parametrized by
three distinct complex numbers $\lambda,a,b\in\mathbb{C}\setminus\\{0,1\\}$.
As $\lambda,a$ or $b$ tends to $0,1$ or $\infty$, or towards one another,
$X_{\lambda,a,b}$ will become singular. In our setting, we fix the other two
parameters $a$ and $b$, and move $\lambda$ only, so the precise asymptotic
coefficients we have obtained depend on both $a$ and $b$.

##  4 Non-separating node: hyperelliptic and general curves

This section is devoted to the proof of Theorem 1.1, as a generalization of
Theorem 3.1 towards the hyperelliptic case. When
$\lambda\in\mathbb{C}\setminus\\{0,1,a_{1},...,a_{2g-2}\\}$, $X_{\lambda}$
defined in (1.3) has genus $g$. To prove Theorem 1.1, we need to analyze on
$X_{\lambda}$ the asymptotics of its $A$-period matrix and $B$-period matrix,
denoted by $A$ and $B$, respectively. Meanwhile, for the normalization
$Y:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=(x-1)P(x)\\}$ of genus $g-1$, denote
its $A$-period matrix and $B$-period matrix by $A_{0}$ and $B_{0}$,
respectively.

######  Lemma 4.1.

Under the same assumptions as in Theorem 1.1, as $\lambda\rightarrow 0$, it
holds that

$A\sim\left(\begin{array}[]{cccccc}\frac{2\pi}{\sqrt{P(0)}}&\operatorname{O}(1)\\\
&A_{0}\\\ \end{array}\right),$

where $*=(\operatorname{O}(\lambda),0,...,0)^{T}$ is a column vector with
$g-1$ rows.

###### Proof.

Firstly, $A_{11}=\int_{\delta_{1}}\omega_{1}$, where $\delta_{1}$ only
contains $0$ and $\lambda$. Change the variables by setting
$t={\lambda}^{-1},s={x}^{-1}$, and we get the dual cycle $\tilde{\delta}_{1}$
which contains $\\{\infty,t\\}$ so that $-\tilde{\delta}_{1}$ contains
$\\{0,1,{a_{k}}^{-1}\\}$, $k=1,...,d$, for $|s|\in(1,|t|)$. As
$\lambda\rightarrow 0$,

$\displaystyle A_{11}$
$\displaystyle\xlongequal[t={\lambda}^{-1}]{s={x}^{-1}}\int_{-\tilde{\delta}_{1}}\frac{ds}{\sqrt{-1}s\sqrt{P(0)}}\left(1+\operatorname{O}\left(\frac{s}{{t}}\right)\right)\left(1+\operatorname{O}({{s^{-1}}})\right)$
$\displaystyle\sim\int_{-\tilde{\delta}_{1}}\frac{ds}{\sqrt{-1}s\sqrt{P(0)}}\left(1+\operatorname{O}({{s^{-1}}})\right)=\int_{-\tilde{\delta}_{1}}\frac{ds}{\sqrt{-1}s\sqrt{P(0)}}=\frac{2\pi}{\sqrt{P(0)}}.$

$\displaystyle A_{21}$
$\displaystyle=\int_{-\tilde{\delta}_{1}}\frac{ds}{\sqrt{-1}s^{2}\sqrt{P(0)}}\left(1+\operatorname{O}\left(\frac{s}{{t}}\right)\right)\left(1+\operatorname{O}({{s^{-1}}})\right)$
$\displaystyle=\int_{-\tilde{\delta}_{1}}\frac{ds}{\sqrt{-1}\sqrt{P(0)}}\left({{s^{-2}}}+\operatorname{O}\left(\frac{1}{{ts^{i-1}}}\right)\right)\left(1+\operatorname{O}({{s^{-1}}})\right)$
$\displaystyle=\int_{-\tilde{\delta}_{1}}\frac{ds}{\sqrt{-1}\sqrt{P(0)}}\left(+\operatorname{O}\left(\frac{1}{{ts}}\right)\right)=\operatorname{O}(\lambda).$

For $3\leq i\leq g$,

$\displaystyle A_{i1}$
$\displaystyle=\int_{-\tilde{\delta}_{1}}\frac{ds}{\sqrt{-1}s^{i}\sqrt{P(0)}}\left(1+\operatorname{O}\left(\frac{s}{{t}}\right)\right)\left(1+\operatorname{O}({{s^{-1}}})\right)$
$\displaystyle=\int_{-\tilde{\delta}_{1}}\frac{ds}{\sqrt{-1}\sqrt{P(0)}}\left({{s^{-i}}}+\operatorname{O}\left(\frac{1}{{ts^{i-1}}}\right)\right)\left(1+\operatorname{O}({{s^{-1}}})\right)=0.$

Secondly, let $\delta_{2}$ contain only $0,\lambda,1$ and $a_{1}$ (not
$a_{2},...,a_{d}$). Then,

$\displaystyle A_{12}$
$\displaystyle=\int_{\delta_{2}}\frac{dx}{\sqrt{x(x-{\lambda})(x-1)P(x)}}$
$\displaystyle=\int_{\delta_{2}}\frac{dx}{x\sqrt{(x-1)P(x)}}\left(1+\operatorname{O}\left(\frac{{\lambda}}{{x}}\right)\right)\sim\int_{\delta_{2}}\frac{dx}{x\sqrt{(x-1)P(x)}}.$

In general, for $A_{i2}$, there is an extra $x^{i-1}$ in the original
integrand above, so

$A_{i2}\sim\int_{\delta_{2}}\frac{x^{i-2}dx}{\sqrt{(x-1)P(x)}}.$

Thirdly, let $\delta_{3}$ contain only $0,\lambda^{2},\lambda,a_{1},a_{2}$ and
$a_{3}$ (not $a_{4},...,a_{d}$). Similarly, $a_{i3}$ is asymptotic to the same
integrand over $\delta_{3}$ instead of over $\delta_{2}$, i.e.,

$A_{i3}\sim\int_{\delta_{3}}\frac{x^{i-2}dx}{\sqrt{(x-1)P(x)}}.$

In general, for $j\geq 2$ and the corresponding homology basis $\delta_{j}$,
it holds that

$A_{ij}\sim\int_{\delta_{j}}\frac{x^{i-2}dx}{\sqrt{(x-1)P(x)}},$

which are exactly the same as the entries of $A_{0}$ when $i\geq 2$.

∎

######  Lemma 4.2.

Under the same assumptions as in Theorem 1.1, as $\lambda\rightarrow 0$, it
holds that

$B\sim\left(\begin{array}[]{cccccc}\frac{-2\sqrt{-1}}{\sqrt{P(0)}}{\log\lambda}&\operatorname{O}(1)\\\
\star&B_{0}\end{array}\right),$

where $\star$ is a column vector whose entries are all
$\frac{-2\sqrt{-1}}{\sqrt{P(0)}}$ with $g-1$ rows.

###### Proof.

Firstly, by (3.3) and (3.4), for $|s|\in(1,|t|)$,

$\displaystyle B_{11}$
$\displaystyle=-2\int_{\lambda}^{1}\frac{dx}{\sqrt{x(x-{\lambda})(x-1)\cdot
P(x)}}=-2\int_{\lambda}^{1}\frac{dx}{\sqrt{x(x-{\lambda})}}\frac{1}{\sqrt{-P(0)}}(1+\operatorname{O}(x))$
$\displaystyle\xlongequal[t={\lambda}^{-1}]{s={x}^{-1}}\frac{2\sqrt{t}}{\sqrt{P(0)}}\int_{1}^{t}\frac{ds}{s\sqrt{s-t}}(1+\operatorname{O}(s^{-1}))\sim\frac{-2}{\sqrt{P(0)}}{\sqrt{-1}}\log\lambda.$

Thus, for $2\leq i\leq g$, by (3.5),

$B_{i1}=-2\int_{\lambda}^{1}\frac{x^{i-1}dx}{\sqrt{x(x-{\lambda})(x-1)\cdot
P(x)}}=\frac{2\sqrt{t}}{\sqrt{P(0)}}\int_{1}^{t}\frac{ds}{s^{i}\sqrt{s-t}}(1+\operatorname{O}(s^{-1}))\sim\frac{-2\sqrt{-1}}{\sqrt{P(0)}}.$

For Column $j$, $2\leq j\leq g$, we use Taylor series expansion of
$\sqrt{x-{\lambda}}^{-1}$ to get that

$B_{ij}\sim-2\int_{a_{2j-3}}^{a_{2j-2}}\frac{x^{i-2}dx}{\sqrt{(x-1)P(x)}},$

which are exactly the same as the entries of $B_{0}$ when $i\geq 2$.

∎

Now we will give a proof of Theorem 1.1.

###### Proof of Theorem 1.1.

By (2.7), in the local coordinate $z=\sqrt{x}$, the Bergman kernel on the
normalization $Y$ of the nodal curve $X_{0}$ in (1.3) is exactly
$\kappa_{0}=k_{0}(z)dz\otimes d\bar{z}$, where

$k_{0}(z)=\frac{4}{|z^{2}(z^{2}-1)P(z^{2})|}\sum_{i,j=1}^{g-1}((\operatorname{Im}Z_{0})^{-1})_{i,j}(z^{i}\overline{z}^{j})^{2}.$
(4.1)

By Lemma 4.1 and the block matrix inversion, we know that

$A^{-1}\sim\left(\begin{array}[]{cccccc}\frac{\sqrt{P(0)}}{2\pi}&\operatorname{O}(1)\\\
\operatorname{O}(\lambda)&A_{0}^{-1}\end{array}\right),$

where both $(\operatorname{O}(1))^{T}$ and $\lim\limits_{\lambda\to
0}\frac{\operatorname{O}(\lambda)}{\lambda}$ are finite column vectors with
$g-1$ rows. Therefore, as $\lambda\to 0$,

$Z=A^{-1}B\sim\left(\begin{array}[]{cccc}\frac{-\sqrt{-1}}{\pi}{\log\lambda}&\operatorname{O}(1)\\\
A_{0}^{-1}\star&A_{0}^{-1}B_{0}\end{array}\right),\quad\operatorname{Im}Z\sim\left(\begin{array}[]{cccc}\frac{\log|\lambda|}{-\pi}&\operatorname{O}(1)\\\
\operatorname{Im}(A_{0}^{-1}\star)&\operatorname{Im}Z_{0}\end{array}\right),$

and

$(\operatorname{Im}Z)^{-1}\sim\left(\begin{array}[]{cccc}\frac{-\pi}{\log|\lambda|}&\operatorname{O}(({\log|\lambda|})^{-1})\\\
(\operatorname{Im}Z_{0})^{-1}\operatorname{Im}(A_{0}^{-1}\star)\frac{\pi}{\log|\lambda|}&(\operatorname{Im}Z_{0})^{-1}\end{array}\right).$

In fact, since $Z$ is symmetric, the off-diagonal block matrices in each
matrix above concerning $Z$ are the transpose of each other. By (2.1) and
(4.1), as $\lambda\to 0$, $\kappa_{X_{\lambda}}\to\kappa_{0}.$ Moreover, in
the local coordinate $z=\sqrt{x}$ near $(0,0)$, it holds that

${k}_{\lambda}(z)-{k}_{0}(z)\sim\frac{4\pi}{|(z^{2}-\lambda)(z^{2}-1)P(z^{2})|}\frac{1-2\operatorname{Re}\sum\limits_{i=1}^{g-1}\left((\operatorname{Im}Z_{0})^{-1}\operatorname{Im}(A_{0}^{-1}\star)\right)_{i}{z}^{2i}}{-\log|\lambda|}$

which yields the conclusion.

∎

If we ignore the asymptotic coefficients, then the leading term growth in
Theorem 1.1 corresponds to [39, Proposition 3.2]. The degeneration of nodal
non-hypereliptic curves was also treated in [39]†††The author is grateful to
Professor Z. Huang for bringing attention the paper [39] during the 2016
Tsinghua Sanya International Mathematics Forum, where a preliminary version of
this work was presented..

##  5 Cusp I: genus-two curves

In this section, we consider a family of genus two curves

$X_{\lambda}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x(x^{2}-{\lambda})(x-a)(x-b)\\},$
(5.1)

where $a,b,\lambda$ are distinct complex numbers satisfying
$0<|\lambda|<|a|<|b|$. As $\lambda\to 0$, $X_{\lambda}$ degenerates to a
singular curve $X_{0}$ with an ordinary cusp. The normalization of $X_{0}$ is
an elliptic curve $\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x(x-a)(x-b)\\}$, whose
period is $\tau$ given in (5.2). Let

$c_{3}:=\int_{0}^{a}\frac{\sqrt{ab}dx}{\sqrt{x(x-a)(x-b)}},\quad
c_{4}:=-2\int_{0}^{1}\frac{(x-1)dx}{\sqrt{x(x-1)(x-2)}}$

be constants depending on $a,b$. The Bergman kernels on $X_{\lambda}$ and on
the normalization of $X_{0}$ are denoted by $\kappa_{X_{\lambda}}$ and
$\kappa_{0}$, respectively. In the local coordinate $z:=\sqrt{x}$ near
$(0,0)$, we write $\kappa_{X_{\lambda}}=k_{\lambda}(z)dz\otimes d\bar{z}$, and
$\kappa_{0}=k_{0}(z)dz\otimes d\bar{z}$. Then, our result on the asymptotic
behaviour of the Bergman kernel $\kappa_{X_{\lambda}}$ with precise
coefficients is stated as follows.

######  Theorem 5.1.

For $X_{\lambda}$ defined by (5.1), as $\lambda\rightarrow 0$, for small
$|z|\neq 0$, it holds that

${k}_{\lambda}(z)\to{k}_{0}(z)\left(\frac{\operatorname{Im}\tau}{|z^{4}|}+1\right),\quad
i.e.,\,\kappa_{X_{\lambda}}\not\to\kappa_{0};$

$\log{k}_{\lambda}(z)-\log{k}_{0}(z)-\log\left(\frac{\operatorname{Im}\tau}{|z^{4}|}+1\right)\sim\frac{\operatorname{Re}\left\\{\lambda^{1/4}\frac{c_{4}}{c_{3}}\right\\}(z^{2}+\overline{z}^{2})}{|z|^{4}+{\operatorname{Im}\tau}},$

where $\tau$ is the period (scalar) of the elliptic curve
$\\{y^{2}=x(x-a)(x-b)\\}$.

To prove Theorem 5.1, we need the following two lemmata by analyzing the
asymptotics of the $A$-period matrix and $B$-period matrix on $X_{\lambda}$.

######  Lemma 5.2.

Under the same assumptions as in Theorem 5.1, as $\lambda\rightarrow 0$, it
holds that

$A\sim\left(\begin{array}[]{cc}\frac{c_{5}}{-\sqrt{ab}}\lambda^{-1/4}&c_{6}\\\
\frac{c_{4}}{-\sqrt{ab}}\lambda^{1/4}&\frac{-2c_{3}}{\sqrt{ab}}\end{array}\right),$

where $c_{5}:=-2\int_{0}^{1}\frac{du}{\sqrt{u(u-1)(u-2)}}$ and $c_{6}$ depends
on $a,b$.

###### Proof of Lemma 5.2.

We estimate the four entries one by one. Firstly, let $\delta_{1}$ contain
only $-\sqrt{\lambda}$ and $0$. By the Cauchy Integral Theorem and Taylor
series expansion, we know that

$\displaystyle A_{11}$
$\displaystyle=-2\int_{-\sqrt{\lambda}}^{0}\frac{dx}{\sqrt{x(x-\sqrt{\lambda})(x+\sqrt{\lambda})(x-a)(x-b)}}$
$\displaystyle=\frac{-2}{-\sqrt{ab}}\int_{-\sqrt{\lambda}}^{0}\frac{dx}{\sqrt{x(x-\sqrt{\lambda})(x+\sqrt{\lambda})}}\left(1+\operatorname{O}(x)\right)$
$\displaystyle\xlongequal[]{x=(u-1)\sqrt{\lambda}}\frac{-2\lambda^{-1/4}}{-\sqrt{ab}}\int_{0}^{1}\frac{du}{\sqrt{u(u-1)(u-2)}}\left(1+\operatorname{O}\left(\sqrt{\lambda}(u-1)\right)\right)$
$\displaystyle\sim\frac{-2\lambda^{-1/4}}{-\sqrt{ab}}\int_{0}^{1}\frac{du}{\sqrt{u(u-1)(u-2)}}=:\frac{c_{5}}{-\sqrt{ab}}\lambda^{-1/4}.$

Secondly,

$\displaystyle A_{21}$
$\displaystyle=\frac{-2\lambda^{1/4}}{-\sqrt{ab}}\int_{0}^{1}\frac{(u-1)du}{\sqrt{u(u-1)(u-2)}}\left(1+\operatorname{O}\left(\sqrt{\lambda}(u-1)\right)\right)$
$\displaystyle\sim\frac{-2\lambda^{1/4}}{-\sqrt{ab}}\int_{0}^{1}\frac{(u-1)du}{\sqrt{u(u-1)(u-2)}}:=\frac{c_{4}}{-\sqrt{ab}}\lambda^{1/4}.$

Thirdly, let $\delta_{2}$ contain only $-\sqrt{\lambda},0,\sqrt{\lambda}$ and
$a$ (not $b$). Then,

$\displaystyle A_{12}=$
$\displaystyle\int_{\delta_{2}}\frac{dx}{\sqrt{x(x-a)(x-b)}}\frac{1}{\sqrt{x}}\left(1+\operatorname{O}\left(\frac{\sqrt{\lambda}}{x}\right)\right)\frac{1}{\sqrt{x}}\left(1+\operatorname{O}\left(\frac{\sqrt{\lambda}}{x}\right)\right)$
$\displaystyle\sim$
$\displaystyle\int_{\delta_{2}}\frac{dx}{x\sqrt{x(x-a)(x-b)}}=:c_{6}.$
$\displaystyle A_{22}$
$\displaystyle=\int_{\delta_{2}}\frac{dx}{\sqrt{x(x-a)(x-b)}}\left(1+\operatorname{O}\left(\frac{\sqrt{\lambda}}{x}\right)\right)$
$\displaystyle\sim\int_{\delta_{2}}\frac{dx}{\sqrt{x(x-a)(x-b)}}=-2\int_{0}^{a}\frac{dx}{\sqrt{x(x-a)(x-b)}}=\frac{-2c_{3}}{\sqrt{ab}}.$

∎

######  Lemma 5.3.

Under the same assumptions as in Lemma 5.2, as $\lambda\rightarrow 0$, it
holds that

$B\sim\frac{1}{\sqrt{-ab}}\left(\begin{array}[]{cc}c_{5}\lambda^{-1/4}&d_{3}\sqrt{-ab}\\\
-c_{4}\lambda^{1/4}&d_{4}\sqrt{-ab}\end{array}\right),$

where $d_{3}:=-2\int_{a}^{b}\frac{dx}{x\sqrt{x(x-a)(x-b)}}$ and
$d_{4}:=-2\int_{a}^{b}\frac{dx}{\sqrt{x(x-a)(x-b)}}$.

###### Proof of Lemma 5.3.

Again, we estimate all the four entries one by one. Firstly, let $\gamma_{1}$
contain only $0$ and $\sqrt{\lambda}$. By Cauchy Integral Theorem, we can get
that

$\displaystyle B_{11}$
$\displaystyle=-2\int_{0}^{\sqrt{\lambda}}\frac{dx}{\sqrt{x(x^{2}-{\lambda})(x-a)(x-b)}}$
$\displaystyle=\frac{-2}{-\sqrt{ab}}\int_{0}^{\sqrt{\lambda}}\frac{dx}{\sqrt{x(x-\sqrt{\lambda})(x+\sqrt{\lambda})}}\left(1+\operatorname{O}(x)\right)$
$\displaystyle\xlongequal[]{x=(1-u)\sqrt{\lambda}}\frac{2\lambda^{-1/4}}{\sqrt{ab}}\int_{1}^{0}\frac{du}{\sqrt{-u(u-1)(u-2)}}\left(1+\operatorname{O}((-u+1)\cdot\sqrt{\lambda})\right)$
$\displaystyle\sim\frac{2\lambda^{-1/4}}{\sqrt{ab}}\int_{0}^{1}\frac{\sqrt{-1}du}{\sqrt{u(u-1)(u-2)}}=:\frac{c_{5}}{\sqrt{-ab}}\lambda^{-1/4}.$

Secondly,

$\displaystyle B_{21}$
$\displaystyle=\frac{2\lambda^{-1/4}}{\sqrt{ab}}\int_{1}^{0}\frac{(-u+1)\sqrt{\lambda}du}{\sqrt{-u(u-1)(u-2)}}\left(1+\operatorname{O}((-u+1)\cdot\sqrt{\lambda})\right)$
$\displaystyle\sim\frac{2\lambda^{1/4}}{\sqrt{ab}}\int_{0}^{1}\frac{(u-1)du}{\sqrt{-u(u-1)(u-2)}}=:\frac{\sqrt{-1}\lambda^{1/4}c_{4}}{\sqrt{ab}}.$

Thirdly, let $\gamma_{2}$ contain only $a$ and $b$. Then, it holds that

$\displaystyle B_{12}=$
$\displaystyle\int_{\gamma_{2}}\frac{dx}{\sqrt{x(x-\sqrt{\lambda})(x+\sqrt{\lambda})(x-a)(x-b)}}$
$\displaystyle=$
$\displaystyle-2\int_{a}^{b}\frac{dx}{\sqrt{x(x-a)(x-b)}}\frac{1}{\sqrt{x}}\left(1+\operatorname{O}\left(\frac{\sqrt{\lambda}}{2x}\right)\right)\frac{1}{\sqrt{x}}\left(1+\operatorname{O}\left(\frac{\sqrt{\lambda}}{2x}\right)\right)$
$\displaystyle\sim$
$\displaystyle-2\int_{a}^{b}\frac{dx}{x\sqrt{x(x-a)(x-b)}}=:d_{3}.$

Lastly,

$B_{22}=\int_{\gamma_{2}}\frac{dx}{\sqrt{x(x-a)(x-b)}}\left(1+\operatorname{O}\left(\frac{\sqrt{\lambda}}{2x}\right)\right)\sim-2\int_{a}^{b}\frac{dx}{\sqrt{x(x-a)(x-b)}}=:d_{4}.$

∎

Combining Lemmata 5.2 and 5.3 with (2.4), we get the asymptotics of the
Bergman kernels.

###### Proof of Theorem 5.1.

Notice that the period is defined as

$\tau:=\left\\{\frac{\int_{a}^{b}\frac{dx}{\sqrt{x(x-a)(x-b)}}}{\int_{0}^{a}\frac{dx}{\sqrt{x(x-a)(x-b)}}}\right\\}=\tau\left(\frac{b}{a}\right),$
(5.2)

where $\tau(\cdot)$ is the inverse of the elliptic modular lambda function. On
the regular part of the cuspidal curve $X_{0}$, by (2.8), in the local
coordinate $z=\sqrt{x}$, the Bergman kernel is exactly
$\kappa_{0}=k_{0}(z)dz\otimes d\bar{z}$, where

$k_{0}(z)=\frac{4}{\operatorname{Im}\tau\cdot\left|(z^{2}-a)(z^{2}-b)\right|}.$
(5.3)

By Lemma 5.2, as $\lambda\to 0$,

$A^{-1}\sim\left(\begin{array}[]{cc}\frac{c_{5}}{-\sqrt{ab}}\lambda^{-1/4}&c_{6}\\\
\frac{c_{4}}{-\sqrt{ab}}\lambda^{1/4}&\frac{-2c_{3}}{\sqrt{ab}}\end{array}\right)^{-1}\sim\frac{ab\lambda^{1/4}}{{2c_{3}}c_{5}}\left(\begin{array}[]{cc}\frac{-2c_{3}}{\sqrt{ab}}&-c_{6}\\\
\frac{c_{4}}{\sqrt{ab}}\lambda^{1/4}&\frac{c_{5}}{-\sqrt{ab}}\lambda^{-1/4}\end{array}\right).$

Let $Z$ denote the period matrix of $X_{\lambda}$. Then, it follows that

$\displaystyle Z=A^{-1}B$
$\displaystyle\sim\frac{{ab}\lambda^{1/4}}{{2c_{3}}c_{5}}\left(\begin{array}[]{cc}\frac{-2c_{3}}{\sqrt{ab}}&-c_{6}\\\
\frac{c_{4}}{\sqrt{ab}}\lambda^{1/4}&\frac{c_{5}}{-\sqrt{ab}}\lambda^{-1/4}\end{array}\right)\frac{1}{\sqrt{-ab}}\left(\begin{array}[]{cc}c_{5}\lambda^{-1/4}&d_{3}\sqrt{-ab}\\\
-c_{4}\lambda^{1/4}&d_{4}\sqrt{-ab}\end{array}\right)$
$\displaystyle\sim\left(\begin{array}[]{cc}\sqrt{-1}&\sqrt{-1}\lambda^{1/4}{\left(d_{3}\sqrt{-ab}-\frac{c_{6}d_{4}ab}{2c_{3}}\right)}{c_{5}^{-1}}\\\
\sqrt{-1}\lambda^{1/4}\frac{c_{4}}{{-c_{3}}}&\tau\end{array}\right),$
$\displaystyle\operatorname{Im}Z\sim$
$\displaystyle\left(\begin{array}[]{cc}1&\operatorname{O}(\lambda^{1/4})\\\
\operatorname{Re}\left\\{\lambda^{1/4}\frac{c_{4}}{{-c_{3}}}\right\\}&\operatorname{Im}\tau\end{array}\right).$

So, as $\lambda\rightarrow 0$,

$(\operatorname{Im}Z)^{-1}\sim\left(\begin{array}[]{cc}1&-({\operatorname{Im}\tau})^{-1}\operatorname{O}(\lambda^{{1}/{4}})\\\
({\operatorname{Im}\tau})^{-1}\operatorname{Re}\left\\{\lambda^{1/4}\frac{c_{4}}{c_{3}}\right\\}&({\operatorname{Im}\tau})^{-1}\end{array}\right).$

By (2.6) and (5.3),

$k_{\lambda}(z)\sim
4\frac{1+({\operatorname{Im}\tau})^{-1}\operatorname{Re}\left\\{\lambda^{1/4}\frac{c_{4}}{c_{3}}\right\\}(\overline{z}^{2}+{z}^{2})+(\operatorname{Im}\tau)^{-1}|z|^{4}}{|(z^{2}-a)(z^{2}-b)(z^{2}-\lambda)(z^{2}-\lambda^{2})|}\to\left(\frac{\operatorname{Im}\tau}{|z^{4}|}+1\right)k_{0}(z),$
(5.4)

which means that $\kappa_{X_{\lambda}}\not\to\kappa_{0}.$ Moreover, in the
local coordinate $z=\sqrt{x}$ near $(0,0)$,

${k}_{\lambda}(z)-\left(\frac{\operatorname{Im}\tau}{|z^{4}|}+1\right)k_{0}(z)\sim\frac{4({\operatorname{Im}\tau})^{-1}\operatorname{Re}\left\\{\lambda^{1/4}\frac{c_{4}}{c_{3}}\right\\}(\overline{z}^{2}+{z}^{2})}{|z^{4}(z^{2}-a)(z^{2}-b)|},$

which yields the conclusion.

∎

##  6 Cusp I: hyperelliptic curves

This section is devoted to the proof of Theorem 1.2. When
$\lambda\in\mathbb{C}\setminus\\{0,a_{1},...,a_{2g-2}\\}$, $X_{\lambda}$
defined in (1.4) has genus $g$. To prove Theorem 1.2, we need to analyze on
$X_{\lambda}$ the asymptotics of its $A$-period matrix and $B$-period matrix,
denoted by $A$ and $B$, respectively. Meanwhile, for the normalization
$Y:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=xP(x)\\}$ of genus $g-1$, denote its
$A$-period matrix and $B$-period matrix by $A_{0}$ and $B_{0}$, respectively.

######  Lemma 6.1.

Under the same assumptions as in Theorem 1.2, as $\lambda\rightarrow 0$, it
holds that

$A\sim\left(\begin{array}[]{cccccc}c_{5}\lambda^{-\frac{1}{4}}\frac{1}{\sqrt{P(0)}}&\operatorname{O}(1)\\\
&A_{0}\\\ \end{array}\right),$

where $*$ is a column vector with $g-1$ rows whose entries are given by (6.1)
below.

###### Proof.

We will estimate all the $g\times g$ elements one by one. Firstly, as
$\lambda\to 0$, it holds that

$\displaystyle A_{11}$
$\displaystyle=-2\int_{-\sqrt{\lambda}}^{0}\frac{dx}{\sqrt{x(x^{2}-{\lambda})P(x)}}=-2\int_{-\sqrt{\lambda}}^{0}\frac{dx}{\sqrt{x(x^{2}-{\lambda})}}\frac{1}{\sqrt{P(0)}}(1+\operatorname{O}(x))$
$\displaystyle\xlongequal[]{x=(u-1)\sqrt{\lambda}}-2\lambda^{-\frac{1}{4}}\int_{0}^{1}\frac{du}{\sqrt{u(u-1)(u-2)}}\frac{(1+\operatorname{O}((u-1)\sqrt{\lambda}))}{\sqrt{P(0)}}$
$\displaystyle\sim-2\lambda^{-\frac{1}{4}}\int_{0}^{1}\frac{du}{\sqrt{u(u-1)(u-2)}}\frac{1}{\sqrt{P(0)}}=:\lambda^{-\frac{1}{4}}\frac{c_{5}}{\sqrt{P(0)}}.$

In general, for $a_{i1}$, $2\leq i\leq g$, there is an extra $x^{i-1}$ in the
original integrand and thus an extra $\sqrt{\lambda}^{i-1}(u-1)^{i-1}$ in the
numerator of the above last expression, so

$A_{i1}\sim-2\int_{0}^{1}\frac{(u-1)^{i-1}du}{\sqrt{u(u-1)(u-2)}}\frac{\sqrt{\lambda}^{\frac{1}{2}+i-2}}{\sqrt{P(0)}}.$
(6.1)

Secondly, let $\delta_{2}$ contain only $-\sqrt{\lambda},0,\sqrt{\lambda}$ and
$a_{1}$ (not $a_{2},...,a_{d}$). Then,

$\displaystyle A_{12}$
$\displaystyle=\int_{\delta_{2}}\frac{dx}{\sqrt{x(x^{2}-\lambda)P(x)}}$
$\displaystyle=\int_{\delta_{2}}\frac{dx}{x\sqrt{xP(x)}}\left(1+\operatorname{O}\left(\frac{{\lambda}}{{x}^{2}}\right)\right)\sim\int_{\delta_{2}}\frac{dx}{x\sqrt{xP(x)}}.$

In general, for $a_{i2}$, there is an extra $x^{i-1}$ in the original
integrand, so

$A_{i2}\sim\int_{\delta_{2}}\frac{x^{i-2}dx}{\sqrt{xP(x)}}.$

Thirdly, let $\delta_{3}$ contain only
$-\sqrt{\lambda},0,\sqrt{\lambda},a_{1},a_{2}$ and $a_{3}$ (not
$a_{4},...,a_{d}$). Similarly, $a_{i3}$ is asymptotic to the same integrand
over $\delta_{3}$ instead of over $\delta_{2}$, i.e.,

$A_{i3}\sim\int_{\delta_{3}}\frac{x^{i-2}dx}{\sqrt{xP(x)}}.$

In general, for $j\geq 2$ and the corresponding homology basis $\delta_{j}$,
it holds that

$A_{ij}\sim\int_{\delta_{j}}\frac{x^{i-2}dx}{\sqrt{xP(x)}},$

which are exactly the same as the entries of $A_{0}$ when $i\geq 2$.

∎

######  Lemma 6.2.

Under the same assumptions as in Theorem 1.2, as $\lambda\rightarrow 0$, it
holds that

$B\sim\left(\begin{array}[]{cccccc}c_{5}\lambda^{-\frac{1}{4}}\frac{\sqrt{-1}}{\sqrt{P(0)}}&\operatorname{O}(1)\\\
\diamondsuit&B_{0}\end{array}\right),$

where $\diamondsuit$ is a column vector whose entries are
$B_{i1}=(-1)^{i-1}\sqrt{-1}A_{i1}$ for $A_{i1}$ in (6.1) and $2\leq i\leq g$.

###### Proof.

For the first column of $B$, we make the change of coordinates (similar to the
proof of Lemma 5.3) by setting $x=(-u+1)\sqrt{\lambda}$, and get for $1\leq
i\leq g$ that

$B_{i1}\sim(-1)^{i-1}\sqrt{-1}a_{i1}.$

For Column $j$, $2\leq j\leq g$, we use Taylor expansion of
$(x^{2}-\lambda)^{-1/2}$ and get that

$B_{ij}\sim-2\int_{a_{2j-3}}^{a_{2j-2}}\frac{x^{i-2}dx}{\sqrt{xP(x)}}.$

∎

Now we will give a proof of Theorem 1.2.

###### Proof of Theorem 1.2.

By (2.8), in the local coordinate $z=\sqrt{x}$, the Bergman kernel on the
normalization $Y$ of the nodal curve $X_{0}$ in (1.4) is exactly
$\kappa_{0}=k_{0}(z)dz\otimes d\bar{z}$, where

$k_{0}(z)=\frac{4}{|z^{4}P(z^{2})|}\sum_{i,j=1}^{g-1}((\operatorname{Im}Z_{0})^{-1})_{i,j}(z^{i}\overline{z}^{j})^{2}.$
(6.2)

By Lemma 6.1 and the block matrix inversion, we know that

$A^{-1}\sim\left(\begin{array}[]{cccccc}\lambda^{\frac{1}{4}}\frac{\sqrt{P(0)}}{c_{5}}&\operatorname{O}(\lambda^{{1}/{4}})\\\
(-A_{0}^{-1}*)\lambda^{\frac{1}{4}}\frac{\sqrt{P(0)}}{c_{5}}&A_{0}^{-1}\end{array}\right),$

where $\lim\limits_{\lambda\to
0}\frac{(\operatorname{O}(\lambda^{\frac{1}{2}}))^{T}}{\lambda^{\frac{1}{2}}}$
is a finite column vectors with $g-1$ rows. Therefore, as $\lambda\to 0$,

$Z=A^{-1}B\sim\left(\begin{array}[]{cccc}\sqrt{-1}&\operatorname{O}(\lambda^{\frac{1}{4}})\\\
A_{0}^{-1}(\diamondsuit-\sqrt{-1}*)&A_{0}^{-1}B_{0}\end{array}\right),\quad\operatorname{Im}Z\sim\left(\begin{array}[]{cccc}1&\operatorname{O}(\lambda^{{1}/{4}})\\\
\operatorname{Im}(A_{0}^{-1}(\diamondsuit-\sqrt{-1}*))&\operatorname{Im}Z_{0}\end{array}\right),$

and

$(\operatorname{Im}Z)^{-1}\sim\left(\begin{array}[]{cccc}1&\operatorname{O}(\lambda^{{1}/{4}})\\\
(\operatorname{Im}Z_{0})^{-1}\operatorname{Im}(A_{0}^{-1}(\diamondsuit-\sqrt{-1}*))&(\operatorname{Im}Z_{0})^{-1}\end{array}\right).$

By (2.3) and (6.2), as $\lambda\to 0$, in the local coordinate $z=\sqrt{x}$
near $(0,0)$, it holds that

${k}_{\lambda}(z)\to\frac{1+\sum_{i,j=1}^{g-1}((\operatorname{Im}Z_{0})^{-1})_{i,j}(z^{i}\overline{z}^{j})^{2}}{|4^{-1}z^{4}P(z^{2})|}={k}_{0}(z)+\frac{4}{|z^{4}P(z^{2})|},$

so $\kappa_{X_{\lambda}}\not\to\kappa_{0}$. Moreover,

${k}_{\lambda}(z)-{k}_{0}(z)-\frac{4}{|z^{4}P(z^{2})|}\sim\frac{-8\operatorname{Re}\sum\limits_{i=1}^{g-1}\left((\operatorname{Im}Z_{0})^{-1}\operatorname{Im}(A_{0}^{-1}(\diamondsuit-\sqrt{-1}*))\right)_{i}{z}^{2i}}{|z^{4}P(z^{2})|},$

which yields the conclusion.

∎

Since $Z$ is symmetric,
$A_{0}^{-1}(\diamondsuit-\sqrt{-1}*)=\operatorname{O}(\lambda^{{1}/{4}})$, and
both the leading and subleading terms in the expansion of
$\kappa_{X_{\lambda}}$ are harmonic with respect to $\lambda$.

##  7 Cusp II: genus-two curves

In this section, we consider a family of genus two curves

$X_{\lambda}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x(x-{\lambda})(x-1)(x-a)(x-b)\\},$
(7.1)

where $a,b,\lambda$ are distinct complex numbers satisfying
$0<|\lambda|<|a|<|b|$. As $\lambda\to 0$, $X_{\lambda}$ degenerates to a
singular curve $X_{0}$ with an ordinary cusp. The normalization of $X_{0}$ is
an elliptic curve $\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=x(x-a)(x-b)\\}$, whose
period is $\tau$ given in (5.2). The Bergman kernels on $X_{\lambda}$ and on
the normalization of $X_{0}$ are denoted by $\kappa_{X_{\lambda}}$ and
$\kappa_{0}$, respectively. In the local coordinate $z:=\sqrt{x}$ near
$(0,0)$, we write $\kappa_{X_{\lambda}}=k_{\lambda}(z)dz\otimes d\bar{z}$, and
$\kappa_{0}=k_{0}(z)dz\otimes d\bar{z}$. Then, our result on the asymptotic
behaviour of the Bergman kernel $\kappa_{X_{\lambda}}$ with precise
coefficients is stated as follows.

######  Theorem 7.1.

For $X_{\lambda}$ defined by (7.1), as $\lambda\rightarrow 0$, it holds that

$\kappa_{X_{\lambda}}\to\kappa_{0}$;

for small $|z|\neq 0$ ,

$\log{k}_{\lambda}(z)-\log{k}_{0}(z)\sim\frac{\pi\cdot\operatorname{Im}\tau}{-\log|\lambda|\cdot|z|^{4}},$

To prove Theorem 7.1, we need the following two lemmata by analyzing the
asymptotics of the $A$-period matrix and $B$-period matrix on $X_{\lambda}$.

######  Lemma 7.2.

Under the same assumptions as in Theorem 7.1, as $\lambda\rightarrow 0$, it
holds that

$A\sim\left(\begin{array}[]{cc}\frac{-2\pi}{\sqrt{ab}\sqrt{\lambda}}&c_{6}\\\
c_{7}\lambda^{3/2}&\frac{-2c_{3}}{\sqrt{ab}}\end{array}\right),$

where $c_{7}:=\frac{-2}{\sqrt{ab}}\int_{0}^{1}\sqrt{v^{-1}-1}dv$ and
$c_{3},c_{6}$ are the same as in Lemma 5.2.

###### Proof of Lemma 7.2.

Consider the integrals

$\int_{0}^{\lambda^{2}}\frac{dx}{\sqrt{x(x-\lambda)(x-\lambda^{2})}}\xlongequal{x=\lambda^{2}(1-v)}\int_{0}^{1}\frac{-\sqrt{-1}\lambda^{-1}dv}{\sqrt{v(v-1)(v-1+\lambda^{-1})}}\sim\frac{-\lambda^{-1}}{2\sqrt{-1}}\int_{\tilde{\gamma}}\frac{dv}{v\lambda^{-1/2}}=\frac{-\pi}{\sqrt{\lambda}},$

where $\tilde{\gamma}$ is a large cycle containing $0,1$, and

$\displaystyle\int_{0}^{\lambda^{2}}\frac{xdx}{\sqrt{x(x-\lambda)(x-\lambda^{2})}}$
$\displaystyle=\frac{-\lambda}{\sqrt{-1}}\int_{0}^{1}\frac{(v-1)dv}{\sqrt{v(v-1)(v-1+\lambda^{-1})}}\sim\frac{-\lambda^{3/2}}{\sqrt{-1}}\int_{0}^{1}\sqrt{\frac{v-1}{v}}dv.$

We estimate the four entries one by one. Firstly, let $\delta_{1}$ only
contain $0$ and $\lambda^{2}$, and similar to the proof of Lemma 5.2 we know
that

$\displaystyle A_{11}$
$\displaystyle=-2\int_{0}^{\lambda^{2}}\frac{dx}{\sqrt{x(x-\lambda)(x-\lambda^{2})}}\frac{1}{\sqrt{-a}}\frac{1}{\sqrt{-b}}\left(1+\frac{x}{2a}+\operatorname{O}(x^{2})\right)\left(1-\frac{x}{2b}+\operatorname{O}(x^{2})\right)$
$\displaystyle\sim\frac{-2}{-\sqrt{ab}}\int_{0}^{\lambda^{2}}\frac{dx}{\sqrt{x(x-\lambda)(x-\lambda^{2})}}\sim\frac{-2\pi}{\sqrt{ab}\sqrt{\lambda}}.$

Secondly,

$A_{21}\sim\frac{2}{\sqrt{ab}}\int_{0}^{\lambda^{2}}\frac{xdx}{\sqrt{x(x-\lambda)(x-\lambda^{2})}}\sim\frac{-2}{\sqrt{ab}}\frac{\lambda^{3/2}}{\sqrt{-1}}\int_{0}^{1}\sqrt{\frac{v-1}{v}}dv=:c_{7}\lambda^{3/2}.$

Thirdly, let $\delta_{2}$ contain only $0,\lambda,\lambda^{2}$ and $a$. Then,
it holds that

$\displaystyle A_{12}=$
$\displaystyle\int_{\delta_{2}}\frac{dx}{\sqrt{x(x-\lambda)(x-\lambda^{2})(x-a)(x-b)}}$
$\displaystyle=$
$\displaystyle\int_{\delta_{2}}\frac{dx}{x\sqrt{x(x-a)(x-b)}}\left(1+\operatorname{O}\left(\frac{\lambda}{x}\right)\right)\left(1+\operatorname{O}\left(\frac{\lambda^{2}}{x}\right)\right)$
$\displaystyle\sim$
$\displaystyle\int_{\delta_{2}}\frac{dx}{x\sqrt{x(x-a)(x-b)}}=:c_{6},$

Lastly,

$A_{22}\sim\int_{\delta_{2}}\frac{dx}{x\sqrt{x(x-a)(x-b)}}=\int_{0}^{a}\frac{dx}{x\sqrt{x(x-a)(x-b)}}.$

∎

######  Lemma 7.3.

Under the same assumptions as in Theorem 7.1, as $\lambda\rightarrow 0$, it
holds that

$B\sim\left(\begin{array}[]{cc}\frac{2\sqrt{-1}\log\lambda}{\sqrt{ab}\sqrt{\lambda}}&d_{3}\\\
\frac{2}{\sqrt{ab}}\sqrt{-\lambda}&d_{4}\end{array}\right),$

where $d_{3},d_{4}$ are the same as in Lemma 5.3.

###### Proof of Lemma 7.3.

By [26] or (3.3), it is known that, as $\lambda\to 0$,

$\int_{\lambda^{2}}^{\lambda}\frac{dx}{\sqrt{x(x-\lambda)(x-\lambda^{2})}}=\frac{1}{\sqrt{\lambda}}\int_{\lambda}^{1}\frac{du}{\sqrt{u(u-1)(u-\lambda)}}\sim\frac{\sqrt{-1}\log\lambda}{\sqrt{\lambda}}.$

Firstly, let $\gamma_{1}$ contain only $\lambda$ and $\lambda^{2}$, and we get
that

$\displaystyle B_{11}$
$\displaystyle=-2\int_{\lambda^{2}}^{\lambda}\frac{dx}{\sqrt{x(x-\lambda)(x-\lambda^{2})(x-a)(x-b)}}$
$\displaystyle=\frac{-2}{-\sqrt{ab}}\int_{\lambda^{2}}^{\lambda}\frac{dx}{\sqrt{x(x-\lambda)(x-\lambda^{2})}}\left(1+\operatorname{O}(x)\right)$
$\displaystyle\sim\frac{2}{\sqrt{ab}}\int_{\lambda^{2}}^{\lambda}\frac{dx}{\sqrt{x(x-\lambda)(x-\lambda^{2})}}\sim\frac{2}{\sqrt{ab}}\frac{\sqrt{-1}\log\lambda}{\sqrt{\lambda}}.$

Secondly, by (3.4), as $\lambda\to 0$,

$B_{21}\sim\frac{2}{\sqrt{ab}}\int_{\lambda^{2}}^{\lambda}\frac{xdx}{\sqrt{x(x-\lambda)(x-\lambda^{2})}}\xlongequal[t=\frac{1}{\lambda}]{x=\frac{\lambda}{s}}\frac{-2}{\sqrt{ab}}\int_{1}^{t}\frac{ds}{s^{2}\sqrt{s-t}}\sim\frac{2}{\sqrt{ab}}\sqrt{-\lambda}.$

Thirdly,

$\displaystyle B_{12}=$
$\displaystyle-2\int_{a}^{b}\frac{dx}{\sqrt{x(x-\lambda)(x-\lambda^{2})(x-a)(x-b)}}$
$\displaystyle=$
$\displaystyle-2\int_{a}^{b}\frac{dx\left(1+\operatorname{O}\left({\lambda}{x^{-1}}\right)\right)\left(1+\operatorname{O}\left({\lambda^{2}}{x^{-1}}\right)\right)}{x\sqrt{x(x-a)(x-b)}}\sim-2\int_{a}^{b}\frac{dx}{x\sqrt{x(x-a)(x-b)}}:=d_{3}.$

Lastly,

$B_{22}\sim-2\int_{a}^{b}\frac{dx}{\sqrt{x(x-a)(x-b)}}:=d_{4}.$

∎

###### Proof of Theorem 7.1.

On the regular part of the cuspidal curve $X_{0}$, (5.3) gives the formulae
for the Bergman kernel $\kappa_{0}=k_{0}(z)|dz|^{2}$ in the local coordinate
$z=\sqrt{x}$. By Lemma 7.2, as $\lambda\to 0$,

$A^{-1}\sim\left(\begin{array}[]{cc}\frac{-2\pi}{\sqrt{ab}\sqrt{\lambda}}&c_{6}\\\
c_{7}\lambda^{3/2}&\frac{-2c_{3}}{\sqrt{ab}}\end{array}\right)^{-1}\sim\frac{ab\sqrt{\lambda}}{{4\pi}{}{c_{3}}}\left(\begin{array}[]{cc}\frac{-2c_{3}}{\sqrt{ab}}&-c_{6}\\\
-c_{7}\lambda^{3/2}&\frac{-2\pi}{\sqrt{ab}\sqrt{\lambda}}\end{array}\right).$

Let $Z$ denote the period matrix of $X_{\lambda}$. Then, it follows that

$Z=A^{-1}B\sim\left(\begin{array}[]{cc}\frac{-\sqrt{-1}\log\lambda}{\pi}&\frac{{{ab}\sqrt{\lambda}}}{4\pi
c_{3}}\left(\frac{-2c_{3}}{\sqrt{ab}}d_{3}-c_{6}d_{4}\right)\\\
\frac{\sqrt{\lambda}}{\sqrt{-1}}c_{3}^{-1}{}{}&\tau\end{array}\right),\quad\operatorname{Im}Z\sim\left(\begin{array}[]{cc}\frac{-\log|\lambda|}{\pi}&\operatorname{O}(\lambda^{\frac{1}{2}})\\\
-\operatorname{Re}\left\\{\sqrt{\lambda}c_{3}^{-1}\right\\}&\operatorname{Im}\tau\end{array}\right)$

So, as $\lambda\rightarrow 0$,

$(\operatorname{Im}Z)^{-1}\sim\left(\begin{array}[]{cc}\frac{\pi}{-\log|\lambda|}&\frac{1}{\log|\lambda|}\operatorname{O}(\lambda^{\frac{1}{2}})\\\
\frac{-\pi}{\log|\lambda|\operatorname{Im}\tau}\operatorname{Re}\left\\{\sqrt{\lambda}c_{3}^{-1}\right\\}&({\operatorname{Im}\tau})^{-1}\end{array}\right).$

By (2.6) and (5.3), $\kappa_{X_{\lambda}}\to\kappa_{0}.$ Moreover, in the
local coordinate $z=\sqrt{x}$ near $(0,0)$,

${k}_{\lambda}(z)-{k}_{0}(z)\sim\frac{4\pi}{|(z^{2}-a)(z^{2}-b)(z^{2}-\lambda)(z^{2}-\lambda^{2})|}\cdot\frac{1+\operatorname{O}(\lambda^{\frac{1}{2}})}{-\log|\lambda|},$

which yields the conclusion.

∎

##  8 Cusp II: hyperelliptic curves

This section is devoted to the proof of Theorem 1.3. When
$\lambda\in\mathbb{C}\setminus\\{0,1,a_{1},...,a_{2g-2}\\}$, $X_{\lambda}$
defined in (1.5) has genus $g$. To prove Theorem 1.3, we need to analyze on
$X_{\lambda}$ the asymptotics of its $A$-period matrix and $B$-period matrix,
denoted by $A$ and $B$, respectively. Meanwhile, for the normalization
$Y:=\\{(x,y)\in\mathbb{C}^{2}\,|\,y^{2}=xP(x)\\}$ of genus $g-1$, denote its
$A$-period matrix and $B$-period matrix by $A_{0}$ and $B_{0}$, respectively.

######  Lemma 8.1.

Under the same assumptions as in Theorem 1.3, as $\lambda\rightarrow 0$, it
holds that

$A\sim\left(\begin{array}[]{cccccc}\frac{2\pi}{\sqrt{\lambda
P(0)}}&\operatorname{O}(1)\\\ &A_{0}\\\ \end{array}\right),$

where $*$ is a column vector with $g-1$ rows whose entries are at most
$\operatorname{O}(\lambda^{3/2}).$

###### Proof.

Firstly, as $\lambda\rightarrow 0$,

$\displaystyle A_{11}$
$\displaystyle=-2\int_{0}^{\lambda^{2}}\frac{dx}{\sqrt{x(x-{\lambda})(x-{\lambda}^{2})}}\frac{1}{\sqrt{P(0)}}(1+\operatorname{O}(x))$
$\displaystyle\sim-2\int_{0}^{\lambda^{2}}\frac{dx}{\sqrt{x(x-{\lambda})(x-{\lambda}^{2})}}\frac{1}{\sqrt{P(0)}}\sim\frac{2\pi}{\sqrt{\lambda
P(0)}}.$
$A_{21}\sim\frac{-2}{\sqrt{P(0)}}\int_{0}^{\lambda^{2}}\frac{xdx}{\sqrt{x(x-{\lambda})(x-{\lambda}^{2})}}\sim\frac{-2}{\sqrt{P(0)}}\frac{-\lambda^{3/2}}{\sqrt{-1}}\int_{0}^{1}\sqrt{\frac{v-1}{v}}dv.$

Here we change the variable by letting $x=\lambda^{2}(1-v)$. In general, for
$2\leq i\leq g$,

$A_{i1}\sim\frac{-2}{\sqrt{P(0)}}\frac{-\lambda^{3/2}}{\sqrt{-1}}\int_{0}^{1}{\lambda}^{2i-4}(1-v)^{i-2}\sqrt{\frac{v-1}{v}}dv=\operatorname{O}(\lambda^{2i-2.5}).$

Secondly, let $\delta_{2}$ contain only $0,\lambda^{2},\lambda$ and $a_{1}$
(not $a_{2},...,a_{d}$). Then,

$\displaystyle A_{12}$
$\displaystyle=\int_{\delta_{2}}\frac{dx}{\sqrt{x(x-{\lambda})(x-{\lambda}^{2})P(x)}}$
$\displaystyle=\int_{\delta_{2}}\frac{dx}{x\sqrt{xP(x)}}\left(1+\operatorname{O}\left(\frac{{\lambda}}{{x}^{2}}\right)\right)\sim\int_{\delta_{2}}\frac{dx}{x\sqrt{xP(x)}}.$

In general, for $A_{i2}$, there is an extra $x^{i-1}$ in the original
integrand above, so

$A_{i2}\sim\int_{\delta_{2}}\frac{x^{i-2}dx}{\sqrt{xP(x)}}.$

Thirdly, let $\delta_{3}$ contain only $0,\lambda^{2},\lambda,a_{1},a_{2}$ and
$a_{3}$ (not $a_{4},...,a_{d}$). Similarly, $A_{i3}$ is asymptotic to the same
integrand over $\delta_{3}$ instead of over $\delta_{2}$, i.e.,

$A_{i3}\sim\int_{\delta_{3}}\frac{x^{i-2}dx}{\sqrt{xP(x)}}.$

Lastly, for $j\geq 2$ and the corresponding homology basis $\delta_{j}$, it
holds that

$A_{ij}\sim\int_{\delta_{j}}\frac{x^{i-2}dx}{\sqrt{xP(x)}},$

which are exactly the same as the entries of $A_{0}$ when $i\geq 2$.

∎

######  Lemma 8.2.

Under the same assumptions as in Theorem 1.3, as $\lambda\rightarrow 0$, it
holds that

$B\sim\left(\begin{array}[]{cccccc}\frac{-2}{\sqrt{P(0)}}\frac{\sqrt{-1}\log\lambda}{\sqrt{\lambda}}&\operatorname{O}(1)\\\
\heartsuit&B_{0}\end{array}\right),$

where $\heartsuit$ is a column vector whose entries are
$\frac{-2\lambda^{i-2}\sqrt{-\lambda}}{\sqrt{P(0)}}$, for $2\leq i\leq g$.

###### Proof.

$\displaystyle B_{11}$
$\displaystyle=-2\int_{\lambda^{2}}^{\lambda}\frac{dx}{\sqrt{x(x-{\lambda})(x-{\lambda}^{2})\cdot
P(x)}}=-2\int_{\lambda^{2}}^{\lambda}\frac{dx}{\sqrt{x(x-{\lambda})(x-{\lambda}^{2})}}\frac{1}{\sqrt{P(0)}}(1+\operatorname{O}(x))$
$\displaystyle\sim-2\int_{\lambda^{2}}^{\lambda}\frac{dx}{\sqrt{x(x-{\lambda})(x-{\lambda}^{2})}}\frac{1}{\sqrt{P(0)}}\sim\frac{-2}{\sqrt{P(0)}}\frac{\sqrt{-1}\log\lambda}{\sqrt{\lambda}}.$

Thus, for $2\leq i\leq g$, by (3.5),

$\displaystyle B_{i1}$
$\displaystyle=-2\int_{\lambda^{2}}^{\lambda}\frac{x^{i-1}dx}{\sqrt{x(x-{\lambda})(x-{\lambda}^{2})}}\frac{1}{\sqrt{P(0)}}(1+\operatorname{O}(x))$
$\displaystyle\sim-2\int_{\lambda^{2}}^{\lambda}\frac{x^{i-1}dx}{\sqrt{x(x-{\lambda})(x-{\lambda}^{2})}}\frac{1}{\sqrt{P(0)}}\xlongequal[t=\frac{1}{\lambda}]{x=\frac{\lambda}{s}}\frac{2\lambda^{i-2}}{\sqrt{P(0)}}\int_{1}^{t}\frac{ds}{s^{i}\sqrt{s-t}}\sim\frac{-2\lambda^{i-2}\sqrt{-\lambda}}{\sqrt{P(0)}}.$

For Column $j$, $2\leq j\leq g$, we use Taylor expansion of
$\sqrt{(x-{\lambda})(x-{\lambda}^{2})}^{-1}$ to get that

$B_{ij}\sim-2\int_{a_{2j-3}}^{a_{2j-2}}\frac{x^{i-2}dx}{\sqrt{xP(x)}}.$

which are exactly the same as the entries of $B_{0}$ when $i\geq 2$.

∎

Now we will give a proof of Theorem 1.3.

###### Proof of Theorem 1.3.

On the regular part of the cuspidal curve $X_{0}$, (6.2) gives the formulae
for the Bergman kernel $\kappa_{0}=k_{0}(z)|dz|^{2}$ in the local coordinate
$z=\sqrt{x}$. By Lemma 8.1 and the block matrix inversion, we know that

$A^{-1}\sim\left(\begin{array}[]{cccccc}\frac{\sqrt{\lambda
P(0)}}{2\pi}&\operatorname{O}(\lambda^{\frac{1}{2}})\\\
\operatorname{O}(\lambda^{2})&A_{0}^{-1}\end{array}\right),$

where both $\lim\limits_{\lambda\to
0}\frac{(\operatorname{O}(\lambda^{\frac{1}{2}}))^{T}}{\lambda^{\frac{1}{2}}}$
and $\lim\limits_{\lambda\to
0}\frac{\operatorname{O}(\lambda^{2})}{\lambda^{2}}$ are finite column vectors
with $g-1$ rows. Therefore, as $\lambda\to 0$,

$Z=A^{-1}B\sim\left(\begin{array}[]{cccc}\frac{-\sqrt{-1}}{\pi}{\log\lambda}&\operatorname{O}(\lambda^{\frac{1}{2}})\\\
A_{0}^{-1}\heartsuit&A_{0}^{-1}B_{0}\end{array}\right),\quad\operatorname{Im}Z\sim\left(\begin{array}[]{cccc}\frac{\log|\lambda|}{-\pi}&\operatorname{O}(\lambda^{\frac{1}{2}})\\\
\operatorname{Im}(A_{0}^{-1}\heartsuit)&\operatorname{Im}Z_{0}\end{array}\right),$

and

$(\operatorname{Im}Z)^{-1}\sim\left(\begin{array}[]{cccc}\frac{-\pi}{\log|\lambda|}&\operatorname{O}(({\log|\lambda|})^{-1}\lambda^{\frac{1}{2}})\\\
(\operatorname{Im}Z_{0})^{-1}\operatorname{Im}(A_{0}^{-1}\heartsuit)\frac{\pi}{\log|\lambda|}&(\operatorname{Im}Z_{0})^{-1}\end{array}\right).$

In fact, since $Z$ is symmetric, the off-diagonal block matrices in each
matrix above concerning $Z$ are the transpose of each other. By (2.5) and
(6.2), as $\lambda\to 0$, $\kappa_{X_{\lambda}}\to\kappa_{0}.$ Moreover, in
the local coordinate $z=\sqrt{x}$ near $(0,0)$, it holds that

${k}_{\lambda}(z)-{k}_{0}(z)\sim\frac{4\pi}{|(z^{2}-\lambda)(z^{2}-\lambda^{2})P(z^{2})|}\frac{1-2\operatorname{Re}\sum\limits_{i=1}^{g-1}\left((\operatorname{Im}Z_{0})^{-1}\operatorname{Im}(A_{0}^{-1}\heartsuit)\right)_{i}{z}^{2i}}{-\log|\lambda|}$

which yields that

$\psi-\log{k}_{0}(z)\sim\frac{\pi}{-\log|\lambda|}\frac{1}{\sum\limits_{i,j=1}^{g-1}((\operatorname{Im}Z_{0})^{-1})_{i,j}(z^{i}\overline{z}^{j})^{2}}.$

∎

##  9 Jacobian varieties

We will give a proof of Theorem 1.5 by using the results obtained in the
proofs of Theorems 1.1, 1.2, 1.3. Let $X_{\lambda}$ be a compact curve of
genus $g\geq 2$, and let $Z$ be its period matrix with respect to some chosen
homology basis. The Jacobian variety of $X_{\lambda}$, which is denoted by
$\text{Jac}(X_{\lambda})$, is then identified with the $g$-dimensional complex
torus $\mathbb{C}^{g}/\mathbb{Z}^{g}+Z\mathbb{Z}^{g}$. It is well known that
the Abel-Jacobi (period) map $X_{\lambda}\to\text{Jac}(X_{\lambda})$ is a
holomorphic embedding, and the Bergman kernel on a smooth algebraic curve is
the pull back of the Euclidean metric from the Jacobian variety via this map.

###### Proof of Theorem 1.5.

By definition (0.1), the Bergman kernel on $\text{Jac}(X_{\lambda})$ can be
written as $\mu_{\lambda}dw\otimes d\bar{w}$, under the coordinate $w$ induced
from $\mathbb{C}^{g}$, where $\mu_{\lambda}=(\det(\operatorname{Im}Z))^{-1}$.
After adding a possible one-point compactification at $\infty$, one may assume
that the curve $X_{\lambda}$ is compact. For $X_{\lambda}$ defined in (1.3) or
(1.5), as $\lambda\to 0$, it holds that

$\det(\operatorname{Im}Z)=\frac{\log|\lambda|}{-\pi}\det(\operatorname{Im}Z_{0})+\operatorname{O}(1)\to+\infty,$

which yields the first part of the conclusion in Theorem 1.5. For
$X_{\lambda}$ defined in (1.4), more careful analysis in the proof of Theorem
1.2 shows that as $\lambda\to 0$,

$\operatorname{Im}Z=\left(\begin{array}[]{cccc}1+\operatorname{O}(\lambda^{{1}/{2}})&\operatorname{O}(\lambda^{{1}/{4}})\\\
\operatorname{O}(\lambda^{{1}/{4}})&\operatorname{Im}Z_{0}+\operatorname{O}(\lambda^{{1}/{2}})\end{array}\right),\quad\det(\operatorname{Im}Z)=\det\operatorname{Im}Z_{0}+\operatorname{O}(\lambda^{{1}/{2}})<+\infty,$

which yields the second part of the conclusion.

∎

### Funding

The research of the author is supported by AMS-Simons travel grant. This work
was supported by KAKENHI and the Grant-in-Aid for JSPS Fellows (No. 15J05093).

### Acknowledgements

This article constitutes essentially the author’s doctoral thesis in Nagoya
University under the advice of Professor Ohsawa.

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