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BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FORCURVES

ROBERT XIN DONG

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

1. Introduction

On an n-dimensional complex manifoldX , the Bergman kernel is a reproducing kernel of the spaceof square integrable holomorphic top forms. For the canonical bundle K, which is the nth exteriorpower of the cotangent bundle Ω on X , the Bergman kernel is defined as

(1.1) κ :=∑

sj ⊗ sj,

where sjj is a complete orthonormal basis of H0(X,K). Thus, the Bergman kernel is a canonicalvolume-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 obtainedby Maitani and Yamaguchi [45] and later by Berndtsson [1,2]. More generally, further important devel-opments on general Stein manifolds and complex projective algebraic manifolds [2, 7, 17, 44, 55, 64, 66]state certain positivity properties of the direct images of the relative canonical bundles, and turn outto have intimate connections with the space of Kähler metrics [3,18,19,65], the invariance of plurigen-era [8,38,53,59,60], and the sharp Ohsawa-Takegoshi theorem [6,9,11,16,33–35,47,52,58]. Conversely,the Ohsawa-Takegoshi theorem with optimal constant was applied in [11, 33] to prove the variationresults of the Bergman kernels.

Consider the familyXπ−−→ ∆,whereX is a complex (n+1)-dimensional Stein manifold fibred over

the unit disc ∆ ⊂ C, and π is a surjective holomorphic map with connected fibers. Moreover, π is aholomorphic submersion over ∆∗ := ∆ \ 0. For λ ∈ ∆∗, the fibres Xλ = π−1(λ) are n-dimensionalStein manifolds, whose Bergman kernel are denoted by κXλ

. Let (z, λ) be the coordinate of Xλ ×∆∗,which is the local trivialization of the fibration, and write κXλ

:= kλ(z)dz⊗dz locally. Then, the abovementioned positivity results imply that ψ := log kλ(z) is subharmonic with respect to λ ∈ ∆∗, i.e.,

∂2ψ

∂λ∂λ≥ 0,

2010 Mathematics Subject Classification. Primary 32A25; Secondary 32G20, 32G15, 14H45, 32Q28, 14D06Key words and phrases. Bergman kernel metric, cusp, degeneration, Hodge theory, hyperelliptic Riemann surface,Jacobian variety, node, Ohsawa-Takegoshi L2 extension, period matrix, positivity of direct image

1

2 ROBERT XIN DONG

if the Bergman kernel is not idetically zero. Here as λ varies, eψ induces on KX/∆ the so-called relativeBergman kernel metric, which is represented by different local functions, given different local trivial-izations (see [7]). When the fibration π has a singular fiberX0, our goal is to quantitatively characterizeat degeneration the asymptotics of ψ, as λ→ 0.

In fact, the study of the asymptotics analysis of integrals is a very classical subject and has beenpursued by many experts in field of Hodge theory [13, 14, 39, 40, 67, 71], especially the nilpotent orbittheorem [32, 56] in the variations of Hodge structures. In [31], the (pluri)subharmonicity in the basedirection of the above ψ was described as “a pleasant surprise” by Griffiths. It is known classically thatthe earlierworks of Griffiths [30,31], Fujita [29], as well as other important results in algebraic geometryincluding [41, 43] played a decisive role in understanding the behavior of ψ. For more background ontheL2-analysis in several complex variables, and its interactionwith Hodge theory, see [5,15,48–51,54].

In the theory of Riemannn surfaces and their moduli space, the degenerations of analytic dif-ferentials have also been extensively studied via a general method called the pinching coordinate(see [28,46,69] and the references therein). The spaces of degenerating Riemannn surfaces correspondto paths in the moduli space leading to the boundary points, and are obtained from compact surfacesby shrinking finitely many closed loops to points, called nodes. Near nodal singularities of generalcurves, Wentworth [68] obtained precise estimates for the Arakelov metric with applications to arith-metic geometry and string theory; Habermann and Jost [36, 37] studied the behavior of the Bergmankernels and their induced L2 metrics on Teichmüller spaces, with a strong motivation in minimal sur-face theory. Specifically, the result in [36] shows that the Bergman kernels on a degenerating family ofcompact Riemann surfaces converge to that on the normalization of the limiting nodal curve, with thesecond term having asymptotic growth (− log |λ|)−1, as λ→ 0.

On the other hand, there exist worse singularities such as cusps, with which the pinching coordinatemethod cannot deal. Boucksom and Jonsson [10] studied the asymptotics of volume forms on degen-erating compact manifolds and established a measure-theoretic version of the Kontsevich-Soibelmanconjecture, which deals with the limiting behavior of a family of Calabi-Yau manifolds approachinga “cusp” in the moduli space boundary. In [20–23], the author obtained quantitative results for theBergman kernels on Legendre and other degenerating families of elliptic curves, by using elliptic func-tions as well as a method based on the Taylor series expansion of Abelian differentials and periodmatrices (see [12]).

In this paper, for the fibers Xλ being algebraic curves or the Jacobian varieties, we determine theprecise asymptotic behaviors of ψ at degeneration, using the classical Taylor expansion method. Ourfirst motivation is to written down explicitly the asymptotic coefficients, which involve the complexstructure information and reflect the geometry of the base varieties and their singularities, for variousfamilies of hyperelliptic curves degenerating to singular ones with nodes or cusps. Our second motiva-tion is to investigate whether similar convergence results hold true for the Bergman kernel on cuspidaldegenerating families of curves, in comparison with [10, 36, 37, 57, 68]. It turns out that in general thisis not the case as seen in Theorem 2.2. However, in Theorem 2.3 we find that the Bergman kernels onsome cuspidal families of curves indeed converge at degeneration. As a last motivation, we apply theresults on higher genus curves to the study of their Jacobians, and generalize the results in [20, 21, 23]toward higher dimensions. To summarize, in our studies of curves, the subharmonic function ψ tendsto be small, not big, near the singular fibers (cf. [7]).

The organization of this paper is as follows. In Section 2 we state our main results on the degenera-tion of the Bergman kernel. In Section 3, we study the Bergman kernel on the normalization of generalalgebraic curves. In Sections 4 and 5, we work on nodal curves. In Sections 6–7 and 8–9, we work oncuspidal curves of types I and II, respectively. Our results on the Jacobian varieties are stated in Section10. An announcement of this paper for curves appeared in [24].

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 3

2. Main results

Throughout this paper, we identify the term Riemann surface with smooth algebraic curve, and letP denote a polynomial of degree d ≥ 2 with complex roots aj such that 1 < |a1| < |a2| < ... < |ad|.For a holomorphic family of Riemann surfaces Xλ parametrized by λ ∈ ∆∗, each fiberwise Bergmankernel is locally written as κXλ

:= kλ(z)dz ⊗ dz, in some coordinate z for some function kλ. Recallthat ψ := log kλ(z) is subharmonic with respect to λ ∈ ∆∗.

2.1. Nodal cases. In affine coordinates, consider a family of hyperelliptic curves

(2.1) Xλ := (x, y) ∈ C2 | y2 = x(x− λ)(x− 1)P (x).

As λ → 0, Xλ degenerates to a singular curve X0 with a non-separating node. The normalization ofX0 is a smooth curve Y := (x, y) ∈ C2 | y2 = (x− 1)P (x), whose A-period matrix and normalizedperiod matrix are denoted byA0 and Z0, respectively. By the L2 removable singularity theorem, Y andthe regular part of X0 have the same Bergman kernel, denoted by κ0. In the local coordinate z :=

√x

near (0, 0), we write κXλ= kλ(z)dz ⊗ dz and κ0 = k0(z)dz ⊗ dz. See (5.1) for the precise formulae of

κ0.

Theorem 2.1. As λ→ 0, it holds that

(i) κXλ→ κ0;

(ii) for 0 6= |z| small,

ψ − log k0(z) ∼π

− log |λ|1− 2Re

∑g−1i=1

(

(ImZ0)−1 Im(A−1

0 ⋆))

iz2i

∑g−1i,j=1((ImZ0)−1)i,j(ziz

j)2,

where ⋆ is a column vector with g − 1 rows whose entries are all −2i√

P (0)−1.

Part (i) is essentially due to Habermann and Jost [36], who in fact have used the the pinching coor-dinate method to study general (possibly non-hypereliptic) algebraic curves degenerating to singularones with separating or non-separating nodes. To write down the explicit asymptotic coefficients inPart (ii), we rely on the study of genus two curves in Theorem 4.1. We remark that in both Theorem2.1 and Theorem 4.1, for each small z 6= 0, ψ regarded as a subharmonic function in λ is bounded nearλ = 0 and thus has Lelong number zero. Moreover, the coefficients of the first and second terms in theexpansions of ψ depend only on the information away from the node.

2.2. Cuspidal cases. Since in [28, 36, 37, 46, 68, 69] only the nodal degeneration was considered, wecontinue investigating whether similar convergence results hold true for the Bergman kernel on cusp-idal degenerating families of curves, and find that in general this is not the case. For the singular curveX0 := (x, y) ∈ C2 | y2 = x3P (x) with an ordinary cusp at (0, 0), its normalization is the smoothcurve Y := (x, y) ∈ C2 | y2 = xP (x), whose A-period matrix and normalized period matrix aredenoted byA0 andZ0, respectively. Then, Y and the regular part ofX0 have the same Bergman kernel,denoted by κ0. In the following two cases, we explore different types of families of hyperelliptic curvesXλ that give rise to X0, as λ → 0. In each case, similarly, in the local coordinate z :=

√x near (0, 0),

we write κXλ= kλ(z)dz ⊗ dz and κ0 = k0(z)dz ⊗ dz. See (7.2) for the precise formulae of κ0.

Case I. We consider the hyperelliptic curves

(2.2) Xλ := (x, y) ∈ C2 | y2 = x(x2 − λ)P (x).

Theorem 2.2. As λ→ 0, for 0 6= |z| small, it holds that

(i)

kλ(z) → k0(z) +4

|z4P (z2)| , i.e., κXλ6→ κ0;

4 ROBERT XIN DONG

(ii)

ψ − log

(

k0(z) +4

|z4P (z2)|

)

∼−2Re

∑g−1i=1

(

(ImZ0)−1 Im(A−1

0 (♣−√−1∗))

)

iz2i

1 +∑g−1

i,j=1((ImZ0)−1)i,j(zizj)2

,

where ♣ and ∗ are column vectors with g − 1 rows involving λ.∗

Case II. We consider the hyperelliptic curves

(2.3) Xλ := (x, y) ∈ C2 | y2 = x(x− λ)(x− λ2)P (x).

Theorem 2.3. As λ→ 0, it holds that

(i) κXλ→ κ0;

(ii) for 0 6= |z| small,

ψ − log k0(z) ∼4π

k0(z)|z4P (z2)|· 1

− log |λ| .

Remarks.

(a) In Theorem 2.2, (ii), the right hand side in the expansion of ψ is harmonic in λ of orderO(λ1/4),which is different from Theorems 2.1 and 2.3 where growth order (− log |λ|)−1 appears. Theharmonicity in the Bergman kernel variation detects the triviality of the holomorphic fiberation,with applications to the Suita conjecture (see [4, 6, 25, 26, 62]).

(b) The proofs of Theorems 2.2 and 2.3 rely on the proofs of Theorems 6.1 and 8.1, respectively,for genus-two curves with more explicit asymptotic coefficients when P = (x − a)(x − b). Inparticular, Theorem 8.1 says that the right hand side in Theorem 2.3, (ii), reduces to

π · Im τ

− log |λ| · |z|4 ,

where τ is the period (scalar) of the elliptic curve (x, y) ∈ C2 | y2 = x(x− a)(x− b) ∪ ∞.(c) In both Cases I and II, for each small z 6= 0, ψ regarded as a subharmonic function in λ 6= 0, is

bounded near λ = 0 and thus has Lelong number zero. In Theorems 2.1 and 2.3, the coefficientsin the two term expansions of ψ depend only on the information away from the singularity.

2.3. Jacobian varieties. The Bergman kernel on a smooth algebraic curve is the pull back via theAbel-Jacobi (period) map of the flat metric on the Jacobian variety. For a curve Xλ of genus g ≥ 1, itsJacobian variety Jac(Xλ) is a g-dimensional complex torus whose Bergman kernel by (1.1) is written asµλdw⊗ dw, under the coordinate w induced from Cg . For a family of curvesXλ in Theorem 2.1, 2.2 or2.3, we determine the asymptotic behaviors of the Bergman kernels on the naturally associated familyof Jacobians Jac(Xλ) as follows.

Theorem 2.4. As λ→ 0, it holds that

logµλ =

− log(− log |λ|) + log π − log det(ImZ0) + O ((log |λ|)−1) , for Xλ in (2.1) or (2.3);

− log det(ImZ0) + O(λ1/2), for Xλ in (2.2).

∗See the definitions of ♣ and ∗ in Lemmata 7.2 and 7.1, respectively.

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 5

3. Bergman kernel on algebraic curves

Consider the complex analytic families of hyperelliptic curvesXλ := (x, y) ∈ C2 | y2 = hλ(x)P (x)

of genus g ≥ 2. Here hλ is a degree 3 polynomial which depend on λ ∈ ∆∗, with distinct rootsof small absolute values. P again is a polynomial of degree d ≥ 2 with complex roots aj such that1 < |a1| < |a2| < ... < |ad|. Without loss of generality, one may always assume that d is odd so thatXλ compact, since∞ can be added to the definition ofXλ in case d is even. For each λ, on the smoothcurve Xλ there exists a globally defined basis, given by the Abelian differentials

(3.1) ωi :=xi−1dx

y, i = 1, ..., g,

for the Hilbert space of (square integrable) holomorphic 1-forms.

3.1. RelationwithRiemannperiodmatrix. It iswell known that in the definition (1.1), the Bergmankernel is independent of the choices of the basis. To write down the Bergman kernel onXλ, one couldobtain from (3.1) the orthonormal basis ϕ1, ϕ2, ... , ϕg after the Gram-Schmidt process by the L2 innerproduct. Alternatively, the Bergman kernel κXλ

can be written down in terms of the normalized periodmatrix, denoted by Z . By the classical construction of hyperelliptic curves, Xλ can be realised as a2-sheeted ramified covering of the Riemann sphere P1 (see [12]). Take the homologous basis δi and γjofH1(Xλ,Z) such that their intersection numbers are δi · δj = 0, γi ·γj = 0, and δi ·γj = δi,j = −γj · δi(here δi,j is the Kronecker δ). Then, for the above ωi ∈ H0(Xλ,Ω), we set

Ai,j :=

∫

δj

ωi, Bi,j :=

∫

δj

γi.

By the Hodge-Riemann bilinear relation, it is known that the matrix (Aij)1≤i,j≤g is invertible. More-over, using the matrix (Bij)1≤i,j≤g, we define the normalized period matrix Z := A−1B. Then, Z issymmetric and has a positive definite imaginary part, i.e., ImZ > 0. For simplicity, we call the ma-trices A,B and Z , the A-period matrix, B-period matrix, and normalized period matrix, respectively.Therefore, the Bergman kernel on Xλ can be equivalently defined as

κXλ:=

g∑

i,j=1

((ImZ)−1)i,j ωi ⊗ ωj.

Under certain local coordinate z specified in the next subsection, we write κXλ= kλ(z)dz ⊗ dz, and

explore the asymptotic behavior of ψ := log kλ(z) near λ = 0 when the curve degenerates.

3.2. Local coordinates near singularities. Near each singularity, we choose particular local coordi-nates to make the computations less involved.

Near a node. A nodal Riemann surface X is a connected complex space such that every point p ∈ Xhas a neighborhood isomorphic to either a disk in C or (x, y) ∈ C2 | |x|, |y| < 1 , xy = 0. In thelatter, p is a node. For the family of hyperelliptic curves in (2.1), as λ→ 0,Xλ degenerates to a singularcurve X0 with a non-separating node. On X0, near the node (0, 0), the local coordinate can be takenas z =

√x, so it holds that x = z2 and dx = 2zdz. Then, in this z-coordinate,

ωi =2z2(i−1)dz

√

(z2 − λ)(z2 − 1)P (z2), i = 1, ..., g,

and the Bergman kernel κXλcan be written as

(3.2)g∑

i,j=1

((ImZ)−1)i,j(zizj)2

4dz ⊗ dz

|z4(z2 − λ)(z2 − 1)P (z2)| =: kλ(z)dz ⊗ dz.

6 ROBERT XIN DONG

Particularly, when P = (x− a)(x− b), the expression (3.2) reduces to

(3.3) 4((ImZ)−1)1,1 + ((ImZ)−1)1,2z

2 + ((ImZ)−1)2,1z2 + ((ImZ)−1)2,2|z|4

|(z2 − 1)(z2 − a)(z2 − b)(z2 − λ)| dz ⊗ dz.

The pinching coordinate could be used for general (possibly non-hypereliptic) algebraic curves degen-erating to a nodal curve, which represents a boundary point of the Deligne-Mumford compactificationof moduli space (see [28, 46, 68, 69]). The non-hyperellipticity ofXλ is equivalent to the non-vanishingof the Gaussian curvature of the Bergman kernel κXλ

. However, in this paper we do not use this ef-fective pinching coordinate method, which works for both non-separating and separating nodal cases,since we need to deal with the cuspidal curves as well.

Near a cusp, Case I. For the family of hyperelliptic curves in (2.2), asλ→ 0,Xλ degenerates to a singularcurve X0 with a cusp. On X0, near (0, 0), the local coordinate can be taken as z =

√x. Similarly,

(3.4) κXλ=

g∑

i,j=1

((ImZ)−1)i,j(zizj)2

4dz ⊗ dz

|z4(z4 − λ) · P (z2)| =: kλ(z)dz ⊗ dz.

Particularly, when P = (x− a)(x− b), the expression (3.4) reduces to

(3.5) 4((ImZ)−1)1,1 + ((ImZ)−1)1,2z

2 + ((ImZ)−1)2,1z2 + ((ImZ)−1)2,2|z|4

|(z2 − a)(z2 − b)(z4 − λ)| dz ⊗ dz.

Near a cusp, Case II. For the family of hyperelliptic curves in (2.3), as λ → 0, Xλ degenerates to asingular curve X0 with a cusp. On X0, near (0, 0), the local coordinate can be taken as z =

√x.

Similarly,

(3.6) κXλ=

g∑

i,j=1

((ImZ)−1)i,j(zizj)2

4dz ⊗ dz

|z4(z2 − λ)(z2 − λ2) · P (z2)| =: kλ(z)dz ⊗ dz.

Particularly, when P = (x− a)(x− b), the expression (3.6) reduces to

(3.7) 4((ImZ)−1)1,1 + ((ImZ)−1)1,2z

2 + ((ImZ)−1)2,1z2 + ((ImZ)−1)2,2|z|4

|(z2 − a)(z2 − b)(z2 − λ)(z2 − λ2)| dz ⊗ dz.

3.3. On the normalization of singular curves. For the nodal singular curveX0 in (2.1), we considerits normalization Y := (x, y) ∈ C2 | y2 = (x−1)P (x), which is a compact curve of genus g−1 = ⌊d

2⌋.

The regular part of X0 is non-compact, and can be seen as either X0 or Y less the node. Let Z0 be theperiod matrix of Y . By the L2 removable singularity theorem, Y and the regular part of X0 have thesame Bergman kernel which can be written under the local coordinate z =

√x as

(3.8) κ0 =

g−1∑

i,j=1

((ImZ0)−1)i,j(z

izj)24dz ⊗ dz

|z2(z2 − 1)P (z2)| ,

where

ωi =x(i−1)dx

√

(x− 1)P (x)=

2z2i−1dz√

(z2 − 1)P (z2), i = 1, ..., g − 1.

For the cuspidal singular curve X0 := (x, y) ∈ C2 | y2 = x3P (x), its normalization is Y :=(x, y) ∈ C2 | y2 = xP (x). Similarly, the Bergman kernel on Y can be written under the coordinatez =

√x as

(3.9) κ0 =

g−1∑

i,j=1

((ImZ0)−1)i,j(z

izj)24dz ⊗ dz

|z4P (z2)| .

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 7

4. Non-separating node: genus-two curves

In this section, we consider a family of genus two curves Xλ := (x, y) ∈ C2 | y2 = x(x − λ)(x−

1)(x−a)(x−b)∪∞, where a, b, λ are distinct complex numbers satisfying 0 < |λ| < 1 < |a| < |b|.As λ → 0, Xλ degenerates to a singular curve X0 with a non-separating node. The normalization ofX0 is an elliptic curve (x, y) ∈ C2 | y2 = (x − 1)(x − a)(x − b) ∪ ∞, whose period is c given in(4.5) and let

c1 :=

∫ a

1

√abdx

√

(x− 1)(x− a)(x− b)

be a constant depending on a, b. The Bergman kernels on Xλ and on the normalization of X0 aredenoted by κXλ

and κ0, respectively. In the local coordinate z :=√x near (0, 0), we write κXλ

=kλ(z)dz ⊗ dz, and κ0 = k0(z)dz ⊗ dz. Then, our result on the asymptotic behaviour of the Bergmankernel κXλ

with precise coefficients is stated as follows.

Theorem 4.1. As λ→ 0, it holds that(i) κXλ

→ κ0;

(ii) for small |z| 6= 0 ,

log kλ(z)− log k0(z) ∼π

− log |λ|Im c+ (z2 + z2) Rec−1

1 |z|4 .

In fact, the results in Section 5 on general hyperelliptic curves largely rely on our proofs in thissection. To prove Theorem 4.1, we need the following two lemmata by analyzing the asymptotics ofthe A-period matrix and B-period matrix on Xλ.

Lemma 4.2. Under the same assumptions as in Theorem 4.1, as λ→ 0, it holds that

A =

(

−2π√ab

c20 −2c1√

ab

)

+O(λ),

where c2 :=∫

δ2dx

x√

(x−1)(x−a)(x−b)and lim

λ→0

O(λ)λ

is a finite matrix.

Proof of Lemma 4.2. We estimate the four entries of A one by one. Firstly, A1,1 =∫

δ1ω1, where δ1 only

contains 0 and λ. Change the variables by setting t = λ−1, s = x−1, and get the dual cycle δ1 whichcontains ∞, t so that −δ1 contains 0, 1, a−1, b−1, for |s| ∈ (1, |t|). As λ→ 0,

A11s=x−1

=====t=λ−1

∫

δ1

−s−2ds√

s−1(s−1 − t−1)(s−1 − 1)(s−1 − a)(s−1 − b)

= −∫

−δ1

−√s√tds

√

ab(s− 1)(s− a−1)(s− b−1)(s− t)

=

∫

−δ1

√sds

√

−ab(s − 1)(s− a−1)(s− b−1)

(

1 +s

2t+O

(

s2

t2

))

=

∫

−δ1

ds

−s√−ab

(

1 +s

2t+O

(

s2

t2

))

(

1 + O(s−1)) (

1 + O(s−1)) (

1 + O(s−1))

=

∫

−δ1

ds

−s√−ab

(

1 +s

2t+O(

s2

t2)

)

(

1 + O(s−1))

=1

−√−ab

∫

−δ1

(

1 + O(t−1)) ds

s=

2π

−√ab

(1 + O(λ)) .

8 ROBERT XIN DONG

Notice that we have used the Maclaurin expansion

(4.1)1√s− a

=

√−a−1

(1 + O (sa−1)) , |s| < |a|;√s−1

(1 + O(s−1a)) , |s| > |a|.

Secondly, look at A21 =∫

δ1ω2 and similarly it holds that

A21 =

∫

−δ1

ds

−s2√−ab

(

1 +s

2t+O

(

s2

t2

))

(

1 + O(s−1))

=1

−√−ab

∫

−δ1

(

1

2t+O(t−2)

)

ds

s=

2π

−√ab

(

λ

2+ O(λ2)

)

.

Thirdly, let δ2 contain 0, λ, 1, a. Then, as λ→ 0, it holds that

A12 =

∫

δ2

dx

x√

(x− 1)(x− a)(x− b)

(

1 +λ

2x+O

(

λ2

x2

))

.

=

∫

δ2

dx

x√

(x− 1)(x− a)(x− b)(1 + O (λ)) .

Lastly,

A22 =

∫

δ2

dx√

(x− 1)(x− a)(x− b)(1 + O(λ)) = −2

∫ a

1

dx√

(x− 1)(x− a)(x− b)(1 + O(λ)) .

Lemma 4.3. Under the same assumptions as in Theorem 4.1, as λ→ 0, it holds that

B ∼(

−2 log λ√−ab d1−2√−ab d2

)

,

where d1 := −2∫ b

adx

x√

(x−1)(x−a)(x−b)and d2 := −2

∫ b

adx√

(x−1)(x−a)(x−b).

Proof of Lemma 4.3 . As t→ ∞, we make use of the following computations (cf. [12]).

(4.2)

∫ t

1

ds

s√s− t

=−2√t

√−1 log

(

√

t

s+

√

t

s− 1

)∣

∣

∣

∣

∣

t

1

∼√−1√t

log t.

(4.3)

∫ t

1

ds

s2√s− t

=

√s− t

ts

∣

∣

∣

∣

t

1

+1

2t

∫ t

1

ds

s√s− t

=−√1− t

t+

√−1

2t√tlog t ∼ −

√−1√t.

In particular, (4.3) yields the boundedness of∫ t

1

√t

s√s− t

O(s−1)ds.

More generally, for any α ≥ 1, as t→ ∞, it holds that

(4.4)

∫ t

1

ds

sα+1√s− t

=

√s− t

αtsα

∣

∣

∣

∣

t

1

+2α− 1

2αt

∫ t

1

ds

sα√s− t

∼ −√−1

α√t.

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 9

Similar to the proof of Lemma 4.2, the four entries of B are estimated one by one. Firstly, by (4.2) and(4.3), for |s| ∈ (1, |t|),

B11 =− 2

∫ 1

λ

dx√

x(x− λ)(x− 1)(x− a)(x− b)

=− 2

∫ 1

t

−√s√tds

√

ab(s− 1)(s− a−1)(s− b−1)(s− t)

=− 2

∫ t

1

√s√tds

√

(s− t)ab

1

s√s

(

1 + O(s−1)) (

1 + O(s−1)) (

1 + O(s−1))

=− 2√t√ab

∫ t

1

ds

s√s− t

(

1 + O(s−1))

∼ −2 log λ√−ab

,

as λ→ 0. Secondly, by (4.3),

B21 = −2√t√ab

∫ t

1

ds

s2√s− t

(

1 + O(s−1))

∼ −2√−ab

.

Finally, similar to A12,

B12 ∼ −2

∫ b

a

dx

x√

(x− 1)(x− a)(x− b),

and

B22 ∼ −2

∫ b

a

dx√

(x− 1)(x− a)(x− b).

Combining Lemmata 4.2 and 4.3 with (3.3), we get the asymptotics of the Bergman kernels.

Proof of Theorem 4.1. Notice that the period is defined as

(4.5) c :=

∫ b

adx√

(x−1)(x−a)(x−b)∫ a

1dx√

(x−1)(x−a)(x−b)

= τ

(

1− b

1− a

)

,

where τ(·) is the inverse of the elliptic modular lambda function. On the normalization of the nodalcurve X0, by (3.8), in the local coordinate z =

√x, the Bergman kernel is exactly κ0 = k0(z)|dz|2,

where

(4.6) k0(z) =4|z|2

(Im c)|(z2 − 1)(z2 − a)(z2 − b)| .

By Lemma 4.2, as λ→ 0,

A−1 ∼(

−2π√ab

c2O(λ) −2c1√

ab

)−1

=

(

−√ab

2π

√abc2

2πd2

O(λ)√ab

−2c1

)

.

Let Z denote the period matrix of Xλ. Then, it follows that

Z = A−1B ∼(

−√−1π

log λ abc2d2+√ab2c1d1

−4πc1

−√−1c1

−1 c

)

, ImZ ∼(

− log |λ|π

−Rec−11

−Rec−11 Im c

)

.

10 ROBERT XIN DONG

So, as λ→ 0, it holds that

(ImZ)−1 ∼(

π− log |λ|

−Rec−1

1π

(log |λ|) Im c−Rec−1

1π

(log |λ|) Im c(Im c)−1

)

.

By (3.3) and (4.6), κXλ→ κ0.Moreover, in the local coordinate z =

√x near (0, 0),

kλ(z)− k0(z) ∼4π

|(z2 − 1)(z2 − a)(z2 − b)(z2 − λ)|1 +

Rec−1

1

Im c(z2 + z2)

− log |λ| ,

which yields the conclusion.

More generally, genus-two curves defined asXλ,a,b := y2 = x(x− 1)(x− λ)(x− a)(x− b)) canbe parametrized by three distinct complex numbers λ, a, b ∈ C \ 0, 1, since the moduli space is 3dimensional. As λ, a or b tends to 0, 1 or ∞, or towards each other,Xλ,a,b will become singular. In oursetting, we fix the other two parameters a and b, and move λ only. The precise asymptotic coefficientswe obtain depend on both a and b, and may be interpretable in terms of moduli theory.

5. Non-separating node: hyperelliptic and general curves

This section is devoted to the proof of Theorem 2.1, as a generalization of Theorem 4.1 towards thehyperelliptic case.† When λ ∈ C \ 0, 1, a1, ..., ad, Xλ defined in (2.1) has genus g = ⌊d

2⌋ + 1. To

prove Theorem 2.1, we need to analyze on Xλ the asymptotics of its A-period matrix and B-periodmatrix, denoted by A and B, respectively. Meanwhile, for the normalization Y := (x, y) ∈ C2 | y2 =(x−1)P (x) of genus g−1, denote itsA-periodmatrix andB-periodmatrix byA0 andB0, respectively.

Lemma 5.1. Under the same assumptions as in Theorem 2.1, as λ→ 0, it holds that

A ∼(

2π√P (0)

O(1)

∗ A0

)

,

where ∗ = (O(λ), 0, ..., 0)T is a column vector with g − 1 rows.

Proof. Firstly, A11 =∫

δ1ω1, where δ1 only contains 0 and λ. Change the variables by setting t =

λ−1, s = x−1, and we get the dual cycle δ1 which contains ∞, t so that −δ1 contains 0, 1, ak−1,k = 1, ..., d, for |s| ∈ (1, |t|). As λ→ 0,

A11s=x−1

=====t=λ−1

∫

−δ1

ds√−1s

√

P (0)

(

1 + O(s

t

))

(

1 + O(s−1))

∼∫

−δ1

ds√−1s

√

P (0)

(

1 + O(s−1))

=

∫

−δ1

ds√−1s

√

P (0)=

2π√

P (0).

A21 =

∫

−δ1

ds√−1s2

√

P (0)

(

1 + O(s

t

))

(

1 + O(s−1))

=

∫

−δ1

ds√−1√

P (0)

(

s−2 +O

(

1

tsi−1

))

(

1 + O(s−1))

=

∫

−δ1

ds√−1√

P (0)

(

+O

(

1

ts

))

= O(λ).

†The author is grateful to Professor Z. Huang for bringing attention the paper [36] during 2016 Tsinghua Sanya Inter-national Mathematics Forum, where a preliminary version of this work was presented.

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 11

For 3 ≤ i ≤ g,

Ai1 =

∫

−δ1

ds√−1si

√

P (0)

(

1 + O(s

t

))

(

1 + O(s−1))

=

∫

−δ1

ds√−1√

P (0)

(

s−i +O

(

1

tsi−1

))

(

1 + O(s−1))

= 0.

Secondly, let δ2 contain only 0, λ, 1 and a1 (not a2, ..., ad). Then,

A12 =

∫

δ2

dx√

x(x− λ)(x− 1)P (x)

=

∫

δ2

dx

x√

(x− 1)P (x)

(

1 + O

(

λ

x

))

∼∫

δ2

dx

x√

(x− 1)P (x).

In general, for Ai2, there is an extra xi−1 in the original integrand above, so

Ai2 ∼∫

δ2

xi−2dx√

(x− 1)P (x).

Thirdly, let δ3 contain only 0, λ2, λ, a1, a2 and a3 (not a4, ..., ad). Similarly, ai3 is asymptotic to the sameintegrand over δ3 instead of over δ2, i.e.,

Ai3 ∼∫

δ3

xi−2dx√

(x− 1)P (x).

In general, for j ≥ 2 and the corresponding homologous basis δj , it holds that

Aij ∼∫

δj

xi−2dx√

(x− 1)P (x),

which are exactly the same as the entries of A0 when i ≥ 2.

Lemma 5.2. Under the same assumptions as in Theorem 2.1, as λ→ 0, it holds that

B ∼(

−2√−1√

P (0)log λ O(1)

⋆ B0

)

,

where ⋆ is a column vector whose entries are all −2√−1√

P (0)with g − 1 rows.

Proof. Firstly, by (4.2) and (4.3), for |s| ∈ (1, |t|),

B11 = −2

∫ 1

λ

dx√

x(x− λ)(x− 1) · P (x)= −2

∫ 1

λ

dx√

x(x− λ)

1√

−P (0)(1 + O(x))

s=x−1

=====t=λ−1

2√t

√

P (0)

∫ t

1

ds

s√s− t

(1 + O(s−1)) ∼ −2√

P (0)

√−1 log λ.

Thus, for 2 ≤ i ≤ g, by (4.4),

Bi1 = −2

∫ 1

λ

xi−1dx√

x(x− λ)(x− 1) · P (x)=

2√t

√

P (0)

∫ t

1

ds

si√s− t

(1 + O(s−1)) ∼ −2√−1

√

P (0).

For Column j, 2 ≤ j ≤ g, we use Taylor series expansion of√x− λ

−1to get that

Bij ∼ −2

∫ a2j−2

a2j−3

xi−2dx√

(x− 1)P (x),

which are exactly the same as the entries of B0 when i ≥ 2.

12 ROBERT XIN DONG

Now we will give a proof of Theorem 2.1.

Proof of Theorem 2.1. By (3.8), in the local coordinate z =√x, the Bergman kernel on the normalization

Y of the nodal curve X0 in (2.1) is exactly κ0 = k0(z)|dz|2, where

(5.1) k0(z) =4

|z2(z2 − 1)P (z2)|

g−1∑

i,j=1

((ImZ0)−1)i,j(z

izj)2.

By Lemma 5.1 and the block matrix inversion, we know that

A−1 ∼( √

P (0)

2πO(1)

O(λ) A−10

)

,

where both (O(1))T and limλ→0

O(λ)λ

are finite column vectors with g − 1 rows. Therefore, as λ→ 0,

Z = A−1B ∼( −

√−1π

log λ O(1)A−1

0 ⋆ A−10 B0

)

, ImZ ∼(

log |λ|−π O(1)

Im(A−10 ⋆) ImZ0

)

,

and

(ImZ)−1 ∼(

−πlog |λ| O((log |λ|)−1)

(ImZ0)−1 Im(A−1

0 ⋆) πlog |λ| (ImZ0)

−1

)

.

In fact, since Z is symmetric, the off-diagonal block matrices in each matrix above concerning Z arethe transpose of each other. By (3.2) and (5.1), as λ → 0, κXλ

→ κ0. Moreover, in the local coordinatez =

√x near (0, 0), it holds that

kλ(z)− k0(z) ∼4π

|(z2 − λ)(z2 − 1)P (z2)|

1− 2Reg−1∑

i=1

(

(ImZ0)−1 Im(A−1

0 ⋆))

iz2i

− log |λ|which yields the conclusion.

If we ignore the asymptotic coefficients, then the leading term growth in Theorem 2.1 corresponds to[36, Proposition 3.2]. See also [12] for the asymptotic behaviors of the periodmatrix. For degenerationsof non-separating nodal type, comparing our result with [36], we do not see any role the hyperellipticityplays in the asymptotic behaviors of the Bergman kernels, and thus believe that Theorem 2.1 shouldgeneralize toward non-hypereliptic algebraic curves.

6. Cusp I: genus-two curves

In this section, we consider a family of genus two curvesXλ := (x, y) ∈ C2 | y2 = x(x2 − λ)(x−a)(x− b)∪∞, where a, b, λ are distinct complex numbers satisfying 0 < |λ| < |a| < |b|. As λ→ 0,Xλ degenerates to a singular curve X0 with an ordinary cusp. The normalization of X0 is an ellipticcurve (x, y) ∈ C2 | y2 = x(x− a)(x− b) ∪ ∞, whose period is τ given in (6.1). Let

c3 :=

∫ a

0

√abdx

√

x(x− a)(x− b), c4 := −2

∫ 1

0

(x− 1)dx√

x(x− 1)(x− 2)

be constants depending on a, b. The Bergmankernels onXλ and on the normalization ofX0 are denotedby κXλ

and κ0, respectively. In the local coordinate z :=√x near (0, 0), we write κXλ

= kλ(z)dz⊗ dz,and κ0 = k0(z)dz⊗ dz. Then, our result on the asymptotic behaviour of the Bergman kernel κXλ

withprecise coefficients is stated as follows.

Theorem 6.1. As λ→ 0, for small |z| 6= 0, it holds that

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 13

(i)

kλ(z) → k0(z)

(

Im τ

|z4| + 1

)

, i.e., κXλ6→ κ0;

(ii)

log kλ(z)− log k0(z)− log

(

Im τ

|z4| + 1

)

∼Re

λ1/4 c4c3

(z2 + z2)

|z|4 + Im τ,

where τ is the period (scalar) of the elliptic curve y2 = x(x− a)(x− b).

To prove Theorem 6.1, we need the following two lemmata by analyzing the asymptotics of theA-period matrix and B-period matrix on Xλ.

Lemma 6.2. Under the same assumptions as in Theorem 6.1, as λ→ 0, it holds that

A ∼(

c5−√abλ−1/4 c6

c4−√abλ1/4 −2c3√

ab

)

,

where c5 := −2∫ 1

0du√

u(u−1)(u−2)and c6 depends on a, b.

Proof of Lemma 6.2. We estimate the four entries one by one. Firstly, let δ1 contain only −√λ and 0.

By the Cauchy Integral Theorem and Taylor series expansion, we know that

A11 = −2

∫ 0

−√λ

dx√

x(x−√λ)(x+

√λ)(x− a)(x− b)

=−2

−√ab

∫ 0

−√λ

dx√

x(x−√λ)(x+

√λ)

(1 + O(x))

x=(u−1)√λ

=========−2λ−1/4

−√ab

∫ 1

0

du√

u(u− 1)(u− 2)

(

1 + O(√

λ(u− 1)))

∼ −2λ−1/4

−√ab

∫ 1

0

du√

u(u− 1)(u− 2)=:

c5

−√abλ−1/4.

Secondly,

A21 =−2λ1/4

−√ab

∫ 1

0

(u− 1)du√

u(u− 1)(u− 2)

(

1 + O(√

λ(u− 1)))

∼ −2λ1/4

−√ab

∫ 1

0

(u− 1)du√

u(u− 1)(u− 2):=

c4

−√abλ1/4.

Thirdly, let δ2 contain only −√λ, 0,

√λ and a (not b). Then,

A12 =

∫

δ2

dx√

x(x− a)(x− b)

1√x

(

1 + O

(√λ

x

))

1√x

(

1 + O

(√λ

x

))

∼∫

δ2

dx

x√

x(x− a)(x− b)=: c6.

14 ROBERT XIN DONG

A22 =

∫

δ2

dx√

x(x− a)(x− b)

(

1 + O

(√λ

x

))

∼∫

δ2

dx√

x(x− a)(x− b)= −2

∫ a

0

dx√

x(x− a)(x− b)=

−2c3√ab.

Lemma 6.3. Under the same assumptions as in Lemma 6.2, as λ→ 0, it holds that

B ∼ 1√−ab

(

c5λ−1/4 d3

√−ab

−c4λ1/4 d4√−ab

)

+,

where d3 := −2∫ b

adx

x√x(x−a)(x−b)

and d4 := −2∫ b

adx√

x(x−a)(x−b).

Proof of Lemma 6.3. Again, we estimate all the four entries one by one. Firstly, let γ1 contain only 0

and√λ. By Cauchy Integral Theorem, we can get that

B11 = −2

∫

√λ

0

dx√

x(x2 − λ)(x− a)(x− b)

=−2

−√ab

∫

√λ

0

dx√

x(x−√λ)(x+

√λ)

(1 + O(x))

x=(1−u)√λ

=========2λ−1/4

√ab

∫ 0

1

du√

−u(u− 1)(u− 2)

(

1 + O((−u+ 1) ·√λ))

∼ 2λ−1/4

√ab

∫ 1

0

√−1du

√

u(u− 1)(u− 2)=:

c5√−ab

λ−1/4.

Secondly,

B21 =2λ−1/4

√ab

∫ 0

1

(−u+ 1)√λdu

√

−u(u− 1)(u− 2)

(

1 + O((−u+ 1) ·√λ))

∼ 2λ1/4√ab

∫ 1

0

(u− 1)du√

−u(u− 1)(u− 2)=:

√−1λ1/4c4√

ab.

Thirdly, let γ2 contain only a and b. Then, it holds that

B12 =

∫

γ2

dx√

x(x−√λ)(x+

√λ)(x− a)(x− b)

=− 2

∫ b

a

dx√

x(x− a)(x− b)

1√x

(

1 + O

(√λ

2x

))

1√x

(

1 + O

(√λ

2x

))

∼− 2

∫ b

a

dx

x√

x(x− a)(x− b)=: d3.

Lastly,

B22 =

∫

γ2

dx√

x(x− a)(x− b)

(

1 + O

(√λ

2x

))

∼ −2

∫ b

a

dx√

x(x− a)(x− b)=: d4.

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 15

Combining Lemmata 6.2 and 6.3 with (3.5), we get the asymptotics of the Bergman kernels.

Proof of Theorem 6.1. Notice that the period is defined as

(6.1) τ :=

∫ b

adx√

x(x−a)(x−b)∫ a

0dx√

x(x−a)(x−b)

= τ

(

b

a

)

,

where τ(·) is the inverse of the elliptic modular lambda function. On the regular part of the cuspidalcurve X0, by (3.9), in the local coordinate z =

√x, the Bergman kernel is exactly κ0 = k0(z)|dz|2,

where

(6.2) k0(z) =4

(Im τ) |(z2 − a)(z2 − b)| .

By Lemma 6.2, as λ→ 0,

A−1 ∼(

c5−√abλ−1/4 c6

c4−√abλ1/4 −2c3√

ab

)−1

∼ abλ1/4

2c3c5

(

−2c3√ab

−c6c4√abλ1/4 c5

−√abλ−1/4

)

.

Let Z denote the period matrix of Xλ. Then, it follows that

Z = A−1B ∼ abλ1/4

2c3c5

(

−2c3√ab

−c6c4√abλ1/4 c5

−√abλ−1/4

)

1√−ab

(

c5λ−1/4 d3

√−ab

−c4λ1/4 d4√−ab

)

∼( √

−1√−1λ1/4

(

d3√−ab − c6d4ab

2c3

)

c−15√

−1λ1/4 c4−c3 τ

)

,

ImZ ∼(

1 O(λ1/4)

Re

λ1/4 c4−c3

Im τ

)

.

So, as λ→ 0,

(ImZ)−1 ∼(

1 −(Im τ )−1O(λ1/4)

(Im τ)−1Re

λ1/4 c4c3

(Im τ )−1

)

.

By (3.7) and (6.2),

(6.3) kλ(z) ∼ 41 + (Im τ )−1Re

λ1/4 c4c3

(z2 + z2) + (Im τ)−1|z|4

|(z2 − a)(z2 − b)(z2 − λ)(z2 − λ2)| →(

Im τ

|z4| + 1

)

k0(z),

which means that κXλ6→ κ0.Moreover, in the local coordinate z =

√x near (0, 0),

kλ(z)−(

Im τ

|z4| + 1

)

k0(z) ∼4(Im τ)−1Re

λ1/4 c4c3

(z2 + z2)

|z4(z2 − a)(z2 − b)| ,

which yields the conclusion.

7. Cusp I: hyperelliptic curves

This section is devoted to the proof of Theorem 2.2. When λ ∈ C \ 0, 1, a1, ..., ad, Xλ definedin (2.2) has genus g = ⌊d

2⌋ + 1. To prove Theorem 2.2, we need to analyze on Xλ the asymptotics

of its A-period matrix and B-period matrix, denoted by A and B, respectively. Meanwhile, for thenormalization Y := (x, y) ∈ C2 | y2 = xP (x) of genus g − 1, denote its A-period matrix andB-period matrix by A0 and B0, respectively.

16 ROBERT XIN DONG

Lemma 7.1. Under the same assumptions as in Theorem 2.2, as λ→ 0, it holds that

A ∼(

c5λ− 1

41√P (0)

O(1)

∗ A0

)

,

where ∗ is a column vector with g − 1 rows whose entries are given by (7.1) below.

Proof. We will estimate all the g × g elements one by one. Firstly, as λ→ 0, it holds that

A11 = −2

∫ 0

−√λ

dx√

x(x2 − λ)P (x)= −2

∫ 0

−√λ

dx√

x(x2 − λ)

1√

P (0)(1 + O(x))

x=(u−1)√λ

========= −2λ−1

4

∫ 1

0

du√

u(u− 1)(u− 2)

(1 + O((u− 1)√λ))

√

P (0)

∼ −2λ−1

4

∫ 1

0

du√

u(u− 1)(u− 2)

1√

P (0)=: λ−

1

4

c5√

P (0).

In general, for ai1, 2 ≤ i ≤ g, there is an extra xi−1 in the original integrand and thus an extra√λi−1

(u− 1)i−1 in the numerator of the above last expression, so

(7.1) Ai1 ∼ −2

∫ 1

0

(u− 1)i−1du√

u(u− 1)(u− 2)

√λ

1

2+i−2

√

P (0).

Secondly, let δ2 contain only −√λ, 0,

√λ and a1 (not a2, ..., ad). Then,

A12 =

∫

δ2

dx√

x(x2 − λ)P (x)

=

∫

δ2

dx

x√

xP (x)

(

1 + O

(

λ

x2

))

∼∫

δ2

dx

x√

xP (x).

In general, for ai2, there is an extra xi−1 in the original integrand, so

Ai2 ∼∫

δ2

xi−2dx√

xP (x).

Thirdly, let δ3 contain only −√λ, 0,

√λ, a1, a2 and a3 (not a4, ..., ad). Similarly, ai3 is asymptotic to the

same integrand over δ3 instead of over δ2, i.e.,

Ai3 ∼∫

δ3

xi−2dx√

xP (x).

In general, for j ≥ 2 and the corresponding homologous basis δj , it holds that

Aij ∼∫

δj

xi−2dx√

xP (x),

which are exactly the same as the entries of A0 when i ≥ 2.

Lemma 7.2. Under the same assumptions as in Theorem 2.2, as λ→ 0, it holds that

B ∼(

c5λ− 1

4

√−1√P (0)

O(1)

♣ B0

)

,

where ♣ is a column vector whose entries are Bi1 = (−1)i−1√−1Ai1 for Ai1 in (7.1) and 2 ≤ i ≤ g.

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 17

Proof. For the first column of B, we make the change of coordinates (similar to the proof of Lemma6.3) by setting x = (−u+ 1)

√λ, and get for 1 ≤ i ≤ g that

Bi1 ∼ (−1)i−1√−1ai1.

For Column j, 2 ≤ j ≤ g, we use Taylor expansion of (x2 − λ)−1/2 and get that

Bij ∼ −2

∫ a2j−2

a2j−3

xi−2dx√

xP (x).

Now we will give a proof of Theorem 2.2.

Proof of Theorem 2.2. By (3.9), in the local coordinate z =√x, the Bergman kernel on the normalization

Y of the nodal curve X0 in (2.2) is exactly κ0 = k0(z)|dz|2, where

(7.2) k0(z) =4

|z4P (z2)|

g−1∑

i,j=1

((ImZ0)−1)i,j(z

izj)2.

By Lemma 7.1 and the block matrix inversion, we know that

A−1 ∼

λ1

4

√P (0)

c5O(λ1/4)

(−A−10 ∗)λ 1

4

√P (0)

c5A−1

0

,

where limλ→0

(O(λ12 ))T

λ12

is a finite column vectors with g − 1 rows. Therefore, as λ→ 0,

Z = A−1B ∼( √

−1 O(λ1

4 )A−1

0 (♣−√−1∗) A−1

0 B0

)

, ImZ ∼(

1 O(λ1/4)Im(A−1

0 (♣−√−1∗)) ImZ0

)

,

and

(ImZ)−1 ∼(

1 O(λ1/4)(ImZ0)

−1 Im(A−10 (♣−

√−1∗)) (ImZ0)

−1

)

.

By (3.4) and (7.2), as λ→ 0, in the local coordinate z =√x near (0, 0), it holds that

kλ(z) →1 +

∑g−1i,j=1((ImZ0)

−1)i,j(zizj)2

|4−1z4P (z2)| = k0(z) +4

|z4P (z2)| ,

so κXλ6→ κ0. Moreover,

kλ(z)− k0(z)−4

|z4P (z2)| ∼−8Re

g−1∑

i=1

(

(ImZ0)−1 Im(A−1

0 (♣−√−1∗))

)

iz2i

|z4P (z2)| ,

which yields the conclusion.

Since Z is symmetric, A−10 (♣ −

√−1∗) = O(λ1/4), and both the leading and subleading terms in

the expansion of κXλare harmonic with respect to λ.

18 ROBERT XIN DONG

8. Cusp II: genus-two curves

In this section, we consider a family of genus two curves Xλ := (x, y) ∈ C2 | y2 = x(x − λ)(x−

λ2)(x − a)(x − b) ∪ ∞, where a, b, λ are distinct complex numbers satisfying 0 < |λ| < |a| < |b|.As λ → 0, Xλ degenerates to a singular curve X0 with an ordinary cusp. The normalization of X0 isan elliptic curve (x, y) ∈ C2 | y2 = x(x − a)(x − b) ∪ ∞, whose period is τ given in (6.1). TheBergman kernels on Xλ and on the normalization of X0 are denoted by κXλ

and κ0, respectively. Inthe local coordinate z :=

√x near (0, 0), we write κXλ

= kλ(z)dz ⊗ dz, and κ0 = k0(z)dz ⊗ dz. Then,our result on the asymptotic behaviour of the Bergman kernel κXλ

with precise coefficients is statedas follows.

Theorem 8.1. As λ→ 0, it holds that(i) κXλ

→ κ0;

(ii) for small |z| 6= 0 ,

log kλ(z)− log k0(z) ∼π · Im τ

− log |λ| · |z|4 ,

To prove Theorem 8.1, we need the following two lemmata by analyzing the asymptotics of theA-period matrix and B-period matrix on Xλ.

Lemma 8.2. Under the same assumptions as in Theorem 8.1, as λ→ 0, it holds that

A ∼(

−2π√ab

√λ

c6c7λ

3/2 −2c3√ab

)

,

where c7 :=−2√ab

∫ 1

0

√v−1 − 1dv and c3, c6 are the same as in Lemma 6.2.

Proof of Lemma 8.2. Consider the integrals

∫ λ2

0

dx√

x(x− λ)(x− λ2)

x=λ2(1−v)========

∫ 1

0

−√−1λ−1dv

√

v(v − 1)(v − 1 + λ−1)∼ −λ−1

2√−1

∫

γ

dv

vλ−1/2=

−π√λ,

where γ is a large cycle containing 0, 1, and

∫ λ2

0

xdx√

x(x− λ)(x− λ2)=

−λ√−1

∫ 1

0

(v − 1)dv√

v(v − 1)(v − 1 + λ−1)∼ −λ3/2√

−1

∫ 1

0

√

v − 1

vdv.

We estimate the four entries one by one. Firstly, let δ1 only contain 0 and λ2, and similar to the proofof Lemma 6.2 we know that

A11 = −2

∫ λ2

0

dx√

x(x− λ)(x− λ2)

1√−a

1√−b

(

1 +x

2a+O(x2)

)(

1− x

2b+O(x2)

)

∼ −2

−√ab

∫ λ2

0

dx√

x(x− λ)(x− λ2)∼ −2π√

ab√λ.

Secondly,

A21 ∼2√ab

∫ λ2

0

xdx√

x(x− λ)(x− λ2)∼ −2√

ab

λ3/2√−1

∫ 1

0

√

v − 1

vdv =: c7λ

3/2.

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 19

Thirdly, let δ2 contain only 0, λ, λ2 and a. Then, it holds that

A12 =

∫

δ2

dx√

x(x− λ)(x− λ2)(x− a)(x− b)

=

∫

δ2

dx

x√

x(x− a)(x− b)

(

1 + O

(

λ

x

))(

1 + O

(

λ2

x

))

∼∫

δ2

dx

x√

x(x− a)(x− b)=: c6,

Lastly,

A22 ∼∫

δ2

dx

x√

x(x− a)(x− b)=

∫ a

0

dx

x√

x(x− a)(x− b).

Lemma 8.3. Under the same assumptions as in Theorem 8.1, as λ→ 0, it holds that

B ∼(

2√−1 log λ√ab

√λ

d32√ab

√−λ d4

)

,

where d3, d4 are the same as in Lemma 6.3.

Proof of Lemma 8.3. By [23] or (4.2), it is known that, as λ→ 0,∫ λ

λ2

dx√

x(x− λ)(x− λ2)=

1√λ

∫ 1

λ

du√

u(u− 1)(u− λ)∼

√−1 log λ√

λ.

Firstly, let γ1 contain only λ and λ2, and we get that

B11 = −2

∫ λ

λ2

dx√

x(x− λ)(x− λ2)(x− a)(x− b)

=−2

−√ab

∫ λ

λ2

dx√

x(x− λ)(x− λ2)(1 + O(x))

∼ 2√ab

∫ λ

λ2

dx√

x(x− λ)(x− λ2)∼ 2√

ab

√−1 log λ√

λ.

Secondly, by (4.3), as λ→ 0,

B21 ∼2√ab

∫ λ

λ2

xdx√

x(x− λ)(x− λ2)

x=λs====

t= 1

λ

−2√ab

∫ t

1

ds

s2√s− t

∼ 2√ab

√−λ.

Thirdly,

B12 =− 2

∫ b

a

dx√

x(x− λ)(x− λ2)(x− a)(x− b)

=− 2

∫ b

a

dx (1 + O (λx−1)) (1 + O (λ2x−1))

x√

x(x− a)(x− b)∼ −2

∫ b

a

dx

x√

x(x− a)(x− b):= d3.

Lastly,

B22 ∼ −2

∫ b

a

dx√

x(x− a)(x− b):= d4.

20 ROBERT XIN DONG

Proof of Theorem 8.1. On the regular part of the cuspidal curve X0, (6.2) gives the formulae for theBergman kernel κ0 = k0(z)|dz|2 in the local coordinate z =

√x. By Lemma 8.2, as λ→ 0,

A−1 ∼(

−2π√ab

√λ

c6c7λ

3/2 −2c3√ab

)−1

∼ ab√λ

4πc3

(

−2c3√ab

−c6−c7λ3/2 −2π√

ab√λ

)

.

Let Z denote the period matrix of Xλ. Then, it follows that

Z = A−1B ∼(

−√−1 log λπ

ab√λ

4πc3

(

−2c3√abd3 − c6d4

)

√λ√−1c−13 τ

)

, ImZ ∼( − log |λ|

πO(λ

1

2 )

−Re√

λc−13

Im τ

)

So, as λ→ 0,

(ImZ)−1 ∼(

π− log |λ|

1log |λ| O(λ

1

2 )

−πlog |λ| Im τ

Re√

λc−13

(Im τ)−1

)

.

By (3.7) and (6.2), κXλ→ κ0.Moreover, in the local coordinate z =

√x near (0, 0),

kλ(z)− k0(z) ∼4π

|(z2 − a)(z2 − b)(z2 − λ)(z2 − λ2)| ·1 + O(λ

1

2 )

− log |λ| ,

which yields the conclusion.

9. Cusp II: hyperelliptic curves

This section is devoted to the proof of Theorem 2.3. When λ ∈ C \ 0, 1, a1, ..., ad, Xλ definedin (2.3) has genus g = ⌊d

2⌋ + 1. To prove Theorem 2.3, we need to analyze on Xλ the asymptotics

of its A-period matrix and B-period matrix, denoted by A and B, respectively. Meanwhile, for thenormalization Y := (x, y) ∈ C2 | y2 = xP (x) of genus g − 1, denote its A-period matrix andB-period matrix by A0 and B0, respectively.

Lemma 9.1. Under the same assumptions as in Theorem 2.3, as λ→ 0, it holds that

A ∼(

2π√λP (0)

O(1)

∗ A0

)

,

where ∗ is a column vector with g − 1 rows whose entries are at most O(λ3/2).

Proof. Firstly, as λ→ 0,

A11 = −2

∫ λ2

0

dx√

x(x− λ)(x− λ2)

1√

P (0)(1 + O(x))

∼ −2

∫ λ2

0

dx√

x(x− λ)(x− λ2)

1√

P (0)∼ 2π√

λP (0).

A21 ∼−2

√

P (0)

∫ λ2

0

xdx√

x(x− λ)(x− λ2)∼ −2√

P (0)

−λ3/2√−1

∫ 1

0

√

v − 1

vdv.

Here we change the variable by letting x = λ2(1− v). In general, for 2 ≤ i ≤ g,

Ai1 ∼−2

√

P (0)

−λ3/2√−1

∫ 1

0

λ2i−4(1− v)i−2

√

v − 1

vdv = O(λ2i−2.5).

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 21

Secondly, let δ2 contain only 0, λ2, λ and a1 (not a2, ..., ad). Then,

A12 =

∫

δ2

dx√

x(x− λ)(x− λ2)P (x)

=

∫

δ2

dx

x√

xP (x)

(

1 + O

(

λ

x2

))

∼∫

δ2

dx

x√

xP (x).

In general, for Ai2, there is an extra xi−1 in the original integrand above, so

Ai2 ∼∫

δ2

xi−2dx√

xP (x).

Thirdly, let δ3 contain only 0, λ2, λ, a1, a2 and a3 (not a4, ..., ad). Similarly,Ai3 is asymptotic to the sameintegrand over δ3 instead of over δ2, i.e.,

Ai3 ∼∫

δ3

xi−2dx√

xP (x).

Lastly, for j ≥ 2 and the corresponding homologous basis δj , it holds that

Aij ∼∫

δj

xi−2dx√

xP (x),

which are exactly the same as the entries of A0 when i ≥ 2.

Lemma 9.2. Under the same assumptions as in Theorem 2.3, as λ→ 0, it holds that

B ∼(

−2√P (0)

√−1 log λ√

λO(1)

♠ B0

)

,

where ♠ is a column vector whose entries are −2λi−2√−λ√

P (0), for 2 ≤ i ≤ g.

Proof.

B11 = −2

∫ λ

λ2

dx√

x(x− λ)(x− λ2) · P (x)= −2

∫ λ

λ2

dx√

x(x− λ)(x− λ2)

1√

P (0)(1 + O(x))

∼ −2

∫ λ

λ2

dx√

x(x− λ)(x− λ2)

1√

P (0)∼ −2√

P (0)

√−1 log λ√

λ.

Thus, for 2 ≤ i ≤ g, by (4.4),

Bi1 = −2

∫ λ

λ2

xi−1dx√

x(x− λ)(x− λ2)

1√

P (0)(1 + O(x))

∼ −2

∫ λ

λ2

xi−1dx√

x(x− λ)(x− λ2)

1√

P (0)

x=λs====

t= 1

λ

2λi−2

√

P (0)

∫ t

1

ds

si√s− t

∼ −2λi−2√−λ

√

P (0).

For Column j, 2 ≤ j ≤ g, we use Taylor expansion of√

(x− λ)(x− λ2)−1

to get that

Bij ∼ −2

∫ a2j−2

a2j−3

xi−2dx√

xP (x).

which are exactly the same as the entries of B0 when i ≥ 2.

Now we will give a proof of Theorem 2.3.

22 ROBERT XIN DONG

Proof of Theorem 2.3. On the regular part of the cuspidal curve X0, (7.2) gives the formulae for theBergman kernel κ0 = k0(z)|dz|2 in the local coordinate z =

√x. By Lemma 9.1 and the block matrix

inversion, we know that

A−1 ∼( √

λP (0)

2πO(λ

1

2 )O(λ2) A−1

0

)

,

where both limλ→0

(O(λ12 ))T

λ12

and limλ→0

O(λ2)λ2

are finite column vectors with g− 1 rows. Therefore, as λ→ 0,

Z = A−1B ∼( −

√−1π

log λ O(λ1

2 )A−1

0 ♠ A−10 B0

)

, ImZ ∼(

log |λ|−π O(λ

1

2 )

Im(A−10 ♠) ImZ0

)

,

and

(ImZ)−1 ∼(

−πlog |λ| O((log |λ|)−1λ

1

2 )

(ImZ0)−1 Im(A−1

0 ♠) πlog |λ| (ImZ0)

−1

)

.

In fact, since Z is symmetric, the off-diagonal block matrices in each matrix above concerning Z arethe transpose of each other. By (3.6) and (7.2), as λ → 0, κXλ

→ κ0. Moreover, in the local coordinatez =

√x near (0, 0), it holds that

kλ(z)− k0(z) ∼4π

|(z2 − λ)(z2 − λ2)P (z2)|

1− 2Reg−1∑

i=1

(

(ImZ0)−1 Im(A−1

0 ♠))

iz2i

− log |λ|which yields that

ψ − log k0(z) ∼π

− log |λ|1

g−1∑

i,j=1

((ImZ0)−1)i,j(zizj)2.

10. Jacobian varieties

In this section, we will use the results obtained in the proofs of Theorems 2.1, 2.2, 2.3 to give aproof of Theorem 2.4. Let Xλ be a compact curve of genus g ≥ 1, and let Z be its period matrix. TheJacobian variety of Xλ is then identified with the g-dimensional complex torus Cg/Zg + ZZg , and isdenoted by Jac(Xλ). It is well known that the Abel-Jacobi (period)mapXλ → Jac(Xλ) is a holomorphicembedding.

Proof of Theorem 2.4. By definition (1.1), the Bergman kernel on Jac(Xλ) can be written as µλdw⊗ dw,under the coordinate w induced from Cg, where µλ = (det(ImZ))−1. For Xλ defined in (2.1) or (2.3),as λ→ 0, it holds that

det(ImZ) =log |λ|−π det(ImZ0) + O(1) → +∞,

which yields the first part of the conclusion in Theorem 2.4. For Xλ defined in (2.2), more carefullyanalysis in the proof of Theorem 2.2 shows that as λ→ 0,

ImZ =

(

1 + O(λ1/2) O(λ1/4)O(λ1/4) ImZ0 +O(λ1/2)

)

, det(ImZ) = det ImZ0 +O(λ1/2) < +∞,

which yields the second part of the conclusion.

BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 23

It is worth mentioning that as the family of curves Xλ in (2.2) degenerate to a cuspidal one, theirJacobian varieties Jac(Xλ) remain being manifolds (i.e., non-degenerate), as λ → 0. However, for twofamilies of curves Xλ in (2.1) and (2.3), their Jacobian varieties Jac(Xλ) indeed degenerate. Theorem2.4 generalizes the results in [20,21,23] on elliptic curves toward higher dimensional Abelian varieties.

Tsuji raised the following question at the 2016 Hayama symposium: can we recover the singularityinformation of the base varieties from the asymptotics of the Bergman kernels? Related to this, furthergeometric interpretation of the asymptotic coefficients is needed (see recent works [27, 42, 61, 63, 70]).

Acknowledgements. This article constitutes essentially the author’s doctoral thesis in Nagoya Universityunder the advice of Professor Ohsawa. This work was supported by the Ideas Plus grant 0001/ID3/2014/63of the Polish Ministry of Science and Higher Education, KAKENHI and the Grant-in-Aid for JSPS Fellows(No. 15J05093).

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Department of Mathematics, University of California, Irvine, CA 92697-3875, USA

Email addresses: xin.dong@uci.edu, 1987xindong@tongji.edu.cn