Boundary asymptotics of the relative Bergman kernel metric for curves
aa r X i v : . [ m a t h . C V ] M a y BOUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FORCURVES
ROBERT XIN DONGAbstract. 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. Introduction
On an n -dimensional complex manifold X , the Bergman kernel is a reproducing kernel of the spaceof square integrable holomorphic top forms. For the canonical bundle K , which is the n th exteriorpower of the cotangent bundle Ω on X , the Bergman kernel is defined as(1.1) κ := X s j ⊗ s j , where { s j } j is a complete orthonormal basis of H ( 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 family X π −−→ ∆ , where X is a complex ( n + 1) -dimensional Stein manifold fibred overthe unit disc ∆ ⊂ C , and π is a surjective holomorphic map with connected fibers. Moreover, π is aholomorphic submersion over ∆ ∗ := ∆ \ { } . For λ ∈ ∆ ∗ , the fibres X λ = π − ( λ ) are n -dimensionalStein manifolds, whose Bergman kernel are denoted by κ X λ . Let ( z, λ ) be the coordinate of X λ × ∆ ∗ ,which is the local trivialization of the fibration, and write κ X λ := k λ ( z ) dz ⊗ d ¯ z locally. Then, the abovementioned positivity results imply that ψ := log k λ ( z ) is subharmonic with respect to λ ∈ ∆ ∗ , i.e., ∂ ψ∂λ∂ ¯ λ ≥ , L extension, period matrix, positivity of direct image if the Bergman kernel is not idetically zero. Here as λ varies, e ψ induces on K X / ∆ 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 fiber X , our goal is to quantitatively characterizeat degeneration the asymptotics of ψ , as λ → .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 earlier works 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 onthe L -analysis in several complex variables, and its interaction with 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 L 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 | ) − , as λ → .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 X λ 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]. OUNDARY 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 let P denote a polynomial of degree d ≥ with complex roots a j such that < | a | < | a | < ... < | a d | .For a holomorphic family of Riemann surfaces X λ parametrized by λ ∈ ∆ ∗ , each fiberwise Bergmankernel is locally written as κ X λ := k λ ( z ) dz ⊗ d ¯ z , in some coordinate z for some function k λ . Recallthat ψ := log k λ ( z ) is subharmonic with respect to λ ∈ ∆ ∗ .2.1. Nodal cases.
In affine coordinates, consider a family of hyperelliptic curves(2.1) X λ := { ( x, y ) ∈ C | y = x ( x − λ )( x − P ( x ) } . As λ → , X λ degenerates to a singular curve X with a non-separating node. The normalization of X is a smooth curve Y := { ( x, y ) ∈ C | y = ( x − P ( x ) } , whose A -period matrix and normalizedperiod matrix are denoted by A and Z , respectively. By the L removable singularity theorem, Y andthe regular part of X have the same Bergman kernel, denoted by κ . In the local coordinate z := √ x near (0 , , we write κ X λ = k λ ( z ) dz ⊗ d ¯ z and κ = k ( z ) dz ⊗ d ¯ z . See (5.1) for the precise formulae of κ . Theorem 2.1. As λ → , it holds that(i) κ X λ → κ ;(ii) for = | z | small, ψ − log k ( z ) ∼ π − log | λ | − P g − i =1 (cid:0) (Im Z ) − Im( A − ⋆ ) (cid:1) i z i P g − i,j =1 ((Im Z ) − ) i,j ( z i z j ) , where ⋆ is a column vector with g − rows whose entries are all − i p P (0) − . 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 = 0 , ψ regarded as a subharmonic function in λ is bounded near λ = 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 curve X := { ( x, y ) ∈ C | y = x P ( x ) } with an ordinary cusp at (0 , , its normalization is the smoothcurve Y := { ( x, y ) ∈ C | y = xP ( x ) } , whose A -period matrix and normalized period matrix aredenoted by A and Z , respectively. Then, Y and the regular part of X have the same Bergman kernel,denoted by κ . In the following two cases, we explore different types of families of hyperelliptic curves X λ that give rise to X , as λ → . In each case, similarly, in the local coordinate z := √ x near (0 , ,we write κ X λ = k λ ( z ) dz ⊗ d ¯ z and κ = k ( z ) dz ⊗ d ¯ z . See (7.2) for the precise formulae of κ . Case I.
We consider the hyperelliptic curves(2.2) X λ := { ( x, y ) ∈ C | y = x ( x − λ ) P ( x ) } . Theorem 2.2. As λ → , for = | z | small, it holds that(i) k λ ( z ) → k ( z ) + 4 | z P ( z ) | , i.e., κ X λ κ ; ROBERT XIN DONG (ii) ψ − log (cid:18) k ( z ) + 4 | z P ( z ) | (cid:19) ∼ − P g − i =1 (cid:0) (Im Z ) − Im( A − ( ♣ − √− ∗ )) (cid:1) i z i P g − i,j =1 ((Im Z ) − ) i,j ( z i z j ) , where ♣ and ∗ are column vectors with g − rows involving λ . ∗ Case II.
We consider the hyperelliptic curves(2.3) X λ := { ( x, y ) ∈ C | y = x ( x − λ )( x − λ ) P ( x ) } . Theorem 2.3. As λ → , it holds that(i) κ X λ → κ ;(ii) for = | z | small, ψ − log k ( z ) ∼ πk ( z ) | z P ( z ) | · − log | λ | . Remarks. (a) In Theorem 2.2, (ii), the right hand side in the expansion of ψ is harmonic in λ of order O( λ / ) ,which is different from Theorems 2.1 and 2.3 where growth order ( − log | λ | ) − 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 | , where τ is the period (scalar) of the elliptic curve { ( x, y ) ∈ C | y = x ( x − a )( x − b ) } ∪ {∞} .(c) In both Cases I and II, for each small z = 0 , ψ regarded as a subharmonic function in λ = 0 , isbounded 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 X λ of genus g ≥ , itsJacobian variety Jac ( X λ ) is a g -dimensional complex torus whose Bergman kernel by (1.1) is written as µ λ dw ⊗ d ¯ w , under the coordinate w induced from C g . For a family of curves X λ in Theorem 2.1, 2.2 or2.3, we determine the asymptotic behaviors of the Bergman kernels on the naturally associated familyof Jacobians Jac ( X λ ) as follows. Theorem 2.4. As λ → , it holds that log µ λ = ( − log( − log | λ | ) + log π − log det(Im Z ) + O ((log | λ | ) − ) , for X λ in (2.1) or (2.3) ; − log det(Im Z ) + O( λ / ) , for X λ in (2.2) . ∗ See the definitions of ♣ and ∗ in Lemmata 7.2 and 7.1, respectively. OUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 5
3. Bergman kernel on algebraic curves
Consider the complex analytic families of hyperelliptic curves X λ := { ( x, y ) ∈ C | y = h λ ( x ) P ( x ) } of genus g ≥ . Here h λ is a degree polynomial which depend on λ ∈ ∆ ∗ , with distinct rootsof small absolute values. P again is a polynomial of degree d ≥ with complex roots a j such that < | a | < | a | < ... < | a d | . Without loss of generality, one may always assume that d is odd so that X λ compact, since ∞ can be added to the definition of X λ in case d is even. For each λ , on the smoothcurve X λ there exists a globally defined basis, given by the Abelian differentials(3.1) ω i := x i − dxy , i = 1 , ..., g, for the Hilbert space of (square integrable) holomorphic 1-forms.3.1. Relation with Riemann period matrix.
It is well known that in the definition (1.1), the Bergmankernel is independent of the choices of the basis. To write down the Bergman kernel on X λ , one couldobtain from (3.1) the orthonormal basis ϕ , ϕ , ... , ϕ g after the Gram-Schmidt process by the L innerproduct. Alternatively, the Bergman kernel κ X λ can be written down in terms of the normalized periodmatrix, denoted by Z . By the classical construction of hyperelliptic curves, X λ can be realised as a2-sheeted ramified covering of the Riemann sphere P (see [12]). Take the homologous basis δ i and γ j of H ( X λ , Z ) such that their intersection numbers are δ i · δ j = 0 , γ i · γ j = 0 , and δ i · γ j = δ i,j = − γ j · δ i (here δ i,j is the Kronecker δ ). Then, for the above ω i ∈ H ( X λ , Ω) , we set A i,j := Z δ j ω i , B i,j := Z δ j γ i . By the Hodge-Riemann bilinear relation, it is known that the matrix ( A ij ) ≤ i,j ≤ g is invertible. More-over, using the matrix ( B ij ) ≤ i,j ≤ g , we define the normalized period matrix Z := A − B . Then, Z issymmetric and has a positive definite imaginary part, i.e., Im Z > . 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 X λ can be equivalently defined as κ X λ := g X i,j =1 ((Im Z ) − ) i,j ω i ⊗ ω j . Under certain local coordinate z specified in the next subsection, we write κ X λ = k λ ( z ) dz ⊗ d ¯ z , andexplore the asymptotic behavior of ψ := log k λ ( 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 ∈ X has a neighborhood isomorphic to either a disk in C or { ( x, y ) ∈ C | | x | , | y | < , xy = 0 } . In thelatter, p is a node. For the family of hyperelliptic curves in (2.1), as λ → , X λ degenerates to a singularcurve X with a non-separating node. On X , near the node (0 , , the local coordinate can be takenas z = √ x , so it holds that x = z and dx = 2 zdz . Then, in this z -coordinate, ω i = 2 z i − dz p ( z − λ )( z − P ( z ) , i = 1 , ..., g, and the Bergman kernel κ X λ can be written as(3.2) g X i,j =1 ((Im Z ) − ) i,j ( z i z j ) dz ⊗ d ¯ z | z ( z − λ )( z − P ( z ) | =: k λ ( z ) dz ⊗ d ¯ z. ROBERT XIN DONG
Particularly, when P = ( x − a )( x − b ) , the expression (3.2) reduces to(3.3) Z ) − ) , + ((Im Z ) − ) , z + ((Im Z ) − ) , z + ((Im Z ) − ) , | z | | ( z − z − a )( z − b )( z − λ ) | dz ⊗ d ¯ z. 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 of X λ is equivalent to the non-vanishingof the Gaussian curvature of the Bergman kernel κ X λ . 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), as λ → , X λ degenerates to a singularcurve X with a cusp. On X , near (0 , , the local coordinate can be taken as z = √ x . Similarly,(3.4) κ X λ = g X i,j =1 ((Im Z ) − ) i,j ( z i z j ) dz ⊗ d ¯ z | z ( z − λ ) · P ( z ) | =: k λ ( z ) dz ⊗ d ¯ z. Particularly, when P = ( x − a )( x − b ) , the expression (3.4) reduces to(3.5) Z ) − ) , + ((Im Z ) − ) , z + ((Im Z ) − ) , z + ((Im Z ) − ) , | z | | ( z − a )( z − b )( z − λ ) | dz ⊗ d ¯ z. Near a cusp, Case II.
For the family of hyperelliptic curves in (2.3), as λ → , X λ degenerates to asingular curve X with a cusp. On X , near (0 , , the local coordinate can be taken as z = √ x .Similarly,(3.6) κ X λ = g X i,j =1 ((Im Z ) − ) i,j ( z i z j ) dz ⊗ d ¯ z | z ( z − λ )( z − λ ) · P ( z ) | =: k λ ( z ) dz ⊗ d ¯ z. Particularly, when P = ( x − a )( x − b ) , the expression (3.6) reduces to(3.7) Z ) − ) , + ((Im Z ) − ) , z + ((Im Z ) − ) , z + ((Im Z ) − ) , | z | | ( z − a )( z − b )( z − λ )( z − λ ) | dz ⊗ d ¯ z. On the normalization of singular curves.
For the nodal singular curve X in (2.1), we considerits normalization Y := { ( x, y ) ∈ C | y = ( x − P ( x ) } , which is a compact curve of genus g − ⌊ d ⌋ .The regular part of X is non-compact, and can be seen as either X or Y less the node. Let Z be theperiod matrix of Y . By the L removable singularity theorem, Y and the regular part of X have thesame Bergman kernel which can be written under the local coordinate z = √ x as(3.8) κ = g − X i,j =1 ((Im Z ) − ) i,j ( z i ¯ z j ) dz ⊗ d ¯ z | z ( z − P ( z ) | , where ω i = x ( i − dx p ( x − P ( x ) = 2 z i − dz p ( z − P ( z ) , i = 1 , ..., g − . For the cuspidal singular curve X := { ( x, y ) ∈ C | y = x P ( x ) } , its normalization is Y := { ( x, y ) ∈ C | y = xP ( x ) } . Similarly, the Bergman kernel on Y can be written under the coordinate z = √ x as(3.9) κ = g − X i,j =1 ((Im Z ) − ) i,j ( z i ¯ z j ) dz ⊗ d ¯ z | z P ( z ) | . OUNDARY 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 X λ := { ( x, y ) ∈ C | y = x ( x − λ )( x − x − a )( x − b ) } ∪ {∞} , where a, b, λ are distinct complex numbers satisfying < | λ | < < | a | < | b | .As λ → , X λ degenerates to a singular curve X with a non-separating node. The normalization of X is an elliptic curve { ( x, y ) ∈ C | y = ( x − x − a )( x − b ) } ∪ {∞} , whose period is c given in(4.5) and let c := Z a √ abdx p ( x − x − a )( x − b ) be a constant depending on a, b . The Bergman kernels on X λ and on the normalization of X aredenoted by κ X λ and κ , respectively. In the local coordinate z := √ x near (0 , , we write κ X λ = k λ ( z ) dz ⊗ d ¯ z , and κ = k ( z ) dz ⊗ d ¯ z . Then, our result on the asymptotic behaviour of the Bergmankernel κ X λ with precise coefficients is stated as follows. Theorem 4.1. As λ → , it holds that(i) κ X λ → κ ;(ii) for small | z | 6 = 0 , log k λ ( z ) − log k ( z ) ∼ π − log | λ | Im c + ( z + z ) Re { c − }| z | . 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 X λ . Lemma 4.2.
Under the same assumptions as in Theorem 4.1, as λ → , it holds that A = − π √ ab c − c √ ab ! + O( λ ) , where c := R δ dxx √ ( x − x − a )( x − b ) and lim λ → λ ) λ is a finite matrix. Proof of Lemma 4.2.
We estimate the four entries of A one by one. Firstly, A , = R δ ω , where δ onlycontains and λ . Change the variables by setting t = λ − , s = x − , and get the dual cycle ˜ δ whichcontains {∞ , t } so that − ˜ δ contains { , , a − , b − } , for | s | ∈ (1 , | t | ) . As λ → , A s = x − ===== t = λ − Z ˜ δ − s − ds p s − ( s − − t − )( s − − s − − a )( s − − b )= − Z − ˜ δ −√ s √ tds p ab ( s − s − a − )( s − b − )( s − t )= Z − ˜ δ √ sds p − ab ( s − s − a − )( s − b − ) (cid:18) s t + O (cid:18) s t (cid:19)(cid:19) = Z − ˜ δ ds − s √− ab (cid:18) s t + O (cid:18) s t (cid:19)(cid:19) (cid:0) s − ) (cid:1) (cid:0) s − ) (cid:1) (cid:0) s − ) (cid:1) = Z − ˜ δ ds − s √− ab (cid:18) s t + O( s t ) (cid:19) (cid:0) s − ) (cid:1) = 1 −√− ab Z − ˜ δ (cid:0) t − ) (cid:1) dss = 2 π −√ ab (1 + O( λ )) . ROBERT XIN DONG
Notice that we have used the Maclaurin expansion(4.1) √ s − a = ( √− a − (1 + O ( sa − )) , | s | < | a | ; √ s − (1 + O( s − a )) , | s | > | a | . Secondly, look at A = R δ ω and similarly it holds that A = Z − ˜ δ ds − s √− ab (cid:18) s t + O (cid:18) s t (cid:19)(cid:19) (cid:0) s − ) (cid:1) = 1 −√− ab Z − ˜ δ (cid:18) t + O( t − ) (cid:19) dss = 2 π −√ ab (cid:18) λ λ ) (cid:19) . Thirdly, let δ contain { , λ, , a } . Then, as λ → , it holds that A = Z δ dxx p ( x − x − a )( x − b ) (cid:18) λ x + O (cid:18) λ x (cid:19)(cid:19) . = Z δ dxx p ( x − x − a )( x − b ) (1 + O ( λ )) . Lastly, A = Z δ dx p ( x − x − a )( x − b ) (1 + O( λ )) = − Z a dx p ( x − x − a )( x − b ) (1 + O( λ )) . (cid:3) Lemma 4.3.
Under the same assumptions as in Theorem 4.1, as λ → , it holds that B ∼ − λ √− ab d − √− ab d ! , where d := − R ba dxx √ ( x − x − a )( x − b ) and d := − R ba dx √ ( x − x − a )( x − b ) . Proof of Lemma 4.3 . As t → ∞ , we make use of the following computations (cf. [12]).(4.2) Z t dss √ s − t = − √ t √− r ts + r ts − !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t ∼ √− √ t log t. (4.3) Z t dss √ s − t = √ s − tts (cid:12)(cid:12)(cid:12)(cid:12) t + 12 t Z t dss √ s − t = −√ − tt + √− t √ t log t ∼ − √− √ t . In particular, (4.3) yields the boundedness of Z t √ ts √ s − t O( s − ) ds. More generally, for any α ≥ , as t → ∞ , it holds that(4.4) Z t dss α +1 √ s − t = √ s − tαts α (cid:12)(cid:12)(cid:12)(cid:12) t + 2 α − αt Z t dss α √ s − t ∼ − √− α √ t . OUNDARY 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 | ) , B = − Z λ dx p x ( x − λ )( x − x − a )( x − b )= − Z t −√ s √ tds p ab ( s − s − a − )( s − b − )( s − t )= − Z t √ s √ tds p ( s − t ) ab s √ s (cid:0) s − ) (cid:1) (cid:0) s − ) (cid:1) (cid:0) s − ) (cid:1) = − √ t √ ab Z t dss √ s − t (cid:0) s − ) (cid:1) ∼ − λ √− ab , as λ → . Secondly, by (4.3), B = − √ t √ ab Z t dss √ s − t (cid:0) s − ) (cid:1) ∼ − √− ab . Finally, similar to A , B ∼ − Z ba dxx p ( x − x − a )( x − b ) , and B ∼ − Z ba dx p ( x − x − a )( x − b ) . (cid:3) 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 := R ba dx √ ( x − x − a )( x − b ) R a dx √ ( x − x − a )( x − b ) = τ (cid:18) − b − a (cid:19) , where τ ( · ) is the inverse of the elliptic modular lambda function. On the normalization of the nodalcurve X , by (3.8), in the local coordinate z = √ x , the Bergman kernel is exactly κ = k ( z ) | dz | ,where(4.6) k ( z ) = 4 | z | (Im c ) | ( z − z − a )( z − b ) | . By Lemma 4.2, as λ → , A − ∼ − π √ ab c O( λ ) − c √ ab ! − = −√ ab π √ abc πd O( λ ) √ ab − c ! . Let Z denote the period matrix of X λ . Then, it follows that Z = A − B ∼ −√− π log λ abc d + √ ab c d − πc −√− c − c ! , Im Z ∼ (cid:18) − log | λ | π − Re { c − }− Re { c − } Im c (cid:19) . So, as λ → , it holds that (Im Z ) − ∼ π − log | λ | − Re { c − } π (log | λ | ) Im c − Re { c − } π (log | λ | ) Im c (Im c ) − ! . By (3.3) and (4.6), κ X λ → κ . Moreover, in the local coordinate z = √ x near (0 , , k λ ( z ) − k ( z ) ∼ π | ( z − z − a )( z − b )( z − λ ) | Re { c − } Im c ( z + z ) − log | λ | , which yields the conclusion. (cid:3) More generally, genus-two curves defined as X λ,a,b := { y = x ( x − x − λ )( x − a )( x − b )) } canbe parametrized by three distinct complex numbers λ, a, b ∈ C \ { , } , since the moduli space is 3dimensional. As λ, a or b tends to , or ∞ , or towards each other, X λ,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 \ { , , a , ..., a d } , X λ defined in (2.1) has genus g = ⌊ d ⌋ + 1 . Toprove Theorem 2.1, we need to analyze on X λ the asymptotics of its A -period matrix and B -periodmatrix, denoted by A and B , respectively. Meanwhile, for the normalization Y := { ( x, y ) ∈ C | y =( x − P ( x ) } of genus g − , denote its A -period matrix and B -period matrix by A and B , respectively. Lemma 5.1.
Under the same assumptions as in Theorem 2.1, as λ → , it holds that A ∼ π √ P (0) O(1) ∗ A ! , where ∗ = (O( λ ) , , ..., T is a column vector with g − rows. Proof.
Firstly, A = R δ ω , where δ only contains and λ . Change the variables by setting t = λ − , s = x − , and we get the dual cycle ˜ δ which contains {∞ , t } so that − ˜ δ contains { , , a k − } , k = 1 , ..., d , for | s | ∈ (1 , | t | ) . As λ → , A s = x − ===== t = λ − Z − ˜ δ ds √− s p P (0) (cid:16) (cid:16) st (cid:17)(cid:17) (cid:0) s − ) (cid:1) ∼ Z − ˜ δ ds √− s p P (0) (cid:0) s − ) (cid:1) = Z − ˜ δ ds √− s p P (0) = 2 π p P (0) .A = Z − ˜ δ ds √− s p P (0) (cid:16) (cid:16) st (cid:17)(cid:17) (cid:0) s − ) (cid:1) = Z − ˜ δ ds √− p P (0) (cid:18) s − + O (cid:18) ts i − (cid:19)(cid:19) (cid:0) s − ) (cid:1) = Z − ˜ δ ds √− p P (0) (cid:18) + O (cid:18) ts (cid:19)(cid:19) = 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.
OUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 11
For ≤ i ≤ g , A i = Z − ˜ δ ds √− s i p P (0) (cid:16) (cid:16) st (cid:17)(cid:17) (cid:0) s − ) (cid:1) = Z − ˜ δ ds √− p P (0) (cid:18) s − i + O (cid:18) ts i − (cid:19)(cid:19) (cid:0) s − ) (cid:1) = 0 . Secondly, let δ contain only , λ, and a (not a , ..., a d ). Then, A = Z δ dx p x ( x − λ )( x − P ( x )= Z δ dxx p ( x − P ( x ) (cid:18) (cid:18) λx (cid:19)(cid:19) ∼ Z δ dxx p ( x − P ( x ) . In general, for A i , there is an extra x i − in the original integrand above, so A i ∼ Z δ x i − dx p ( x − P ( x ) . Thirdly, let δ contain only , λ , λ, a , a and a (not a , ..., a d ). Similarly, a i is asymptotic to the sameintegrand over δ instead of over δ , i.e., A i ∼ Z δ x i − dx p ( x − P ( x ) . In general, for j ≥ and the corresponding homologous basis δ j , it holds that A ij ∼ Z δ j x i − dx p ( x − P ( x ) , which are exactly the same as the entries of A when i ≥ . (cid:3) Lemma 5.2.
Under the same assumptions as in Theorem 2.1, as λ → , it holds that B ∼ − √− √ P (0) log λ O(1) ⋆ B ! , where ⋆ is a column vector whose entries are all − √− √ P (0) with g − rows. Proof.
Firstly, by (4.2) and (4.3), for | s | ∈ (1 , | t | ) , B = − Z λ dx p x ( x − λ )( x − · P ( x ) = − Z λ dx p x ( x − λ ) 1 p − P (0) (1 + O( x )) s = x − ===== t = λ − √ t p P (0) Z t dss √ s − t (1 + O( s − )) ∼ − p P (0) √− λ. Thus, for ≤ i ≤ g , by (4.4), B i = − Z λ x i − dx p x ( x − λ )( x − · P ( x ) = 2 √ t p P (0) Z t dss i √ s − t (1 + O( s − )) ∼ − √− p P (0) . For Column j , ≤ j ≤ g , we use Taylor series expansion of √ x − λ − to get that B ij ∼ − Z a j − a j − x i − dx p ( x − P ( x ) , which are exactly the same as the entries of B when i ≥ . (cid:3) 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 X in (2.1) is exactly κ = k ( z ) | dz | , where(5.1) k ( z ) = 4 | z ( z − P ( z ) | g − X i,j =1 ((Im Z ) − ) i,j ( z i z j ) . By Lemma 5.1 and the block matrix inversion, we know that A − ∼ √ P (0)2 π O(1)O( λ ) A − ! , where both (O(1)) T and lim λ → λ ) λ are finite column vectors with g − rows. Therefore, as λ → , Z = A − B ∼ (cid:18) −√− π log λ O(1) A − ⋆ A − B (cid:19) , Im Z ∼ (cid:18) log | λ |− π O(1)Im( A − ⋆ ) Im Z (cid:19) , and (Im Z ) − ∼ − π log | λ | O((log | λ | ) − )(Im Z ) − Im( A − ⋆ ) π log | λ | (Im Z ) − ! . 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 → , κ X λ → κ . Moreover, in the local coordinate z = √ x near (0 , , it holds that k λ ( z ) − k ( z ) ∼ π | ( z − λ )( z − P ( z ) | − g − P i =1 (cid:0) (Im Z ) − Im( A − ⋆ ) (cid:1) i z i − log | λ | which yields the conclusion. (cid:3) 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 period matrix. 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 curves X λ := { ( x, y ) ∈ C | y = x ( x − λ )( x − a )( x − b ) } ∪ {∞} , where a, b, λ are distinct complex numbers satisfying < | λ | < | a | < | b | . As λ → , X λ degenerates to a singular curve X with an ordinary cusp. The normalization of X is an ellipticcurve { ( x, y ) ∈ C | y = x ( x − a )( x − b ) } ∪ {∞} , whose period is τ given in (6.1). Let c := Z a √ abdx p x ( x − a )( x − b ) , c := − Z ( x − dx p x ( x − x − be constants depending on a, b . The Bergman kernels on X λ and on the normalization of X are denotedby κ X λ and κ , respectively. In the local coordinate z := √ x near (0 , , we write κ X λ = k λ ( z ) dz ⊗ d ¯ z ,and κ = k ( z ) dz ⊗ d ¯ z . Then, our result on the asymptotic behaviour of the Bergman kernel κ X λ withprecise coefficients is stated as follows. Theorem 6.1. As λ → , for small | z | 6 = 0 , it holds that OUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 13 (i) k λ ( z ) → k ( z ) (cid:18) Im τ | z | + 1 (cid:19) , i.e., κ X λ κ ; (ii) log k λ ( z ) − log k ( z ) − log (cid:18) Im τ | z | + 1 (cid:19) ∼ Re n λ / c c o ( z + z ) | z | + Im τ , where τ is the period (scalar) of the elliptic curve { y = x ( x − a )( x − b ) } . To prove Theorem 6.1, we need the following two lemmata by analyzing the asymptotics of the A -period matrix and B -period matrix on X λ . Lemma 6.2.
Under the same assumptions as in Theorem 6.1, as λ → , it holds that A ∼ c −√ ab λ − / c c −√ ab λ / − c √ ab ! , where c := − R du √ u ( u − u − and c depends on a, b . Proof of Lemma 6.2.
We estimate the four entries one by one. Firstly, let δ contain only −√ λ and .By the Cauchy Integral Theorem and Taylor series expansion, we know that A = − Z −√ λ dx q x ( x − √ λ )( x + √ λ )( x − a )( x − b )= − −√ ab Z −√ λ dx q x ( x − √ λ )( x + √ λ ) (1 + O( x )) x =( u − √ λ ========= − λ − / −√ ab Z du p u ( u − u − (cid:16) (cid:16) √ λ ( u − (cid:17)(cid:17) ∼ − λ − / −√ ab Z du p u ( u − u −
2) =: c −√ ab λ − / . Secondly, A = − λ / −√ ab Z ( u − du p u ( u − u − (cid:16) (cid:16) √ λ ( u − (cid:17)(cid:17) ∼ − λ / −√ ab Z ( u − du p u ( u − u −
2) := c −√ ab λ / . Thirdly, let δ contain only −√ λ, , √ λ and a (not b ). Then, A = Z δ dx p x ( x − a )( x − b ) 1 √ x √ λx !! √ x √ λx !! ∼ Z δ dxx p x ( x − a )( x − b ) =: c . A = Z δ dx p x ( x − a )( x − b ) √ λx !! ∼ Z δ dx p x ( x − a )( x − b ) = − Z a dx p x ( x − a )( x − b ) = − c √ ab . (cid:3) Lemma 6.3.
Under the same assumptions as in Lemma 6.2, as λ → , it holds that B ∼ √− ab (cid:18) c λ − / d √− ab − c λ / d √− ab (cid:19) + , where d := − R ba dxx √ x ( x − a )( x − b ) and d := − R ba dx √ x ( x − a )( x − b ) . Proof of Lemma 6.3.
Again, we estimate all the four entries one by one. Firstly, let γ contain only and √ λ . By Cauchy Integral Theorem, we can get that B = − Z √ λ dx p x ( x − λ )( x − a )( x − b )= − −√ ab Z √ λ dx q x ( x − √ λ )( x + √ λ ) (1 + O( x )) x =(1 − u ) √ λ ========= 2 λ − / √ ab Z du p − u ( u − u − (cid:16) − u + 1) · √ λ ) (cid:17) ∼ λ − / √ ab Z √− du p u ( u − u −
2) =: c √− ab λ − / . Secondly, B = 2 λ − / √ ab Z ( − u + 1) √ λdu p − u ( u − u − (cid:16) − u + 1) · √ λ ) (cid:17) ∼ λ / √ ab Z ( u − du p − u ( u − u −
2) =: √− λ / c √ ab . Thirdly, let γ contain only a and b . Then, it holds that B = Z γ dx q x ( x − √ λ )( x + √ λ )( x − a )( x − b )= − Z ba dx p x ( x − a )( x − b ) 1 √ x √ λ x !! √ x √ λ x !! ∼ − Z ba dxx p x ( x − a )( x − b ) =: d . Lastly, B = Z γ dx p x ( x − a )( x − b ) √ λ x !! ∼ − Z ba dx p x ( x − a )( x − b ) =: d . (cid:3) OUNDARY 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 := R ba dx √ x ( x − a )( x − b ) R a dx √ x ( x − a )( x − b ) = τ (cid:18) ba (cid:19) , where τ ( · ) is the inverse of the elliptic modular lambda function. On the regular part of the cuspidalcurve X , by (3.9), in the local coordinate z = √ x , the Bergman kernel is exactly κ = k ( z ) | dz | ,where(6.2) k ( z ) = 4(Im τ ) | ( z − a )( z − b ) | . By Lemma 6.2, as λ → , A − ∼ c −√ ab λ − / c c −√ ab λ / − c √ ab ! − ∼ abλ / c c − c √ ab − c c √ ab λ / c −√ ab λ − / ! . Let Z denote the period matrix of X λ . Then, it follows that Z = A − B ∼ abλ / c c − c √ ab − c c √ ab λ / c −√ ab λ − / ! √− ab (cid:18) c λ − / d √− ab − c λ / d √− ab (cid:19) ∼ √− √− λ / (cid:16) d √− ab − c d ab c (cid:17) c − √− λ / c − c τ ! , Im Z ∼ λ / )Re n λ / c − c o Im τ ! . So, as λ → , (Im Z ) − ∼ − (Im τ ) − O( λ / )(Im τ ) − Re n λ / c c o (Im τ ) − ! . By (3.7) and (6.2),(6.3) k λ ( z ) ∼ τ ) − Re n λ / c c o ( z + z ) + (Im τ ) − | z | | ( z − a )( z − b )( z − λ )( z − λ ) | → (cid:18) Im τ | z | + 1 (cid:19) k ( z ) , which means that κ X λ κ . Moreover, in the local coordinate z = √ x near (0 , , k λ ( z ) − (cid:18) Im τ | z | + 1 (cid:19) k ( z ) ∼ τ ) − Re n λ / c c o ( z + z ) | z ( z − a )( z − b ) | , which yields the conclusion. (cid:3)
7. Cusp I: hyperelliptic curves
This section is devoted to the proof of Theorem 2.2. When λ ∈ C \ { , , a , ..., a d } , X λ definedin (2.2) has genus g = ⌊ d ⌋ + 1 . To prove Theorem 2.2, we need to analyze on X λ the asymptoticsof its A -period matrix and B -period matrix, denoted by A and B , respectively. Meanwhile, for thenormalization Y := { ( x, y ) ∈ C | y = xP ( x ) } of genus g − , denote its A -period matrix and B -period matrix by A and B , respectively. Lemma 7.1.
Under the same assumptions as in Theorem 2.2, as λ → , it holds that A ∼ c λ − √ P (0) O(1) ∗ A ! , where ∗ is a column vector with g − rows whose entries are given by (7.1) below. Proof.
We will estimate all the g × g elements one by one. Firstly, as λ → , it holds that A = − Z −√ λ dx p x ( x − λ ) P ( x ) = − Z −√ λ dx p x ( x − λ ) 1 p P (0) (1 + O( x )) x =( u − √ λ ========= − λ − Z du p u ( u − u −
2) (1 + O(( u − √ λ )) p P (0) ∼ − λ − Z du p u ( u − u −
2) 1 p P (0) =: λ − c p P (0) . In general, for a i , ≤ i ≤ g , there is an extra x i − in the original integrand and thus an extra √ λ i − ( u − i − in the numerator of the above last expression, so(7.1) A i ∼ − Z ( u − i − du p u ( u − u − √ λ + i − p P (0) . Secondly, let δ contain only −√ λ, , √ λ and a (not a , ..., a d ). Then, A = Z δ dx p x ( x − λ ) P ( x )= Z δ dxx p xP ( x ) (cid:18) (cid:18) λx (cid:19)(cid:19) ∼ Z δ dxx p xP ( x ) . In general, for a i , there is an extra x i − in the original integrand, so A i ∼ Z δ x i − dx p xP ( x ) . Thirdly, let δ contain only −√ λ, , √ λ, a , a and a (not a , ..., a d ). Similarly, a i is asymptotic to thesame integrand over δ instead of over δ , i.e., A i ∼ Z δ x i − dx p xP ( x ) . In general, for j ≥ and the corresponding homologous basis δ j , it holds that A ij ∼ Z δ j x i − dx p xP ( x ) , which are exactly the same as the entries of A when i ≥ . (cid:3) Lemma 7.2.
Under the same assumptions as in Theorem 2.2, as λ → , it holds that B ∼ c λ − √− √ P (0) O(1) ♣ B ! , where ♣ is a column vector whose entries are B i = ( − i − √− A i for A i in (7.1) and ≤ i ≤ g . OUNDARY 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 ≤ i ≤ g that B i ∼ ( − i − √− a i . For Column j , ≤ j ≤ g , we use Taylor expansion of ( x − λ ) − / and get that B ij ∼ − Z a j − a j − x i − dx p xP ( x ) . (cid:3) 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 X in (2.2) is exactly κ = k ( z ) | dz | , where(7.2) k ( z ) = 4 | z P ( z ) | g − X i,j =1 ((Im Z ) − ) i,j ( z i z j ) . By Lemma 7.1 and the block matrix inversion, we know that A − ∼ λ √ P (0) c O( λ / )( − A − ∗ ) λ √ P (0) c A − , where lim λ → λ )) T λ is a finite column vectors with g − rows. Therefore, as λ → , Z = A − B ∼ (cid:18) √− λ ) A − ( ♣ − √− ∗ ) A − B (cid:19) , Im Z ∼ (cid:18) λ / )Im( A − ( ♣ − √− ∗ )) Im Z (cid:19) , and (Im Z ) − ∼ (cid:18) λ / )(Im Z ) − Im( A − ( ♣ − √− ∗ )) (Im Z ) − (cid:19) . By (3.4) and (7.2), as λ → , in the local coordinate z = √ x near (0 , , it holds that k λ ( z ) → P g − i,j =1 ((Im Z ) − ) i,j ( z i z j ) | − z P ( z ) | = k ( z ) + 4 | z P ( z ) | , so κ X λ κ . Moreover, k λ ( z ) − k ( z ) − | z P ( z ) | ∼ − g − P i =1 (cid:0) (Im Z ) − Im( A − ( ♣ − √− ∗ )) (cid:1) i z i | z P ( z ) | , which yields the conclusion. (cid:3) Since Z is symmetric, A − ( ♣ − √− ∗ ) = O( λ / ) , and both the leading and subleading terms inthe expansion of κ X λ are harmonic with respect to λ .
8. Cusp II: genus-two curves
In this section, we consider a family of genus two curves X λ := { ( x, y ) ∈ C | y = x ( x − λ )( x − λ )( x − a )( x − b ) } ∪ {∞} , where a, b, λ are distinct complex numbers satisfying < | λ | < | a | < | b | .As λ → , X λ degenerates to a singular curve X with an ordinary cusp. The normalization of X isan elliptic curve { ( x, y ) ∈ C | y = x ( x − a )( x − b ) } ∪ {∞} , whose period is τ given in (6.1). TheBergman kernels on X λ and on the normalization of X are denoted by κ X λ and κ , respectively. Inthe local coordinate z := √ x near (0 , , we write κ X λ = k λ ( z ) dz ⊗ d ¯ z , and κ = k ( z ) dz ⊗ d ¯ z . Then,our result on the asymptotic behaviour of the Bergman kernel κ X λ with precise coefficients is statedas follows. Theorem 8.1. As λ → , it holds that(i) κ X λ → κ ;(ii) for small | z | 6 = 0 , log k λ ( z ) − log k ( z ) ∼ π · Im τ − log | λ | · | z | , To prove Theorem 8.1, we need the following two lemmata by analyzing the asymptotics of the A -period matrix and B -period matrix on X λ . Lemma 8.2.
Under the same assumptions as in Theorem 8.1, as λ → , it holds that A ∼ − π √ ab √ λ c c λ / − c √ ab ! , where c := − √ ab R √ v − − dv and c , c are the same as in Lemma 6.2. Proof of Lemma 8.2.
Consider the integrals Z λ dx p x ( x − λ )( x − λ ) x = λ (1 − v ) ======== Z −√− λ − dv p v ( v − v − λ − ) ∼ − λ − √− Z ˜ γ dvvλ − / = − π √ λ , where ˜ γ is a large cycle containing , , and Z λ xdx p x ( x − λ )( x − λ ) = − λ √− Z ( v − dv p v ( v − v − λ − ) ∼ − λ / √− Z r v − v dv. We estimate the four entries one by one. Firstly, let δ only contain and λ , and similar to the proofof Lemma 6.2 we know that A = − Z λ dx p x ( x − λ )( x − λ ) 1 √− a √− b (cid:16) x a + O( x ) (cid:17) (cid:16) − x b + O( x ) (cid:17) ∼ − −√ ab Z λ dx p x ( x − λ )( x − λ ) ∼ − π √ ab √ λ . Secondly, A ∼ √ ab Z λ xdx p x ( x − λ )( x − λ ) ∼ − √ ab λ / √− Z r v − v dv =: c λ / . OUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 19
Thirdly, let δ contain only , λ, λ and a . Then, it holds that A = Z δ dx p x ( x − λ )( x − λ )( x − a )( x − b )= Z δ dxx p x ( x − a )( x − b ) (cid:18) (cid:18) λx (cid:19)(cid:19) (cid:18) (cid:18) λ x (cid:19)(cid:19) ∼ Z δ dxx p x ( x − a )( x − b ) =: c , Lastly, A ∼ Z δ dxx p x ( x − a )( x − b ) = Z a dxx p x ( x − a )( x − b ) . (cid:3) Lemma 8.3.
Under the same assumptions as in Theorem 8.1, as λ → , it holds that B ∼ √− λ √ ab √ λ d √ ab √− λ d ! , where d , d are the same as in Lemma 6.3. Proof of Lemma 8.3.
By [23] or (4.2), it is known that, as λ → , Z λλ dx p x ( x − λ )( x − λ ) = 1 √ λ Z λ du p u ( u − u − λ ) ∼ √− λ √ λ . Firstly, let γ contain only λ and λ , and we get that B = − Z λλ dx p x ( x − λ )( x − λ )( x − a )( x − b )= − −√ ab Z λλ dx p x ( x − λ )( x − λ ) (1 + O( x )) ∼ √ ab Z λλ dx p x ( x − λ )( x − λ ) ∼ √ ab √− λ √ λ . Secondly, by (4.3), as λ → , B ∼ √ ab Z λλ xdx p x ( x − λ )( x − λ ) x = λs ==== t = λ − √ ab Z t dss √ s − t ∼ √ ab √− λ. Thirdly, B = − Z ba dx p x ( x − λ )( x − λ )( x − a )( x − b )= − Z ba dx (1 + O ( λx − )) (1 + O ( λ x − )) x p x ( x − a )( x − b ) ∼ − Z ba dxx p x ( x − a )( x − b ) := d . Lastly, B ∼ − Z ba dx p x ( x − a )( x − b ) := d . (cid:3) Proof of Theorem 8.1.
On the regular part of the cuspidal curve X , (6.2) gives the formulae for theBergman kernel κ = k ( z ) | dz | in the local coordinate z = √ x . By Lemma 8.2, as λ → , A − ∼ − π √ ab √ λ c c λ / − c √ ab ! − ∼ ab √ λ πc − c √ ab − c − c λ / − π √ ab √ λ ! . Let Z denote the period matrix of X λ . Then, it follows that Z = A − B ∼ −√− λπ ab √ λ πc (cid:16) − c √ ab d − c d (cid:17) √ λ √− c − τ ! , Im Z ∼ − log | λ | π O( λ ) − Re n √ λc − o Im τ ! So, as λ → , (Im Z ) − ∼ π − log | λ | | λ | O( λ ) − π log | λ | Im τ Re n √ λc − o (Im τ ) − ! . By (3.7) and (6.2), κ X λ → κ . Moreover, in the local coordinate z = √ x near (0 , , k λ ( z ) − k ( z ) ∼ π | ( z − a )( z − b )( z − λ )( z − λ ) | · λ ) − log | λ | , which yields the conclusion. (cid:3)
9. Cusp II: hyperelliptic curves
This section is devoted to the proof of Theorem 2.3. When λ ∈ C \ { , , a , ..., a d } , X λ definedin (2.3) has genus g = ⌊ d ⌋ + 1 . To prove Theorem 2.3, we need to analyze on X λ the asymptoticsof its A -period matrix and B -period matrix, denoted by A and B , respectively. Meanwhile, for thenormalization Y := { ( x, y ) ∈ C | y = xP ( x ) } of genus g − , denote its A -period matrix and B -period matrix by A and B , respectively. Lemma 9.1.
Under the same assumptions as in Theorem 2.3, as λ → , it holds that A ∼ π √ λP (0) O(1) ∗ A ! , where ∗ is a column vector with g − rows whose entries are at most O( λ / ) . Proof.
Firstly, as λ → , A = − Z λ dx p x ( x − λ )( x − λ ) 1 p P (0) (1 + O( x )) ∼ − Z λ dx p x ( x − λ )( x − λ ) 1 p P (0) ∼ π p λP (0) .A ∼ − p P (0) Z λ xdx p x ( x − λ )( x − λ ) ∼ − p P (0) − λ / √− Z r v − v dv. Here we change the variable by letting x = λ (1 − v ) . In general, for ≤ i ≤ g , A i ∼ − p P (0) − λ / √− Z λ i − (1 − v ) i − r v − v dv = O( λ i − . ) . OUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 21
Secondly, let δ contain only , λ , λ and a (not a , ..., a d ). Then, A = Z δ dx p x ( x − λ )( x − λ ) P ( x )= Z δ dxx p xP ( x ) (cid:18) (cid:18) λx (cid:19)(cid:19) ∼ Z δ dxx p xP ( x ) . In general, for A i , there is an extra x i − in the original integrand above, so A i ∼ Z δ x i − dx p xP ( x ) . Thirdly, let δ contain only , λ , λ, a , a and a (not a , ..., a d ). Similarly, A i is asymptotic to the sameintegrand over δ instead of over δ , i.e., A i ∼ Z δ x i − dx p xP ( x ) . Lastly, for j ≥ and the corresponding homologous basis δ j , it holds that A ij ∼ Z δ j x i − dx p xP ( x ) , which are exactly the same as the entries of A when i ≥ . (cid:3) Lemma 9.2.
Under the same assumptions as in Theorem 2.3, as λ → , it holds that B ∼ − √ P (0) √− λ √ λ O(1) ♠ B ! , where ♠ is a column vector whose entries are − λ i − √− λ √ P (0) , for ≤ i ≤ g . Proof. B = − Z λλ dx p x ( x − λ )( x − λ ) · P ( x ) = − Z λλ dx p x ( x − λ )( x − λ ) 1 p P (0) (1 + O( x )) ∼ − Z λλ dx p x ( x − λ )( x − λ ) 1 p P (0) ∼ − p P (0) √− λ √ λ . Thus, for ≤ i ≤ g , by (4.4), B i = − Z λλ x i − dx p x ( x − λ )( x − λ ) 1 p P (0) (1 + O( x )) ∼ − Z λλ x i − dx p x ( x − λ )( x − λ ) 1 p P (0) x = λs ==== t = λ λ i − p P (0) Z t dss i √ s − t ∼ − λ i − √− λ p P (0) . For Column j , ≤ j ≤ g , we use Taylor expansion of p ( x − λ )( x − λ ) − to get that B ij ∼ − Z a j − a j − x i − dx p xP ( x ) . which are exactly the same as the entries of B when i ≥ . (cid:3) Now we will give a proof of Theorem 2.3.
Proof of Theorem 2.3.
On the regular part of the cuspidal curve X , (7.2) gives the formulae for theBergman kernel κ = k ( z ) | dz | in the local coordinate z = √ x . By Lemma 9.1 and the block matrixinversion, we know that A − ∼ √ λP (0)2 π O( λ )O( λ ) A − ! , where both lim λ → λ )) T λ and lim λ → λ ) λ are finite column vectors with g − rows. Therefore, as λ → , Z = A − B ∼ (cid:18) −√− π log λ O( λ ) A − ♠ A − B (cid:19) , Im Z ∼ (cid:18) log | λ |− π O( λ )Im( A − ♠ ) Im Z (cid:19) , and (Im Z ) − ∼ − π log | λ | O((log | λ | ) − λ )(Im Z ) − Im( A − ♠ ) π log | λ | (Im Z ) − ! . 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 → , κ X λ → κ . Moreover, in the local coordinate z = √ x near (0 , , it holds that k λ ( z ) − k ( z ) ∼ π | ( z − λ )( z − λ ) P ( z ) | − g − P i =1 (cid:0) (Im Z ) − Im( A − ♠ ) (cid:1) i z i − log | λ | which yields that ψ − log k ( z ) ∼ π − log | λ | g − P i,j =1 ((Im Z ) − ) i,j ( z i z j ) . (cid:3)
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 X λ be a compact curve of genus g ≥ , and let Z be its period matrix. TheJacobian variety of X λ is then identified with the g -dimensional complex torus C g / Z g + Z Z g , and isdenoted by Jac ( X λ ) . It is well known that the Abel-Jacobi (period) map X λ → Jac ( X λ ) is a holomorphicembedding. Proof of Theorem 2.4.
By definition (1.1), the Bergman kernel on Jac ( X λ ) can be written as µ λ dw ⊗ d ¯ w ,under the coordinate w induced from C g , where µ λ = (det(Im Z )) − . For X λ defined in (2.1) or (2.3),as λ → , it holds that det(Im Z ) = log | λ |− π det(Im Z ) + O(1) → + ∞ , which yields the first part of the conclusion in Theorem 2.4. For X λ defined in (2.2), more carefullyanalysis in the proof of Theorem 2.2 shows that as λ → , Im Z = (cid:18) λ / ) O( λ / )O( λ / ) Im Z + O( λ / ) (cid:19) , det(Im Z ) = det Im Z + O( λ / ) < + ∞ , which yields the second part of the conclusion. (cid:3) OUNDARY ASYMPTOTICS OF THE RELATIVE BERGMAN KERNEL METRIC FOR CURVES 23
It is worth mentioning that as the family of curves X λ in (2.2) degenerate to a cuspidal one, theirJacobian varieties Jac ( X λ ) remain being manifolds (i.e., non-degenerate), as λ → . However, for twofamilies of curves X λ in (2.1) and (2.3), their Jacobian varieties Jac ( X λ ) 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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