Non-completeness of the Arakelov-induced metric on moduli space of curves

NON-COMPLETENESS OF THE ARAKELOV-INDUCED METRIC ON MODULI SPACE OF CURVES J. JORGENSON AND J. KRAMER 1. Introduction 1.1. Weil-Petersson metric on the moduli space of curves. The classical theory of the moduli space of compact Riemann surfaces of a fixed genus g > 1 includes the investigation of the Weil-Petersson metric. Briefly, consider any compact Riemann surface Xof genus g > 1, which represents a point in the moduli space Mgof such objects. Then, the tangent space to Xin Mgis naturally dual to H0(X, Ω2 X), the C-vector space of holomorphic quadratic differentials on X. Let µbe a smooth, positive (1,1)-form on X corresponding to a smooth metric on X. For any ω, η ∈H0(X, Ω2 X), the classical Weil-Petersson inner product is defined by hω, ηi=i 2Z X (ω∧η)·µ−1.(1) By duality, a metric on the tangent space of Mgat the point represented by Xis induced by (1). When taking µto be the (1,1)-form µhyp associated to the hyperbolic metric on X, one obtains the classically studied Weil-Petersson metric on Mg; basic questions such as the completeness, curvature, and K¨ahlerian aspects of the Weil-Petersson metric on Mgare well-known (see [5], e.g., and references therein). 1.2. Arakelov metrics on Riemann surfaces. Much of the existing literature focuses on the Weil-Petersson metric on Mg, which is induced from the family of hyperbolic metrics on the underlying universal family of Riemann surfaces. Indeed, from the uniformization theorem, the family of hyperbolic metrics is a natural candidate from which one can form the associated Weil-Petersson metric on Mg. However, other points of view yield The first author acknowledges support from grants from the NSF and PSC-CUNY. The second author thanks the Centre de Recerca Matem`atica (CRM) in Barcelona for its support and hospitality. 1 2 J. JORGENSON AND J. KRAMER different families of metrics, and each family of metrics has an associated Weil-Petersson metric on Mg. The family of canonical metrics µcan arises from classical algebraic geometry when embedding Xinto its Jacobian variety via the Abel-Jacobi map, and the articles [3], [4] initiated the study of the corresponding Weil-Petersson metric on Mg. From arithmetic algebraic geometry, one has the family of Arakelov metrics µAr, as defined in [1], which provides yet another candidate of metrics from which one can form a Weil-Petersson metric on Mg. In [3] and [4], the authors posed the problem of developing a means by which one can study the resulting Arakelov-induced metric on Mg. Specifically, it is asked if the Arakelov-induced metric on the moduli space Mgis complete or not, which in the case of the classical Weil-Petersson metric was first answered in [13] and [11]. 1.3. The main result. The purpose of the present paper is to answer the problem posed in [3] and [4] concerning the Arakelov-induced metric on Mg. The main result we prove in this article is the following. Let Xt(t∈C,|t|<1) be a family of compact Riemann surfaces of fixed genus g > 1 degenerating at t= 0, and, hence, approaching the Deligne- Mumford boundary of the moduli space Mg; let µAr,t, resp. µhyp,t be the corresponding family of (1,1)-forms associated to the Arakelov, resp. hyperbolic metric. Then, we have lim t→0µsup z∈XtµµAr,t(z) µhyp,t(z)¶¶= 0 . As a corollary of this result, we prove that the Arakelov-induced metric on Mgis not complete. 1.4. Outline of the paper. In section 2, we establish notation and recall various known results, most importantly for our purposes here is section 2.4, which expresses φAr = log(µAr/µhyp), the conformal factor relating the Arakelov metric to the hyperbolic metric, in terms of the hyperbolic heat kernel, a result which has been proved in [9]. In section 3, we study each quantity in the formula for φAr. Ultimately, we bound φAr using known results for the hyperbolic heat kernel on a degenerating family of compact Riemann surfaces of genus g > 1, most notably from [10]. Finally, in section 4, we show how our main theorem answers the question of completeness of the Arakelov-induced metric on Mg. NON-COMPLETENESS OF THE ARAKELOV-INDUCED METRIC 3 2. Background material 2.1. Hyperbolic and canonical metrics. Let Xbe a compact Riemann surface of genus g > 1. By the uniformization theorem, we can identify X complex analytically with the quotient space Γ\H, where Γ is a Fuchsian subgroup of the first kind of PSL2(R) acting by fractional linear transformations on the upper half-plane H={z∈C|Im(z)>0}. In a slight abuse of notation, we will throughout this article identify Xwith a fundamental domain (say, a Ford domain, bounded by geodesic paths) and identify points on Xwith their pre-images in H. Let µdenote a smooth, positive (1,1)-form on Xcorresponding to a smooth metric on X. We write volµ(X) for the volume of Xwith respect to µ. In particular, we let µhyp denote the (1,1)-form corresponding to the hyperbolic metric on X, which is compatible with the complex structure of X, and has constant negative curvature equal to minus one. As a shorthand, we write vhyp for the hyperbolic volume volµhyp (X); we recall that vhyp = 4π(g−1). Let H0(X, Ω1 X) denote the C-vector space of holomorphic 1-forms equipped with the Petersson inner product given by hω, ηi=i 2Z X ω∧η¡ω, η ∈H0(X, Ω1 X)¢. By choosing an orthonormal basis {ω1, . . . , ωg}of H0(X, Ω1 X) with respect to the Petersson inner product, the canonical metric of Xhas the associated (1,1)-form µcan =1 g·i 2 g X j=1 ωj∧ωj. We note that the canonical metric measures the volume of Xto be one, and, furthermore, µcan is independent of the choice of orthonormal basis of H0(X, Ω1 X). We write ∆hyp for the hyperbolic Laplacian on X, which acts on smooth functions, and we let {φn}∞ n=0 denote an orthonormal basis of eigenfunctions of ∆hyp on Xwith corresponding eigenvalues 0 = λ0< λ1≤λ2≤. . . , i.e., ∆hypφn=λnφn(n= 0,1,2, . . .). 2.2. Green’s functions and Arakelov metrics. We denote the Green’s function associated to the metric µby gµ. It is a function on X×X 4 J. JORGENSON AND J. KRAMER characterized by the two properties dzdc zgµ(z, w) + δw(z) = µ(z) volµ(X), Z X gµ(z, w)µ(z) = 0 (w∈X). Assuming that z, w are points on X, which are sufficiently close, our convention for the Green’s function is such that the sum gµ(z, w) + log |z−w|2 is bounded as wapproaches z. By means of the function Gµ= exp(gµ), we now define the residual metric k·kµ,res on the canonical line bundle Ω1 X associated to µby kdzk2 µ,res = lim w→z¡Gµ(z, w)·|z−w|2¢. The (1,1)-form associated to the residual metric is then given by µres(z) = i 2·dz∧d¯z kdzk2 µ,res . In particular, following [1], we define the Arakelov metric form µAr to be the (1,1)-form associated to the residual metric of the canonical metric; the corresponding metric on Ω1 Xis denoted by k·kAr. The conformal factor φAr is the C∞-function on Xdefined by the equation µAr =eφAr µhyp . 2.3. Heat kernels and heat traces. The heat kernel KH(t;z, w) on H (t∈R>0;z, w ∈H) is given by the formula KH(t;z, w) = KH(t;ρ) = √2e−t/4 (4πt)3/2 ∞ Z ρ re−r2/4t pcosh(r)−cosh(ρ)dr , where ρ=dH(z, w) denotes the hyperbolic distance between zand w. Evidentially, KH(t;z, w)>0 for t > 0 and z, w ∈H. The heat kernel Khyp(t;z, w) associated to X(t∈R>0;z, w ∈X) is defined by averaging over the elements of Γ, namely Khyp(t;z, w) = X γ∈Γ KH(t;z, γw). For t > 0 and z∈X, we let HKhyp(t;z) = Khyp(t;z, z)−KH(t; 0) = X γ∈Γ γ6=id KH(t;z, γz). NON-COMPLETENESS OF THE ARAKELOV-INDUCED METRIC 5 Let us define the function F(z) = ∞ Z 0 µHKhyp(t;z)−1 vhyp ¶dt . (2) From standard results regarding the asymptotic behavior of the heat kernel for t→0 and t→ ∞, the integral defining Fconverges for all z∈X. Since Xis compact, the heat kernel admits the spectral expansion Khyp(t;z, w) = ∞ X n=0 φn(z)φn(w)e−λnt, which can be shown to converge uniformly and absolutely on X. The hyperbolic heat trace HTrKhyp(t) (t∈R>0) is now given by HTrKhyp(t) = Z X HKhyp(t;z)µhyp(z). The logarithmic derivative of the Selberg zeta function ZSel(s) associated to Xis defined by the Gauss transform of the hyperbolic heat trace, namely Z0 Sel ZSel (s) = (2s−1) ∞ Z 0 HTrKhyp(t)e−s(s−1)tdt , which converges for Re(s(s−1)) >0 and, when combined with the spectral expansion of the heat kernel, can be used to obtain the analytic continuation of ZSel(s). Of particular interest is the constant cSel = lim s→1µZ0 Sel ZSel (s)−1 s−1¶, which is analogous to the classical η-function as realized via the Kronecker limit formula. From [7], Lemma 4.2, we recall that one has the explicit formula for cSel in terms of the hyperbolic heat trace, namely cSel = 1 + ∞ Z 0 (HTrKhyp(t)−1) dt= ∞ Z 0 (HTrKhyp(t)−1 + e−t) dt . 2.4. Arakelov invariants using hyperbolic geometry. Numerous authors have studied the analytic invariants from the Arakelov theory of algebraic curves with the goal of obtaining expressions for the various quantities, including: the canonical metric, the canonical Green’s function, the Arakelov metric, and Faltings’s delta function. Beginning with [6] and [12], all authors relied on algebraic geometry and used the Riemann theta function as the fundamental object from which various identities were derived. 6 J. JORGENSON AND J. KRAMER In [9], new relations were derived relating the analytic invariants of the Arakelov theory of curves to the hyperbolic heat kernel. For the present paper, we will work with the following result, first proved in [9] (see also [8]). If Xis any compact Riemann surface of genus g > 1, then, in the notation of the previous section, we have φAr(z)=−4πµ1−1 g¶F(z)−π g2Z X F(z)∆hypF(z)µhyp(z)−cSel −1 g(g−1) −log(4) . (3) The proof of (3) follows from more fundamental formulas which relate µcan to µhyp and gcan to ghyp; these proofs are given in [8], [9], and will not be repeated here. Also, we note that other results from [9] yield explicit expressions for the canonical Green’s function, Faltings’s delta function, and the canonical metric form in terms of the hyperbolic heat kernel. 2.5. Degenerating families of Riemann surfaces. By a degenerating family of compact Riemann surfaces Xt(t∈D={t∈C||t|<1}), we mean a holomorphic map of Dinto the stably (Deligne-Mumford) compactified moduli space such that the restriction of this map to the punctured disc D\{0}is a holomorphic map into the moduli space Mg. The fiber over the origin t= 0 of the embedded disc is a noded algebraic curve. We refer the reader to [2] for an explicit construction of degenerating families of Riemann surfaces. Although the discussion in [2] focuses on degenerating families with a single developing node, the methodology immediately extends to the general setting of an arbitrary number of developing nodes. If each surface in the family Xtis given its unique hyperbolic metric, then we call the family a degenerating family of compact hyperbolic Riemann surfaces. The family can be parameterized either by the complex parameter t∈Dor the vector `of lengths of pinching hyperbolic geodesics. We refer the reader to [6], [10], and references therein for further discussion. 3. Asymptotic behavior of the Arakelov conformal factor In this section we estimate the terms in (3), which involve the hyperbolic heat kernel. 3.1. Lemma. For any compact Riemann surface Xof genus g > 1, we have the bound Z X F(z)∆hypF(z)µhyp(z)≥0. NON-COMPLETENESS OF THE ARAKELOV-INDUCED METRIC 7 Proof. Since Xis compact, we can expand Fin terms of the orthonormal basis of eigenfunctions {φn}∞ n=0 of ∆hyp with the non-negative eigenvalues {λn}∞ n=0, i.e., F(z) = ∞ X n=0 anφn(z), from which we derive ∆hypF(z) = ∞ X n=1 λnanφn(z), taking into account that λX,0= 0. Therefore, we have the equality Z X F(z)∆hypF(z)µhyp(z) = ∞ X n=1 λna2 n, from which the assertion in the statement of the lemma follows since λn≥0 for all n.¤ 3.2. Lemma. For any compact Riemann surface Xof genus g > 1, we have the bound F(z)≥ − 1 vhyp − ∞ Z 1 KH(t; 0) dt . Proof. If t≥1, we have the bound HKhyp(t;z)−1 vhyp =Khyp(t;z, z)−1 vhyp −KH(t; 0) = ∞ X n=1 φn(z)2e−λnt−KH(t; 0) ≥ −KH(t; 0) , which simply uses the positivity of the sum of squares of eigenfunctions. If 0< t ≤1, we trivially have HKhyp(t;z)−1 vhyp ≥ − 1 vhyp , which, using the positivity of KH(t;z, w), follows immediately from the positivity of HKhyp(t;z). With these bounds, the result immediately follows from the integral (2), which defines F(z). ¤ 8 J. JORGENSON AND J. KRAMER 3.3. Corollary. For any compact Riemann surface Xof genus g > 1, we have the bound log µµAr(z) µhyp(z)¶=φAr(z)≤4πµ1−1 g¶ ∞ Z 1 KH(t; 0) dt−cSel g(g−1)+1 g−1−log(4) . Proof. By combining the bounds in Lemma 3.1 and Lemma 3.2 with (3), we immediately arrive at the inequality φAr(z)≤4πµ1−1 g¶  1 vhyp + ∞ Z 1 KH(t; 0) dt −cSel −1 g(g−1) −log(4) . The bound follows by combining the constants, namely by showing that 4πµ1−1 g¶1 vhyp +1 g(g−1) =1 g−1, which comes from vhyp = 4π(g−1). ¤ 3.4. Lemma. Let Xtbe a degenerating family of compact hyperbolic Riemann surfaces of genus g > 1. Writing cSel,t for the constants cSel associated to Xt, we have lim t→0cSel,t =∞. Proof. The lemma is a direct consequence of [10], Theorem 4.5, p. 657, which we will briefly describe for the sake of completeness. Writing Xt= Γt\H, and given εgeo >0, resp. εev >0, we let Ggeo ={γ∈Γt|γ6= id,primitive, non-conjugate, `γ< εgeo},resp. Gev ={λn,t |λn,t eigenvalues of ∆hyp on Xt,0< λn,t < εev}; here `γdenotes the hyperbolic length of the primitive geodesic associated to γ∈Γt. Choosing εgeo sufficiently small, and εev <1/4, the above cited theorem of [10] gives the following asymptotic formula for the quantity cSel,t, as tapproaches zero: cSel,t = ∞ X n=1 X γ∈Ggeo `γ en`γ−1+X λn,t∈Gev 1 λn,t + Oεgeo,εev (1). We recall from [10] (and the references therein) that as tapproaches zero, each `γapproaches zero for γ∈ Ggeo. Now, there exists x0>0 such that NON-COMPLETENESS OF THE ARAKELOV-INDUCED METRIC 9 for 0 < x < x0, the inequality ex−1≤2xholds. With this in mind, we obtain the bound cSel,t ≥X γ∈Ggeo X n<x0/`γ 1 2n+X λn,t∈Gev 1 λn,t + Oεgeo,εev (1). Using the positivity of the eigenvalues and the divergence of the harmonic series, the proof of the lemma is now complete. ¤ 3.5. Theorem. Let Xtbe a degenerating family of compact hyperbolic Riemann surfaces of genus g > 1. Writing µhyp,t, resp. µAr,t for the (1,1)- forms µhyp, resp. µAr associated to Xt, we have lim t→0µsup z∈XtµµAr,t(z) µhyp,t(z)¶¶= 0 . Proof. Combine Corollary 3.3 and Lemma 3.4. ¤ 3.6. Remark. The analysis given in [6] and [12] do not seem sufficient to prove Theorem 3.5. More specifically, although the asymptotic behavior of µAr,t is established for points which are bounded away from the developing nodes, the results in [6] and [12] do not establish bounds for µAr,t/µhyp,t in neighborhoods of the developing nodes. As a consequence, it seems that Theorem 3.5 does not follow easily, if at all, from the results in [6] and [12]. 4. The Arakelov-induced metric on moduli space We show in this section how Theorem 3.5 can be used to study the metric on the moduli space Mgnear the Deligne-Mumford boundary. 4.1. Arakelov-induced metric on the moduli space. As before, let X denote a compact Riemann surface of genus g > 1. Then, the C-vector space H0(X, Ω2 X) of holomorphic quadratic differentials on Xhas dimension (3g− 3). Its dual space H0(X, Ω2 X)∨consists of harmonic Beltrami differentials; it represents the tangent space to the point corresponding to Xin the appropriate Teichm¨uller space Tg. The Arakelov-induced inner product h·,·i on H0(X, Ω2 X)∨is defined by hµ, νi=i 2Z X (µ∧ν)·µAr ¡µ, ν ∈H0(X, Ω2 X)∨¢. To be precise, the Arakelov metric µAr on Xinduces a metric on the Teichm¨uller space Tgof marked Riemann surfaces of genus g > 1; we use the term “marked” to denote that a canonical basis of the homology group H1(X, Z) has been chosen. However, the Arakelov metric is independent of