Projective embedding of stably degenerating sequence of hyperbolic Riemann surfaces

Let $\mathcal{M}_{g}$ be the moduli of smooth compact Riemann surfaces of genus $g$. When $g\geq 2$, the Deligne-Mumford compactification [1] $\overline{\mathcal{M}_{g}}$ is the moduli of stable curves. Each smooth curve of genus $g$ carries an unique Poincaré metric with constant Gaussian curvature $-1$. If $C\in\overline{\mathcal{M}_{g}}$ is a singular stable curve, then by removing the nodes, the smooth part carries an unique complete hyperbolic metric with constant Gaussian curvature $-1$. And if a holomorphic family $\pi:\mathcal{C}\to{\mathbf{D}}$ of compact smooth curves $C_{t}$ degenerate to $C=C_{0}$, then the hyperbolic metrics is continuous on the vertical line bundle [12].

In this article, from the point view of the quantization framework by Donaldson [2, 3], we are interested in the convergence of the pluri-canonical Bergman embeddings of the family of hyperbolic surfaces in the complex projective spaces. More precisely, let $(C_{j},g_{j})$ be a sequence of genus $g\geq 2$ Riemann surfaces with Riemannian metric $g_{j}$ of constant Gaussian curvature $-1$, that converges, in the topology of pointed Gromov-Hausdorff, to a Punctured Riemann surface $(C_{0},g_{0})$(not necessarily connected) with a complete Riemannian metric $g_{0}$ of constant Gaussian curvature $-1$. Let $K_{C_{j}}$ denote the canonical bundle of $C_{j}$, then $K_{C_{j}}$ is endowed with a Hermitian metric $h_{j}$ defined by the Kähler form $\omega_{j}$ associated to $g_{j}$. We consider the Bergman space $\mathcal{H}_{j,k}$ consisting of $L^{2}$-integrable holomorphic sections of $K_{C_{j}}^{k}$. Then $\mathcal{H}_{j,k}$ is a finite-dimensional Hermitian space with the Hermitian product defined by

where, by abuse of notation, we still use $h_{j}$ to denote the induced Hermitian metric on $K_{C_{j}}^{k}$. For $k$ large enough, a basis of $\mathcal{H}_{j,k}$ will induce a Kodaira embedding of $C_{j}$ to ${\mathbb{C}}{\mathbb{P}}^{N_{k}},$ where $N_{k}=\dim\mathcal{H}_{j,k}-1$ is independent of $j\geq 1$. For $j=0$, the dimension of $\mathcal{H}_{j,k}$ is smaller than that of $j>0$. It is natural to consider the embedding induced by an orthonormal basis for $\mathcal{H}_{j,k}$, which can be considered as a bridge from Kähler geometry to complex geometry [4, 10]. It is worth mentioning that after this article is finished, the author learned that Dong[5] recently proved that if a smooth family of hyperelliptic curves degenerate to a nodal curve, then their Bergman kernels also converges to the Bergman kernel of the nodal curve.

As the Gaussian curvature is $-1$, the degeneration of metrics can only be ”pinching a nontrivial loop”, namely a sequence of surfaces with growingly thiner and longer handles, with the central loops degenerating to points. So $C_{0}$ has $d$ pairs of punctures, which will be called ends. And for $k$ large enough,the dimension of $\mathcal{H}_{0,k}$ equals $N_{k}+1-d$. Now we can state our main theorem.

It is interesting to mention that during the process of taking limit, the pair of linear ${\mathbb{C}}{\mathbb{P}}^{1}$’s are developed as a pair of bubbles. Also, we should mention that $k$ depends only on the geometry of $C_{0}$, and does not need to be too big by the results in [8].

The proof of this theorem makes heavy use of the methods we developed from [10] to [8]. And just as in [3], the main point is basically proving the convergence of the Bergman kernels. And we hope this result may shine a light on the study of the degeneration of higher dimensional projective manifolds [6, 7, 9].

The structure of this article is as follows. We will first quickly recall the necessary background for this article. Then we will calculate in the model for the thin handles, or ”the collar”, of the Riemann surfaces close to the limit. And in the end, we will finish the proof of the convergence of the pluri-canonical Bergman embeddings.

Acknowledgements. The author would like to thank Professor Song Sun for many very helpful discussions.

The model ${\mathbf{D}}^{*}$ with the Poincaré metric

Take the local section of the canonical bundle $e=\frac{dz}{z}$, the local potential is

We use the notation $\tau=-\log|z|$, so $\varphi_{P}=-\log(2\tau^{2})$. We are interested in the $L^{2}$-norm of the sections $z^{a}e^{k+1}$ of $K_{{\mathbf{D}}^{*}}^{k+1}$, $a\in{\mathbb{Z}}^{+}$. So we have the following integrals

We have

We denote by $g_{a}(\tau)=-2a\tau+2k\log\tau$, then $g_{a}^{\prime\prime}(t)=-\frac{2k}{\tau^{2}}$. So $g_{a}(\tau)$ is a concave function which attains its only maximum at $\tau_{a}=\frac{k}{a}$. We will use the following basic lemma from [8].

We can use Laplace’s method and the lemma above to estimate

Of course, we can directly calculate the integral to get

But the idea of mass concentration is key to our arguments. The Bergman kernel of ${\mathbf{D}}^{*}$ is then

Let $C_{0}$ be a punctured Riemann surface obtained by removing $2d$ points $\{p_{\alpha}\}_{1\leq\alpha\leq 2d}$ from a compact Riemann surface. $C_{0}$ is endowed with a complete Poincaré metric $\omega$ with constant Gaussian curvature $-1$. $\omega$ defines a Hermitian metric $h$ on the canonical bundle $K_{C_{0}}$. Then, for any positive integer $k$, we denote by $\mathcal{H}_{k}$ the space of holomorphic sections of $K_{C_{0}}^{k}$ that are $L^{2}$-integrable, namely

For each $p_{\alpha}$, there is a neighborhood $U_{\alpha}$ with local coordinate $z$ so that $\omega=\omega_{P}$ on $U_{\alpha}\backslash p_{\alpha}$. We can assume that $U_{\alpha}$ contains the points satisfying $|z|\leq R_{\alpha}$. We note that the injective radius at the points $|z|=R_{\alpha}$ is about $\frac{\pi}{4(\log R_{\alpha})^{2}}$. For simplicity, we let $R$ be the minimum of the $R_{\alpha}$’s, $1\leq\alpha\leq 2d$. Clearly, for the complement of $\cup_{1\leq\alpha\leq 2d}U_{\alpha}$ in $C_{0}$, there is a positive lower bound $\lambda_{0}$ for the injective radius. The basic conclusion of [8] is that for $k$ large enough, in the ”inside” of $U_{\alpha}$ where $\tau=-\log|z|>\sqrt{k+1}$, the Bergman kernel $\rho_{k+1}$ is very much like $\rho_{0,k+1}$, which is dominated by the terms $c_{a}|z|^{2a}$ for $a<k^{3/4}$. In particular, when $\tau\geq k$, $\rho_{0,k+1}$ is dominated by $c_{1}|z|^{2}$. The sections for $C_{0}$ corresponding to $z^{a}$ in the model ${\mathbf{D}}^{*}$ is constructed as follows. We let $z_{\alpha}$ denote the local coordinate $z$ on $U_{\alpha}$. Let $e_{\alpha}=\frac{dz_{\alpha}}{\alpha}$ be the local frame of $K_{C_{0}}$. Then $z_{\alpha}^{a}e_{\alpha}^{k+1}$, $a\geq 1$, are local sections of $K_{C_{0}}^{k+1}$. We choose and fix a cut-off function $\chi(r)$ that equals $1$ for $r<R/2$ and that equals 0 for $r>2R/3$. Then we denote by $\chi_{\alpha}$ the function $\chi(|z_{\alpha}|)$ defined on $C_{0}$. Then $\chi_{\alpha}z_{\alpha}^{a}e_{\alpha}^{k+1}$ is a global smooth $L^{2}$-integrable section of $K_{C_{0}}^{k+1}$. We then take the orthogonal projection of this section into the space $\mathcal{H}_{k+1}$, and then normalize the holomorphic section to be of norm 1, obtaining a section $s_{\alpha,a}\in\mathcal{H}_{k+1}$. We denote by

For $k$ large enough, within $r<R/4$, the sections $s_{\alpha,a}\approx\sqrt{c_{a}}z^{a}$ with relative error less than $\frac{1}{k^{2}}$.

We choose and fix an orthonormal basis $W_{0}=\{s_{j}\}$ for the orthogonal complement $V_{0}^{\perp}\subset\mathcal{H}_{k+1}$.

To obtain global sections of $L^{k}$ from local ones, we will need to use Hörmander’s $L^{2}$ estimate. The following lemma is well-known, see for example [11].

In the setting of this article, for a curve $C_{j}$, $j\geq 0$, with line bundle $K_{C_{j}}^{k+1}$, the constant is $k$, independent of $j$.

The model ${\mathbb{C}}^{*}_{\varepsilon}=({\mathbb{C}}^{*},\omega_{\varepsilon})$, where

where $f_{\varepsilon}$ is a function depending only on $|z|$, satisfying the following conditions

• $\bullet$

  $f_{\varepsilon}>0$;

• $\bullet$

  $f_{\varepsilon}(1)=\varepsilon^{2}$;

• $\bullet$

  $f^{\prime}_{\varepsilon}(1)=0$;

• $\bullet$

  $\Delta\log f_{\varepsilon}=\frac{2f_{\varepsilon}}{|z|^{2}}$;

Clearly, such $f_{\varepsilon}$ exists and is unique. Also, our choice of $f_{\varepsilon}$ makes the metric have constant Gaussian Curvature $-1$. We denote by $t=\log|z|$, then by abuse of the notation, we consider $f_{\varepsilon}$ as a function of $t$. Then we have

For simplicity, we will use $f(t)$ to denote $f_{\varepsilon}$. Therefore, we have

The first fundamental form of the metric is

Clearly, by the requirements on $f$, the circle $\{z||z|=1\}$ is a geodesic. Then we use the arc-length parameter $u$ for the $t$-curves. Then by the curvature condition, we have

and $\frac{dt}{du}=\frac{1}{\varepsilon\cosh u}$. Therefore, we have

So we have

Therefore when $\cosh u$ is large, we have the following estimation

So it is natural to use the notations

We also use the frame $e=\frac{dz}{z}$ for the canonical bundle, so we have

We are interested in the $L^{2}$-norm of the sections $z^{a}e^{k+1}$ of $K_{{\mathbb{C}}^{*}}^{k+1}$, $a\in{\mathbb{Z}}$. So we have the following integrals

We have

We denote by $g_{a}(t)=2at-k\log f$, then $g_{a}^{\prime\prime}(t)=-2kf(t)$. So $g_{a}(t)$ is a concave function which attains its only maximum at $t_{a}$. Write $u_{a}=u(t_{a})$, we have

We will assume that $\varepsilon$ is very small compared to $k^{-k}$. So $\sinh u_{a}=\frac{a}{k\varepsilon}$ is very large. So $f(t_{a})>\frac{a^{2}}{k^{2}}$, and we can use Laplace’s method and lemma 2.1 to estimate

And we have that the mass of $I_{\varepsilon,a}$ is concentrated within the neiborhood $\{|t-t_{a}|<\frac{\sqrt{k}\log k}{a}\}$ with relative error less than $k^{-\log k+3/2}$. Also, when $\varepsilon$ is small,

Therefore, the power series $\sum_{a>0}\frac{|z|^{2a}}{I_{\varepsilon,a}}$ is very close to a multiple of the power series

Recall that the power series in the expression of $\rho_{0,k+1}$ is also

So by the same argument in [8], for $t\in[0,t_{1}]$ the Bergman kernel is dominated by $[\frac{|z|^{2}}{I_{\varepsilon,1}}+\frac{1}{I_{\varepsilon,0}}](\frac{2}{f})^{k+1}$, and, by symmetry, for $t\in[t_{-1},0]$ the Bergman kernel is dominated by $[\frac{|z|^{-2}}{I_{\varepsilon,-1}}+\frac{1}{I_{\varepsilon,0}}](\frac{2}{f})^{k+1}$. In particular, we have the following

Assume $C_{j}$ converges to $C_{0}$ in the pointed Gromov-Hausdorff topology. For $j$ big enough, $C_{j}$ has exactly $d$ closed geodesics whose arc length is less than $\lambda_{0}/4$. We denote these circles by $\gamma_{j,\alpha}$, $1\leq\alpha\leq d$, and arc length of $\gamma_{j,\alpha}$ by $\varepsilon_{j,\alpha}$. By rearranging the points $p_{\alpha}$, we can assume that $2\pi\varepsilon_{j,\alpha}$ converges to the pair $(p_{\alpha},p_{\alpha+d})$ as $j\to\infty$. Also for $j$ large enough, there a neighborhood $U_{j,\alpha}$, usually referred to as a collar , of each $\gamma_{j,\alpha}$ which is homeomorphic to an annulus. We define a map

with $\varepsilon=\varepsilon_{j,\alpha}$, as following. Fix an isometry $\lambda$ of $\gamma_{j,\alpha}$ to the circle $|z|=1$ in ${\mathbb{C}}^{*}_{\varepsilon}$. Then passing through each point $q$ on $\gamma_{j,\alpha}$, there is an unique geodesic $l_{q}$ orthogonal to $\gamma_{j,\alpha}$. Then we define $h_{j,\alpha}$ to be the map that sends each such geodesic $l_{q}$ to the geodesic passing through $\lambda(q)$ and being orthogonal to the unit circle, preserving $\lambda$ and the orientation. Since both surfaces have constant Gaussian curvature $-1$, $h_{j,\alpha}$ is an isometry so long as the geodesics $l_{q}$ do not intersect each other. But since the curvature is negative, by the Gauss-Bonnet theorem, these geodesics can not intersect within $U_{j,\alpha}$. Therefore, $h_{j,\alpha}$ is also holomorphic. so we can use the coordinate $z$ from ${\mathbb{C}}^{*}_{\varepsilon}$ as the holomorphic coordinate of $U_{j,\alpha}$. By switching $p_{\alpha}$ and $p_{\alpha+d}$ if necessary, we can assume that the part $|z|>1$ of $U_{j,\alpha}$ converges to a neighborhood of $p_{\alpha}$ and the part $|z|<1$ to that of $p_{\alpha+d}$. We can assume that $U_{j,\alpha}=\{1/M\leq|z|\leq M\}$ and for $j$ large enough, we can assume that the injective radius at $|z|=M$ is larger than $\frac{\pi}{4(\log 3R/4)^{2}}$. We denote by $U_{j,\alpha}^{+}$ the part of $U_{j,\alpha}$ with $|z|>1$, similarly $U_{j,\alpha}^{-}$ the part with $|z|<1$. We then define a map

by sending $\varepsilon\cosh u$ to $\frac{1}{2\tau}$ while preserving the circles $\{u=\textit{constant}\}$. Clearly, we are only preserving the length of the circles. By symmetry, we also have

By our assumption on the injective radius, the image of $\varphi_{j,\alpha}$ contains the circle $|z_{\alpha}|=\frac{3R}{4}$. On $U_{\alpha}$, the first fundamental form is

So the pull back

is almost isometric to the metric

when $u$ is large. In particular, for the part where $\varepsilon\cosh u\geq\frac{-1}{\log\varepsilon}$, $\varphi_{j,\alpha}$ converges to an isometry when $j\to\infty$.

Let $U_{\alpha}(r)$ denote the subset of $U_{\alpha}$ consists of the points $|z_{\alpha}|<r$. Let $F=C_{0}\backslash\cup_{1\leq\alpha\leq 2d}U_{\alpha}(\frac{2R}{3})$, and let $\psi_{j}:F\to C_{j}$ be the diffeomorphism with its image. Since $\psi_{j}$ converges to an isometry as $j\to\infty$, we can glue $\psi_{j}^{-1}$ with the $\varphi_{j,\alpha}$’s, by rotating $\varphi_{j,\alpha}$ if necessary, for $j$ large enough, to get a map

with the following properties:

• $\bullet$

  $G_{j}$ is a diffeomorphism of $C_{j}\backslash\cup\gamma_{j,\alpha}$ with its image.

• $\bullet$

  $G_{j}=\varphi_{j,\alpha}$ for $p\in\varphi_{j,\alpha}^{-1}U_{\alpha}(\frac{2R}{3})$, $1\leq\alpha\leq 2d$.

• $\bullet$

  $G_{j}$ is almost an isometry on $C_{j}\backslash\cup_{1\leq\alpha\leq 2d}\varphi_{j,\alpha}^{-1}U_{\alpha}(\frac{2R}{3})$, and converges to an isometry when $j\to\infty$.

For any conformal metric, the compatible complex structure $J$ is just a counterclockwise rotation by $\frac{\pi}{2}$. We see that almost isometry implies almost holomorphic. Therefore $G_{j}^{-1*}K_{C_{j}}$ converges to $K_{C_{0}}$ as subbundles of $T_{C_{0}}\otimes{\mathbb{C}}$. More precisely, let $J_{j}$ be the complex structure compatible with the Riemannian metric $g_{j}$, if the point-wise norm

then we have

for some constant $\lambda$ independent of $p$ and $g$. We call the supremum above the pointwise distance from $J_{j}$ to $J$. Moreover, if $g_{j}$ converges to $g$ in $C^{2}$-norm, then $J_{j}$ converges to $J$ in $C^{2}$-norm also. If we denote by $T_{J}$ the holomorphic tangent space with respect to $J$, then the orthogonal projection of $T_{J_{j}}$ to $T_{J}$ is close to an isometry if $J_{j}$ is close to $J$. We identify $T_{J_{j}}$ with $T_{j}$ via this orthogonal projection, similarly $K_{j}=T_{J_{j}}^{*}$ with $K=T_{J}^{*}$, which we will also call an orthogonal projection, for simplicity. Since the metric on the canonical bundle is defined by the Kähler form $\omega$, and $\omega_{j}$ converges to $\omega$, we have that the Chern connection $\nabla_{j}$ on $K_{j}$ converges to the Chern connection $\nabla$ on $K$.

By assigning value 1 on $\gamma_{j,\alpha}$, we glue together the pull back $G_{j}^{*}\chi_{\alpha}$ and $G_{j}^{*}\chi_{\alpha+d}$ to get a function denoted by $\tilde{\chi}_{\alpha}$, for $1\leq\alpha\leq d$. On each $\varphi_{j,\alpha}$, we also consider $\tilde{\chi}_{\alpha}z^{a}$ as global smooth sections of $K^{k+1}_{C_{j}}$. Then we repeat the construction of $V_{0}$ by normalizing the orthogonal projection of $\tilde{\chi}_{\alpha}z^{a}$ onto $\mathcal{H}_{j,k+1}$, and denote the result section by $s_{j,\alpha,a}$, $|a|<k^{3/4}$. Then we denote by

We should remark here that the choice of the upper bound $k^{3/4}$ is not necessary, it is purely a habit from [10]. Notice that the number of sections of $V_{j}$ is larger than that of $V_{0}$ by the number $d$. Those extra sections are $\{s_{j,\alpha,0}\}_{1\leq\alpha\leq d}$. We consider $s_{j,\alpha,a}$ as a smooth section of $G_{j}^{-1*}K_{C_{j}}^{k+1}$ on $image(G_{j})$. We then define a piecewise smooth section $\tilde{s}_{j,\alpha,a}$ of $K_{C_{0}}^{k+1}$ which equals the orthogonal projection of $s_{j,\alpha,a}$ to $K_{C_{0}}^{k+1}$ on $image(G_{j})$, and equals 0 in the complement. For simplicity, we will say that $s_{j,\alpha,a}$ converges in some topology if $\tilde{s}_{j,\alpha,a}$ converges in that topology.