Gromov-Hausdorff limits of flat Riemannian surfaces

Posted on January 27, 2019 by Dima

tags: degenerations, weight function, Gromov-Hausdorff limits

Let $$X$$ be a Riemannian surface of genus $$g \geq 1$$. A holomorphic 1-form $$\Omega$$ on $$X$$ has $$2g-2$$ zeroes and putting $$\omega=\dfrac{i}{2}(\Omega \wedge \bar \Omega)$$ we obtain a Kähler metric on the complement of the zeroes of $$\Omega$$ (it is pseudo-Kähler viewed as a metric on the entire $$X$$, since $$\omega$$ is degenerate at at the points where $$\Omega$$ has zeroes). This metric is flat since picking local coordinates $$\operatorname{Re}z$$, $$\operatorname{Im}z$$, where $$z$$ is a local holomorphic coordinate, one observes that $$X$$ is locally isometric (away from zeroes of $$\Omega$$) to $$\mathbb{C}$$ with the flat metric.

Assume we now have a family of such surfaces $$X_t$$ where $$t \in B$$, and that we endow each $$X_t$$ with a pseudo-Kähler metric of the form $$\dfrac{i}{2}{\Omega_t \wedge \bar\Omega_t}$$ where $$\Omega$$ is a relative holomorphic 1-form on $$X \to B$$. As $$t$$ tends towards a point $$O \in B$$, the shape of the Riemannian manifold $$(X_t, \omega_t)$$ changes and we would like to understand if it tends towards some limit shape. More precisely, we will consider the Gromov-Hausdorff limit of $$X_t$$ as $$t \to O$$.

Let $$X, Y$$ be two subsets of a metric space $$Z$$, the the Hausdorff distance between $$X$$ and $$Y$$ is the infimum of positive numbers $$\epsilon$$ such that $$X$$ is contained in the $$\epsilon$$-neighbourhood of $$Y$$ (the union of open balls of raidius $$\epsilon$$ with the center in $$Y$$) and vice versa. The Gromov-Hausdorff distance between two metric spaces $$X$$ and $$Y$$ is the infimum of Hausdorff distances between $$X$$ and $$Y$$ over all possible isometric embeddings of $$X$$ and $$Y$$ into a third metric space $$Z$$.

Ultrapowers and Los theorem

An ultrafilter $$U$$ on a set $$A$$ is a collection of subsets closed under intersections and supersets such that for any subset $$I \subset A$$ either $$I \in U$$ or $$A \setminus I \in U$$. Consider a countable non-trivial ultra-power of $$\mathbb{R}$$, $${}^* \mathbb{R}$$. What it means is that we choose some ultrafilter $$U$$ on $$\mathbb{N}$$ that contains all cofinite sets (such an ultrafilter exists by the axiom of choice) and we consder the factor $$\prod_{i \in \mathbb{N}} \mathbb{R}/\sim$$ by the equivalence relation $(x_i) \sim (y_i) \textrm{ iff } \{ i \in \mathbb{N}\mid x_i=y_i \} \in U$ The quotient is a ring: we can apply the ring operations coordinate-wise and one checks that the equivalence class of the result does not depend on the representatives picked using the definition of the ultrafilter. Moreover, $$\mathbb{R}$$ admits a diagonal embedding into $${}^* \mathbb{R}$$; its image is called standard reals. Now one can observe that $${}^* \mathbb{R}$$ is actually a field. Indeed, let $$[(x_i)]$$ be some element of $${}^*\mathbb{R}$$ that is not equal to 0. Then there must be $$I \in U$$ such that for all $$i \in I$$, $$x_i \neq 0$$. Let $$y_i=x_i^{-1}$$ for $$i \in I$$ and let $$y_i$$ be anything for $$i \notin I$$. One then checks that $$(x_i \cdot y_i) \sim (1)$$ since $$\{ i \in \mathbb{N}\mid x_i y_i = 1\} \supset I$$ and ultrafilter is closed under supersets.

There is a more streamlined way of checking various properties of ultraproducts:

Theorem (Loś). Let $$\varphi(x_1, \ldots, x_n)$$ be a formula of first-order logic with free variables $$x_1, \ldots, x_n$$ (with $$n$$ possibly 0). Then $$\prod_U \mathbb{R}\models \varphi([(x^1_i)], \ldots, [(x^n_i)])$$ if and only if $$\{ i \in \mathbb{N}\mid \mathbb{R}\models \varphi(x^1_i, \ldots, x^n_i)\} \in U$$.

Thus the formula $$\forall x \exists y x \cdot y = 1$$ in the language of rings $$L_{ring} = (+, \times, 0, 1)$$ is true in $$\mathbb{R}$$ since it is a field, and by Los’ theorem it is also true in $${}^* \mathbb{R}$$.

A real closed field $$R$$ can be characterized via one of the following equivalent statements (due to Artin and Schreier, not to confuse with the Artin-Schreier theory of cyclic extensions of degree $$p$$ in positive charectristic!):

• $$[R^{alg}:R] < \infty$$ (equivalently, $$R^{alg} = R(\sqrt{-1})$$);
• the relation “$$x \sim y$$ iff there exists $$z$$ such that $$x - y = z^2$$” is a total order

It is a fun exercise (for someone starting out in model theory at least) to show that that the first condition can be expressed by countably many first-order formulas, while the second one translates quite straightforwardly to the formula $$\forall x \forall y \exists z (xy=z^2) \lor (xy=-z^2)$$. Either way, we conclude, by Los theorem, that $${}^* \mathbb{R}$$ is a real closed field.

Let $$\mathcal{O}$$ be the convex hull of $$\mathbb{R}$$ in $${}^* \mathbb{R}$$, or in other words, the union of all intervals $$[a,b]$$ in $${}^* \mathbb{R}$$ such that $$a,b \in \mathbb{R}$$. One can easily check that $$\mathcal{O}$$ is a value ring, indeed, for any $$x \in {}^* \mathbb{R}$$ if $$x \notin \mathcal{O}$$ then $$|x| > n$$ for any $$n \in \mathbb{N}$$, so $$|x^{-1}| < 1/n$$, so clearly $$x \in \mathcal{O}$$. The standard part map $$st: \mathcal{O}\to \mathbb{R}$$ maps elements of the form $$a + \epsilon$$ to $$a$$, where $$a\in \mathbb{R}$$, $$|\epsilon| < 1/n$$ for all $$n \in \mathbb{N}$$, is obviously a homomorphism of rings, and its kernel is the maximal ideal in $$\mathcal{O}$$. While its well-definedness is immediate (clearly, it is not the case that $$|a-b| < 1/n$$ if $$a \neq b$$ and $$a,b \in \mathbb{R}$$), to see that it is defined on all of $$\mathcal{O}$$, observe that if $$st$$ is not defined on $$x$$ then for any $$a \in \mathbb{R}$$ there $$|x-a| \geq 1/n$$ for some $$n$$, or in other words $$|x|$$ is bigger than any standard real. It follows that cannot belong to $$\mathcal{O}$$ since any elemnt of $$\mathcal{O}$$ is bounded by some standard real.

Furthermore, the quotient $${}^*\mathbb{R}^\times/\mathcal{O}^\times = {}^*\mathbb{R}^\times/(\mathcal{O}\setminus\mathfrak{m})$$ is non-canonically isomorphic to $$(\mathbb{R},+)$$. Indeed, pick an element $$t \in \mathfrak{m}$$ and construct the following map: $$v_t([(x_i)] = st [(\log_{t_i} |x_i|)]|$$. It is well-defined: indeed, for any $$x \in \mathcal{O}^\times=\mathcal{O}\setminus\mathfrak{m}$$, one checks that $$|x|^n < 1/t$$, since $$1/t$$ is bigger than any standard real by our choice. So $$[(\log_{t_i} |x_i|)] < 1/n$$ and hence belongs . The choice of $$t$$ only changes the scaling: $$v_{t'}(x) = v_t(x) / v_t(t')$$ for $$t'$$ distinct from $$t$$.

Let $${}^*\mathbb{C}$$ be the algebraic closure of $${}^*\mathbb{R}$$, then one checks that there is a unique valuation ring $$\mathcal{O}_{{}^* \mathbb{C}} = \{ x \in {}^* \mathbb{C}\mid |x| \in \mathcal{O}\} \subset \mathbb{C}$$ that extends $$\mathcal{O}\subset \mathbb{R}$$.

Gromov-Hausdroff limit via ultraproducts

Let $$f: (X,d) \to (Y,d'')$$ be a map between metric spaces, then the distortion of $$f$$ is $\mathrm{dist} f = \inf |d(x,y) - d'(f(x), f(y))|$ A subset $$Z$$ of a metric space $$X$$ is called an $$\epsilon$$-net if $$X$$ is equal to the $$\epsilon$$-neighbourhood of $$Z$$. A map $$f: X \to Y$$ is called $$\epsilon$$-isometry, if $$\mathrm{dist} f < \epsilon/2$$ and $$f(X)$$ is an $$\epsilon/2$$-net.

Lemma. Let $$f: X \to Y$$ be an $$\epsilon$$-isometry, then $$d_{GH}(X,Y) < \epsilon$$.

Let $$K_1:K$$ be a finite field extension; recall that for a variety $$X$$ over a field $$K_1$$ the Weil restriction of $$W_{K_1/K}(X)$$ is the variety $$Y$$ over $$K$$ that represents the functor $$Res_{K_1/K} X: K-Sch \to Sets$$: $$S \mapsto X(S \otimes_K K_1)$$, in particular, $$Y(K) \cong X(K_1)$$.

Let $$X \to B$$ be as above, and let $$X_{\mathbb{R}} \to B_{\mathbb{R}}$$ be the induced Weil restriction for the field extension $$[\mathbb{C}:\mathbb{R}]$$. Pick a generator $$t$$ of the maximal ideal of $$\mathcal{O}_{B,O}$$. Let $$\widehat{\mathcal{O}_{B,O}} \to \mathbb{C}((t))$$ be some isomorphism, and let $$\mathrm{Spec}\mathbb{C}((t)) \to B$$ be the induced morphism of schemes. Embed $$\mathbb{C}((t)) \hookrightarrow {}^* \mathbb{C}$$ so that $$\mathbb{C}[[t]] \subset \mathcal{O}_{{}^* \mathbb{C}}$$ and $$v(t)=1$$. For any $$\alpha \in \mathbb{C}$$ such that $$|\alpha|=1$$ consider the composition of isomorphism $$\widehat{\mathcal{O}_{B,O}} \xrightarrow{\sim} \mathbb{C}((t))$$ mentioned above with the automorphism of $$\mathbb{C}((t))$$ that sends $$t$$ to $$\alpha \cdot t$$, and with the fixed embedding $$\mathbb{C}((t)) \to {}^* \mathbb{C}$$, and call $$\eta^\alpha: \mathrm{Spec}{}^* \mathbb{C}\to B$$ the corresponding $${}^* \mathbb{C}$$-valued point of $$B$$,. Let $$\eta^\alpha_\mathbb{R}: {}^* \mathbb{R}\to B$$ be the $${}^* \mathbb{R}$$-valued points of $$B_\mathbb{R}$$ that correspond to $$\eta^\alpha$$ via the identification $$B({}^* \mathbb{C}) \cong B_{\mathbb{R}}({}^* \mathbb{R})$$. Denote respective fibres $$\overline{X}^\alpha$$ and $$\overline{X}^\alpha_{{}^* \mathbb{R}}$$ and note that $$\overline{X}^\alpha({}^* \mathbb{C})$$ is naturally identified with $$\overline{X}^\alpha_\mathbb{R}({}^* \mathbb{R})$$.

Recall that a semi-algebraic subset of a variety $$Z$$ defined over a real-closed field $$R$$ is a set of $$R$$-points of $$Z$$ that satisfy a boolean combination of polynomial equalities and inequalities (I refer to the Chapter 7 the book of Bochnak, Coste and Roy on real algebraic geometry for the necesseray foundations material). For our purposes it suffices to know that for a semi-algebraic subset $$O \subset Z$$ the notion of a set of $$R'$$-points of $$O$$ makes sense over any real-closed field extension $$R' \supset R$$, and that given a semi-algebaric set $$O \subset X_\mathbb{R}\to B_\mathbb{R}$$ and a map $$\mathrm{Spec}{}^* \mathbb{R}\to B_\mathbb{R}$$, one can define the semialgebraic subset $$\overline{O}= O \otimes_{\mathbb{R}} {}^* \mathbb{R}$$ (essentialy by substituting the corresponding variable in the polynomial equations and inequalities defining $$O$$ by a value in $${}^* \mathbb{R}$$, but in a more invariant way), a subset of $$\overline{X}_\mathbb{R}$$.

Let $$O \subset X_\mathbb{R}$$ be a semi-algebraic set such that its projection on $$B_\mathbb{R}$$ contains a punctured neighbourhood of $$O$$. Denote by $$\overline{O}^\alpha$$ the semi-algebraic subset of $$\overline{X}^\alpha$$ which is the fibre of $$O$$ over $$\eta_\alpha$$.

Lemma. Let $$U \subset \overline{X}^1$$ be a non-Archimedean semi-algebraic subset of $$\overline{X}$$, and denote its Galois conjugates $$U^\alpha \subset \overline{X}^\alpha$$. Let $$O \subset X_\mathbb{R}$$ be a semi-algebraic set such that $$\overline{O}^\alpha({}^* \mathbb{R}) \subset U^\alpha({}^* \mathbb{C}) \subset \overline{X}^\alpha({}^* \mathbb{C})$$ for all $$\alpha \in \mathbb{C}$$, $$|\alpha| = 1$$. Let $$W$$ be a Zariski open neighbourhood of $$X_O$$ such that $$W_\eta \supset U$$ and let $$f$$ be a regular function on $$W$$. Assume one of the following

• $$\inf_{x \in U} v(f(x)) > 0$$
• $$\inf_{x \in U} v(f(x)) \geq 0$$

Then, respectively,

• $$\sup_{x \in O_s} |f(x,s)| \to 0$$ as $$s \to 0$$
• $$\exists C_1\ \sup_{x \in O_s} |f(x,s)| \leq C_1$$ for $$|s|$$ sufficiently small

Proof. It is clear that the premises of the lemma hold also for the conjugates $$U^\alpha$$.

Assume that $$\inf_{x \in U} v(f(x)) > 0$$ but there exists $$\epsilon > 0$$ such that for all $$\delta > 0$$ there exists $$s_\delta$$, $$|s_\delta| < \delta$$ such that $$|f(x,s)| > \epsilon$$.

The formula $\varphi_{\epsilon,n} (s) = \exists x \in O_\mathbb{R}|f(x,s)| > \epsilon \land |s| < 1/n$ then is satisfiable for all values of $$n$$ and therefore, if $${}^* \mathbb{R}$$ is saturated enough, there exists a point $$s^* \in B({}^* \mathbb{R})$$ such that $$\varphi_{\epsilon,n}(s^*)$$ for all $$n \in \mathbb{N}$$. Without loss of generality we may assume then that $$(O_\mathbb{R})_{s^*}({}^* \mathbb{R}) \cong \overline{O}^\alpha({}^* \mathbb{R})$$ for some $$\alpha$$, and so there exists $$x \in \overline{O}^\alpha({}^* \mathbb{R})$$ such that $$|f(x)| > \epsilon$$. But then it would mean that $$v(f(x)) \leq 0$$ for some $$x \in U^\alpha$$ which contradicts the premise.

The second claim is proved similarly.

$$\Box$$.

Assume now that $$\sim$$ be a definable equivalence relation on $$X({}^* \mathbb{C})$$ that is locally given by $x \sim y \textrm{ iff } v(f(x)) = v(f(y))$ for some holomorphic function $$f$$. Assume that for any semi-algebraic set $$O$$ such that for any $${}^* \mathbb{C}$$-valued points $$x,y \in O$$, $$d(x,y) \to 0$$, and assume that the diameter is bounded as $$s \to 0$$.

Lemma. For any equivalence classes $$[x], [y] \in \overline{X}({}^* \mathbb{C})$$ the limit $$d_s(x_s, y_s)$$ as $$x_s \in [x], y_s \in [y]$$ and $$s \to 0$$ is well-defined.

Let the limit function on $$\overline{X}/\sim$$ be $$\bar d$$.

Proposition. The metric space $$(\overline{X}/\sim, \operatorname{dd})$$ is the Gromov-Hausdorff limit of $$X_s$$ as $$s \to 0$$.

Proof sketch. For any $$\epsilon$$, consider an $$\epsilon/2$$-net $$F \subset \overline{X}$$ so that $$d(F, \overline{X}/\sim) < \epsilon/2$$. We need to show that $$d(X_s, F) < \epsilon/2$$ for $$s$$ sufficiently small.

Let $$F=\{F_1, \ldots, F_n\}$$. Pick semialgebraic sets $$W_i \subset X$$ such that $$W_i \cap \overline{X}\subset [F_i]$$, then for points $$x \in W_i, y \in W_j$$, $$d(x,y) \to d(F_i, F_j)$$ as $$s \to 0$$. Therefore, for $$|s|$$ small enough $$dist(X_s, F) < \epsilon/2$$.

Berkovich spaces, dual comlpexes and weight function

In order to describe the limit we will need some background on the geometry of Berkovich spaces and curves in particular.

Let $$K$$ be a field complete with respect ot a non-Archimedean absolute value $$|\cdot|$$, and let $$R = \{x \in K \mid |x| \leq 1\}$$ be the value ring. Given a variety $$X/K$$ we can construct the Berkovich analytification $X^{an} = \{ (x, ||\cdot|| \mid x \in X, ||\cdot||: K(x) \to \mathbb{R}\}$ (here $$x$$ is the schematic point, $$||\cdot||$$ is a multiplicative semi-norm on the residue field, where semi- means that it is allowed to take zero value on non-zero elementns of the field. The topology on $$X^{an}$$ is the weakest such that the evaluation maps $$|f|: U^{an} \to \mathbb{R}, (x, ||\cdot||_\xi) \mapsto ||f(x)||_\xi$$ (for $$f \in H^0(U, \mathcal{O}_X)$$ for all Zariski opens $$U \subset X$$) are continuous.

A model $$Y$$ of $$X$$ is a flat $$R$$-scheme such that $$Y \otimes_R K \cong X$$. We are going to define the specialization map $$sp: X^{an} \to Y_s$$ (denotng by $$Y_s$$ the fibre of $$Y$$ over the closed point of $$R$$) to be the map that sends the point $$\xi=(x, ||\cdot||) \in X^{an}$$ to the point of $$Y_s$$ designated by the morphism $$(x, ||\cdot||)$$ in the diagram below. In this diagram the morphism from the residue field of the point $$\xi$$ to $$X$$ is lifted to the morphism from $$\mathrm{Spec}\mathcal{H}(xi)$$ (such a lift exists if $$X$$ is proper by the valuative criterion of properness), and the image of the closed point of this scheme is $$\operatorname{sp}(\xi)$$. $\begin{array}{ccc} \mathrm{Spec}\ \mathcal{H}(\xi) & \to^\xi & X \\ \downarrow & & \downarrow \\ \mathrm{Spec}\ \mathcal{H}(\xi)^\circ & - \to^{\xi^\circ} & Y \\ \uparrow & \nearrow_{\operatorname{sp}(\xi)} & \uparrow\\ \mathrm{Spec}\ \tilde{\mathcal{H}}(\xi) & \to & Y_s \\ \end{array}$

Assuming that $$Y$$ is a model of $$X$$ and $$Y_s = \sum_{i=1}^n N_i Y_i$$, and $$\eta_1, \ldots, \eta_n$$ are the generic points of the irreducible components $$Y_1, \ldots, Y_n$$, the points $$\operatorname{sp}^{-1}(\eta_1), \ldots \operatorname{sp}^{-1}(\eta_n)$$ are the verticee of the dual intersection complex of $$Y_s$$ naturally embedded into $$X^{an}$$. The image of the dual intersection complex is called the skeleton of $$Y$$, $$\operatorname{Sk}(Y)$$. Moreover, if $$Y_s$$ is a strictly normal crossing divisor then there exists deformation retraction of $$X^{an}$$ onto $$\operatorname{Sk}(Y)$$.

If $$X$$ is a curve over an algebraically closed non-Archimedean field $$K$$ then by semi-stable reduction theorem there always exists an model $$Y$$ of $$X$$ with the snc special fibre, and $$\Sigma_Y = \{\operatorname{sp}^{-1}(\eta_1), \ldots \operatorname{sp}^{-1}(\eta_n)\}$$ form a semi-stable vertex set, that is, $X^{an} = \bigsqcup_{i=1}^n A_i \sqcup \bigsqcup_{j=1}^m B_j$ where $$A_i \cong \{ x \in (\mathbb{A}^1)^{an} \mid r_i < |x| < s_i \}$$ and $$B_j = \{x\in (\mathbb{A}^1)^{an} | |x| < s_j \}$$. The skeleton of $$Y$$ can be described as $\operatorname{Sk}(Y) = \Sigma_Y \sqcup \bigsqcup \operatorname{Sk}(A_i)$ where $$\operatorname{Sk}(A_i)$$ are defined as follows.

The points of an anulus are classified into the so-called points of types I, II, III and IV (and if $$K$$ has a property of being “spherically complete”, there is no type IV points). Points of type I-III are in bijective corespondence with closed balls in $$(\mathbb{A}^1)^{an}$$: $||f||_{\xi_{x_0,r}} = \max_{||x-x_0|| \leq r} |f(x)|$ The skeleton of $$A_i$$ is defined to be the setof opoints $$\{\xi_{0,\rho}\}_{r < \rho < s}$$. There is a natural metric on $$A_i$$ in which $$\operatorname{Sk}(A_i)$$ is an interal of lengh $$\log r - \log s$$.

Shape of the limit

If $$Y$$ is an snc odel of $$X$$ and $$\eta \in Y_s$$ is the generic point of a componont divisor, $$\operatorname{sp}^{-1}(\eta)$$ is called the divisorial point. Such points are dense in $$X^{an}$$. Weight functio, associated to a (pluri-)canonocial form $$\Omega$$ is defined on divisorial points as follows: $\operatorname{wt}_{\Omega}(\operatorname{sp}^{-1}(\eta_i) = (1 + \operatorname{ord}_{Y_i}(\operatorname{div}_Y(\Omega))/N_i$ (where $$Y_s = \sum N_i Y_i$$ is the decomposition of the central fibre).

Theorem (Nicaise-Xu) This definition does not depend on the partucular model $$Y$$.

The function $$\operatorname{wt}_\Omega$$ is extended to the whole of $$X^{an}$$ by continuity.

A different definition, due to Temkin is as follows $\operatorname{wt}_\Omega(x) = 1 - \log \inf_{\Omega_{\mathcal{H}(x)} = \sum a_i db_i} \max_i |a_i| |b_i|$ where $$\Omega_{\mathcal{H}(x)}$$ is the image of $$\Omega$$ in the Kahler module of the residue field $$\mathcal{H}(x)$$.

On the unit disc one can observe that $$\operatorname{wt}_dx(x) = 1 - \log r(x)$$, where $$r(x)$$ is the radius function, i.e. the radius of the smallest ball containing $$x$$: $r(x) = \inf_{|x-x_0| < \rho} \rho$ For what follows, denote the minimality locus of $$\operatorname{wt}_\Omega$$ as $$\operatorname{Sk}_\Omega(X)$$.

Theorem (Nicaise-Xu, Temkin) $$\operatorname{Sk}_\Omega(X) \subset \operatorname{Sk}(Y)$$ for any snc model $$Y$$.

Now back to our problem, describing the Gromov-Hausdorff limit of a degeneration of curves $$X$$ with the metric cooked up from the 1-form $$\Omega$$. Pick an snc model $$Y$$ such that $$\operatorname{div}(\Omega)$$ does not intersect the nodes of $$Y_s$$ an define the equivalence relation on $$\operatorname{Sk}(Y) \subset X^{an}$$: $x \sim y \textrm{ iff } \exists \textrm{ path } \gamma: [0,1] \to \operatorname{Sk}(X) \textrm{ such that } \gamma^{-1}(\operatorname{Sk}_\Omega(X))$ For all $$\operatorname{Sk}(A_i) \subset \operatorname{Sk}(Y)$$ define $d_i = \left\{\begin{array}{ll} |c| & \textrm{ if } \Omega = c t^k y_i^{-1} (1+f_i) dy_i, k=\min_{x\in X^{an}} \operatorname{wt}_\Omega(x)\\ 0 & \textrm{ otherwise } \end{array} \right.$ and consider $$\operatorname{Sk}(Y)/\sim$$ with the standad metric with edge lengths multiplied by $$d_i$$.

Theorem (S.) The metric graph $$\operatorname{Sk}(Y)/\sim$$ (with the metric normalized so that its diameter is 1) is the Gromov-Hausdorff limit of $$(\overline{X}, i/2 (\Omega_t \wedge \bar\Omega_t)$$ as $$t \to 0$$.