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!):

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

Then, respectively,

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.


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 \(t \to 0\).