Masur’s Divergence for Higher Dimensional Tori and Kummer Surfaces

Given a surface $S_{g}$ with genus $g>1$, its Teichmüller space $\mathrm{Teich}(S_{g})$ is defined as complex structures on $S_{g}$ up to isotopy, which is equivalent to the space of hyperbolic metrics on $S_{g}$ up to isotopy. The Teichmüller distance on the Teichmüller space is given by the logarithmic of the infimum dilatation among maps between the two complex structures: $d_{Teich}(X,Y)=\inf_{f:X\to Y}\log K_{f}$ where the dilatation $K_{f}$ of a map $f$ is given by $K_{f}=\sup_{p\in X}\frac{|f_{z}|+|f_{\bar{z}}|}{|f_{z}|-|f_{\bar{z}}|}$ which at every point describes the ratio of the major and minor axes of the ellipse given by the image of infinitesimal circles under $f$. The Teichmüller distance roughly measures how much two conformal structures are different.

Teichmüller showed that for every diffeomorphism on $S_{g}$, there is a unique map (differentiable outside finitely many points) in its isotopy class, called the Teichmüller map, minimizing the dilatation, thus achieving the Teichmüller distance.

The Teichmüller map is described explicitly by holomorphic quadratic differentials. For a Teichmüller map $f:X\to Y$, there are two holomorphic quadratic differentials $q$ on $X$ and $p$ on $Y$, such that under the natural coordinates of $q$ and $p$ that writes $q=dz^{2}$ and $p=dw^{2}$, $f$ is of the form $\Re w=\sqrt{K_{f}}\Re z,\Im w=\frac{1}{\sqrt{K_{f}}}\Im z$. Given a quadratic differential $q$ on $X$, taking the horizontal foliations and vertical foliations to be parallel to the $x$ and $y$ axes in the natural coordinates of $q$ and $p$, we see that the Teichmüller map simply contracts the vertical foliations and expands the horizontal foliations.

The Teichmüller flow is given as the geodesic flow on the Teichmüller space under the Teichmüller distance. We also call it the geodesic flow on Teichmüller space and denote it by $Ra_{t}(X,q)$ where the quadratic differential $q$ is identified with the cotangent space of $\mathrm{Teich}(S_{g})$. The Teichmüller flow gives a set of complete geodesics on the Teichmüller space, whose end points can be identified with the horizontal and vertical foliations of the quadratic differentials.

The moduli space of $S_{g}$ is defined as the complex structures on $S_{g}$ up to diffeomorphisms, so it is a quotient of the Teichmüller space by the mapping class group $\Gamma=\mathrm{Diff}(S_{g})/\mathrm{Diff}_{0}(S_{g})$ where $\mathrm{Diff}_{0}(S_{g})$ is the group of diffeomorphisms on $S_{g}$ and $\mathrm{Diff}_{0}(S_{g})$ is the identity component of the diffeomorphism group, or diffeomorhpisms of $S_{g}$ isotopic to identity. The mapping class group acts properly discontinuously on the Teichmüller space, so the geodesic flows descends to the moduli space.

Masur’s divergence established a relationship between unique ergodicity of foliations and the dynamics of the geodesic flow on the moduli space.

Recently, there have been many interesting results making analogies between Riemann surfaces and higher dimensional objects. For example, Cantat [Can14] established an analogy of stable and unstable foliations for pseudo-Anosov mappings, proving that that there exists stable and unstable currents contracted and expanded by the hyperbolic automorphisms of K3 surface. Filip [Fil20] made an analogy between periodic billiard trajectories on translation surfaces and special Lagrangian tori on K3 surfaces and counted the growth of the number of special Lagrangian tori in a twistor family.

In this article, we are interested in extending the definitions and properties of foliations and the Teichmüller flow to higher dimensional subjects. We start with the case of complex tori.

Unlike the case of Riemann surfaces, for higher dimensional objects, the space of complex structures up to isotopy and the space of flat (hyperbolic) metrics up to isotopy are different objects. For the complex torus, and hyperkähler manifolds (including K3 surfaces), the moduli space of complex structures is even highly non-Hausdorff. In fact, almost all complex structures are ergodic in the moduli space [Ver15]. Thus to study dynamics on the moduli space and to achieve geometric results related to the dynamics, the non-Hausdorff complex moduli space may not be the appropriate analogy.

In this article, we consider instead the moduli space of Kähler metrics on a differentiable manifold $M$ which is the set of equivalent Kähler structures of $M$ up to isotopy. We define this as

where $(X,I,\omega)$ are taken over Kähler manifolds $X$ equipped with complex structure $I$ and Kähler form $\omega$ such that $X$ is diffeomorphic to $M$; the equivalence relation is defined as $(X,I,\omega)\sim(Y,J,\nu)$ if there exists a homeomorphism $f:X\to Y$ such that $f^{*}J=I$ and $[f^{*}\nu]=[\omega]\in H^{2}(X,\mathbb{R})$.

For a complex torus, up to an isometry of Kähler structures, we may view the complex torus as $\mathbb{C}^{n}/\Lambda$ for a lattice $\Lambda$ of $\mathbb{C}^{n}$ with the standard Kähler metric $\omega_{0}=\sum_{i=1}^{n}dx_{i}\wedge dy_{i}$ on $\mathbb{C}^{n}$. The moduli space of Kähler metrics of a $2n$-torus $\mathbb{T}^{2n}$ is $\mathrm{Kah}(\mathbb{T}^{2n})=\mathrm{SL}(2n,\mathbb{Z})\backslash\mathrm{SL}(2n,\mathbb{R})/\mathrm{U}(n)$.

The group $\mathrm{SL}(2n,\mathbb{R})$ acts on $\mathrm{SL}(2n,\mathbb{Z})\backslash\mathrm{SL}(2n,\mathbb{R})$ by right multiplication. This descends to the moduli space of Kähler structures of complex torus. More concretely, for a torus $X=\mathbb{C}^{n}/\Lambda(g)$, we identify the lattice $\Lambda(g)$ with a matrix $g\in\mathrm{SL}(2n,\mathbb{R})$ by taking a basis of $\Lambda(g)$ to be the row vectors of $g$ and identifying $\mathbb{C}^{n}$ with $\mathbb{R}^{2n}$ by $(z_{1}=x_{1}+iy_{2},...,z_{n}=x_{n}+iy_{n})\mapsto(x_{1},y_{1},...,x_{n},y_{n})$.

We define the geodesic flow $Ra_{t}$ on the moduli space to be image of the action of $a_{t}=\mathrm{diag}(e^{-t},e^{t},...,e^{-t},e^{t})$ on $\mathrm{SL}(2n,\mathbb{Z})\backslash\mathrm{SL}(2n,\mathbb{R})$ in the moduli space $\mathrm{Kah}(\mathbb{T}^{2n})$, given as $Ra_{t}(\mathbb{C}^{n}/\Lambda(g))=\mathbb{C}^{n}/\Lambda(ga_{t})$. The geodesic flow is said to be recurrent if it visits a compact subset of the moduli space infinitely many times.

We define the horizontal and vertical foliations on $\mathbb{C}^{n}/\Lambda$ to be $n$-dimensional foliations on the torus whose leaves are the subspaces in $\mathbb{C}^{n}$ parallel to the $x$-axis and $y$-axis respectively. A foliation is said to be uniquely ergodic if it only has one transverse measure up to a non-zero scalar multiplication. This is roughly saying that the foliation is equally distributed on the manifold. In this way, the geodesic flow acts on the torus $\mathbb{C}^{n}/\Lambda$ by contracting the horizontal foliations and expanding the vertical foliations.

Then we have the following analogy for Masur’s criterion.

This illustrates an interesting relationship between dynamical property of the geodesic flow on the geometric property of foliations on $X$.

We remark that there are, of course, many ways to define geodesic flows on $\mathrm{SL}(2n,\mathbb{R})$, and we choose $a_{t}=\mathrm{diag}(e^{-t},e^{t},...,e^{-t},e^{t})$ so that it acts on foliations similar to the way Teichmüller flow does for complex dimension 1. In fact, any action with exactly $n$ eigenvalues larger than 1 and $n$ eigenvalues smaller than 1 would have similar properties, one simply needs to define the horizontal and vertical foliations to be the respective eigen-spaces.

For real dimension $2$ where $X=\mathbb{T}^{2}$, Masur’s divergence is simply given by the fact that non-rational foliations are uniquely ergodic. (If a foliation is not uniquely ergodic, then a geodesic curve of $X$ must be a leaf of the foliation, so the length of the geodesic tends to zero along the geodesic flow $Ra_{t}X$ which implies that $Ra_{t}X$ must be divergent. ) For higher dimensions, the failure of unique ergodicity of the horizontal foliation does not imply the existence of a closed leaf of the foliation, so Theorem 1.2 cannot be simply deduced by the fact (See Theorem 4.1) that non-rational straight line flows on torus are uniquely ergodic.

Furthermore, the converse of Theorem 1.2 is not true for higher dimensional complex tori, as is the case for Riemann manifolds. For example, on the torus $\mathbb{C}^{2}/\Lambda(g)$ where $\Lambda(g)$ is a lattice in $\mathbb{C}^{2}$ spanned by the row vectors of $g=\begin{pmatrix}1&0\\ i&0\\ \sqrt{3}+\sqrt{5}i&1+\sqrt{2}i\\ 0&i\end{pmatrix}$, where $i=\sqrt{-1}$, the horizontal foliation is uniquely ergodic but the geodesic flow starting at $g$ is divergent. For proof see Proposition 4.4.

We next extend the result to other complex manifolds. The simplest extension is to Kummer surfaces. A Kummer surface is defined as the minimal resolution of a complex torus $X$ mod involution. More explicitly, taking $\tau$ to be the involution on $\mathbb{C}^{2}/\Lambda(g)$ descending from the inversion map $(z_{1},z_{2})\to(-z_{1},-z_{2})$ on $\mathbb{C}^{2}$, the Kummer surface $\mathrm{Kum}(g)$ is given by the blow-up of $(\mathbb{C}^{2}/\Lambda(g))/\tau$ at the 16 fixed points of $\tau$. We denote by $C_{j},j=1,...,16$ the $16$ exceptional curves given by the blow-ups of the fixed points. The orbifold $(\mathbb{C}^{2}/\Lambda(g))/\tau$ before blow-up is called the exceptional Kummer surface. Kummer surfaces are examples of K3 surfaces.

For Kummer surfaces, we consider the moduli space of degenerate Kähler forms, i.e. Kähler forms defined outside some exceptional curves, denoted by $\overline{\mathrm{Kah}}(M_{0})$ (see Section 5 for more details on this) where $M_{0}$ is the differentiable manifold underlying Kummer surfaces. Moreover, we define the geodesic flow in the moduli space of Kummer surfaces to be given by that of the torus, i.e. $Ra_{t}\mathrm{Kum}(g)=\mathrm{Kum}(ga_{t})$. We define the horizontal foliation and vertical foliation of a Kummer surface to be the image of the horizontal and vertical foliations of the torus.

Thus using our result for torus, we also establish the analogy of Masur’s divergence to Kummer surfaces.

The moduli space of degenerate Kähler metrics of a Kummer surface, which is the same as the moduli space of degenerate Kähler metrics of K3 surfaces, can be explicitly calculated via the period map, using the Torelli theorem.

We also have the following necessary condition for a horizontal foliation on the Kummer surface to be uniquely ergodic.

We remark that the cohomology classes $\eta_{1}$, $\eta_{2}$ are on the boundary of the period domain and our result suggests that they can be thought of as representing the vertical and horizontal foliations. This gives an analogy of the identification of the boundary of Teichmüller space with horizontal and vertical foliations.

The form of the geodesic flow seems to suggest a generalization to general K3 surfaces. However, in [McM02] McMullen constructed examples of hyperbolic automorphisms of K3 surfaces that admits Siegel disks. The work of Cantat and Dupont [CD20] (for the projective case) and Filip and Tosatti [FT19] (for the general case) on Kummer rigidity also suggests that foliation preserved by automorphisms of K3 surfaces cannot have full Lebesgue measure except for the case of Kummer surfaces we discussed in this article. Thus, it may be too optimistic to hope for horizontal and vertical foliations on general K3, even outside divisors. Nevertheless, these results do not rule out the possibility of geodesic flows not containing automorphisms to have full measure horizontal and vertical foliations. Moreover, if we hope the geodesic flows to contain automorphism as in translation surfaces, there is still hope of developing the theory for laminations instead of foliations to be preserved by “geodesic flows”.

In Section 2 we recall the preliminaries on moduli spaces. In Section 3 we define the geodesic flow and foliations for complex torus. In Section 4 we prove Theorem 1.2 for torus and discuss the converse. In Section 5, we briefly recall the definitions and properties of K3 surfaces and Kummer surfaces, especially the period map and the generalized moduli spaces. In Section 6, we apply the results on torus to Kummer surfaces and calculate explicit form and properties of the geodesic flow.

First we briefly introduce the definition of measured foliations.

We shall be interested in very nice foliations, namely the horizontal and vertical foliations on $\mathbb{T}^{2n}$, they are Lagrangian, or even special Lagrangian foliations. First we briefly introduce the definitions (See section 7, part 1 of [GHJ03] for detailed definition and examples on this).

Now we define the horizontal and vertical foliations on complex tori in the following way. This is motivated by the definition of horizontal and vertical foliations on Riemann surfaces which is given by the natural coordinates.

We consider the universal covering of the torus. For $g=(g_{ij})_{1\leq i,j\leq 2n}\in\mathrm{SL}(2n,\mathbb{R})$ we take $\Lambda(g)$ to be the lattice in $\mathbb{C}^{n}$ given by the row vectors of

where $i=\sqrt{-1}$. We take $\mathbb{C}^{n}\to\mathbb{C}^{n}/\Lambda(g)=X$ the universal covering of the complex torus $X$ given by $g$. The leaves of the horizontal foliation $\mathcal{F}^{h}$ on $X$ given by $g$ are the images of $\{(x_{1}+iy_{1},...,x_{n}+iy_{n})\in\mathbb{C}^{2}:y_{1}=c_{1},...,y_{n}=c_{n}\}\subset\mathbb{C}^{n}$ in $X$ and the transverse measure is given by $d\mu_{h}=dy_{1}\wedge...\wedge dy_{n}$. Respectively, the vertical foliation $\mathcal{F}^{v}$ is given by $\{(x_{1}+iy_{1},...,x_{n}+iy_{n})\in\mathbb{C}^{2}:x_{1}=c_{1},...,x_{n}=c_{n}\}$ in $X$ with transverse measure $d\mu_{v}=dx_{1}\wedge...\wedge dx_{n}$. The horizontal and vertical foliations are transverse Lagrangian foliations on $X$, where the Kähler form is given by $\omega_{0}$ where $\omega_{0}=\sum dx_{i}\wedge dy_{i}$ is the standard one on $\mathbb{C}^{n}$.

We note that the horizontal and vertical foliations are taken with respect to $g\in\mathrm{SL}(2n,\mathbb{R})$. If we have a complex torus $X$ with a flat Kähler metric $\omega$ to begin with and require that $(X,\omega)=(\mathbb{C}^{n}/\Lambda(g),\omega_{0})$, then there is a $\mathrm{U}(n)$ set of possible choices for $g$ and a $\mathrm{U}(n)/\mathrm{O}(n)$ set of choices for the horizontal and vertical foliations. In fact, in the case $n=2$, every linear special Lagrangian foliation in a flat Kähler torus can be given as the horizontal foliation for some choice of $g\in\mathrm{SL}(2n,\mathbb{R})$ (c.f. Proposition 6.1).

The geodesic flow is given as expansion of the vertical foliations and contraction of the horizontal foliations. This is motivated by the action of Teichmüller flow on natural coordinates. We may also write this more explicitly in the language of a homogeneous flow.

Take $X=\mathbb{C}^{n}/\Lambda(g)$ and consider the horizontal and vertical foliations of $X$ given by $\mathbb{C}^{n}\to X$, we see that the geodesic flow $R_{a_{t}}$ preserves the horizontal and vertical foliations and while contracting and expanding the transverse measures.

We may project the geodesic flow down to the moduli spaces of the torus by projecting $\mathrm{SL}(2n,\mathbb{Z})\backslash\mathrm{SL}(2n,\mathbb{R})$ down to the Kähler moduli space $\mathrm{Kah}(\mathbb{T}^{2n})\simeq\mathrm{SL}(2n,\mathbb{Z})\backslash\mathrm{SL}(2n,\mathbb{R})/\mathrm{U}(n)$, the Kähler moduli space of complex torus. More explicitly, this is given by the right multiplication $Ra_{t}((\mathbb{C}^{n},\omega_{0})/\Lambda(g))=(\mathbb{C}^{n},\omega_{0})/\Lambda(ga_{t})$, where $\omega_{0}$ is the standard Kähler form on $\mathbb{C}^{n}$.

Here we introduce the definition and basic properties and Kummer surface and K3 surface, including the Torelli theorem and the denseness of Kummer surface in the complex moduli space of K3.

From now on we denote the differentiable manifold underlying a K3 surface by $M_{0}$. For any K3 surface $X$, we define a marking $f$ of $X$ to be a diffeomorphism $f:M_{0}\to X$. A marking $f$ identifies the homology space $H_{2}(X,\mathbb{Z})$ with $H_{2}(M_{0},\mathbb{Z})$ through $f_{*}$.

The relation between the period domain, the Teichmüller space and the moduli space of complex structures of K3 surfaces is described by the Torelli theorem.

Noting that mapping class group is generated by change of markings on the Teichmüller space, the Torelli theorem give us the following description of the moduli space.

If we add the Kähler structure into consideration, the Kähler class and the Hodge structure together determine a positive 3-plane in the real cohomology space together with a direction in it specifying the Kähler class. So we have a map from the moduli space of Kähler structures to the space of positive 3-Grassmannian with a specified unit vector up to change of marking. We write this as $\mathrm{Kah}(K3)\to\Gamma\backslash SO(3,19,\mathbb{R})/(SO(2)\times SO(19))$.

However, this map is not surjective since a Kähler class cannot have zero intersection with curves. One way to solve this is to allow for degenerate Kähler forms, i.e. non-degenerate symplectic 2-forms that are in the positive cone but on the boundaries of the Kähler cone.

The following proposition shows that after adding in the degenerate Kähler forms, every positive 3-plane in the real cohomology space can indeed be realized by some K3 surface.

Adding the change of markings into consideration, we have the following.

This gives a homogeneous description of the moduli space of generalized Kähler froms of K3 surfaces. Note that this space is Hausdorff while the complex moduli space of K3 surfaces is not. Elements in the generalized period domain $\mathrm{KPer}$ leaves the compact sets exactly when the positive $3$-plane approaches the light cone in the real cohomology space.

Kummer surfaces is a special class of K3 surfaces with a very concrete construction.

Indeed, we may check that the image of the holomorphic 2-form $dz\wedge dw$ extends to a non-vanishing holomorphic 2-form on the Kummer surface and that the Kummer surface is simply connected. Therefore it is a K3 surface.

Next we fix a marking on Kummer surfaces for simplicity for calculation.

For any $g\in\mathrm{SL}(4,\mathbb{R})$, we identify its row vectors with complex vectors in $\mathbb{C}^{2}$ to obtain a lattice $\Lambda(g)$, and take a marked Kummer surface $(\phi_{g},\mathrm{Kum}(g))$ in the following way. For the identity element $I\in\mathrm{SL}(4,\mathbb{R})$ we fix a marking $\phi_{I}:\mathrm{Kum}(I)\to M_{0}$, and for any $g\in\mathrm{SL}(4,\mathbb{R})$, we take $\phi_{g}=g_{*}\phi_{I}$ where we consider $g$ acting on the left on $\mathbb{C}^{2}$ as a homeomorphism $\mathbb{C}^{2}/\Lambda(I)\to\mathbb{C}^{2}/\Lambda(g)$ and $g_{*}$ is the corresponding homeomorphism between the Kummer surfaces. We denote by $C_{i}$ the image of the 16 exceptional curves of $\mathrm{Kum}(I)$ in $M_{0}$ under $\phi_{I}$, then by the way we defined $\phi_{g}$ they are also the image of the 16 exceptional curves of $\mathrm{Kum}(g)$ under $\phi_{g}$. We denote by $\omega_{g}$ the degenerate Kähler form on the Kummer surface $\mathrm{Kum}(g)$ given by the standard Kähler form $\omega_{0}$ on $\mathbb{C}^{2}$. Then under this special marking $(\phi_{g})^{*}(\omega_{g})$ is constant on $M_{0}$.

We define the Kummer lattice $K$ to be the smallest primitive sublattice of $H^{2}(X,\mathbb{Z})$ that contains $C_{i},1\leq i\leq 16$.