EMS Magazine2021 / No. 119
MAG119DOI 10.4171/MAG-4

Geometry and dynamics on Riemann and K3 surfaces

Simion Filip
Surfaces are some of the simplest yet geometrically rich manifolds. Geometric structures on surfaces illuminate their topology and are useful for studying dynamical systems on surfaces. We illustrate below how some of these concepts blend together, and relate them to algebraic geometry.

1 Geometry of surfaces

In this section we give a brief overview of some geometric facts regarding Riemann surfaces and K3 surfaces. Although both are called “surfaces”, Riemann surfaces are examples of algebraic curves, while K3 surfaces are genuine algebraic surfaces. This means that considering the complex points, Riemann surfaces are complex 11-dimensional while K3 surfaces are complex 22-dimensional. We will give some explicit examples and then describe geometric structures that live on these surfaces. In the Riemann case we are concerned with flat geometry (with singularities) while in the K3 case we consider Ricci-flat Kähler metrics. Moduli spaces of these geometric structures play an important role in the results of Section 2 below.

Riemann surfaces. One can describe a compact Riemann surface by giving the algebraic equations that cut it out in some ambient space. For example, consider

X ⁣:y2=x(x51),Ω=dxy.\displaystyle X\colon y^{2}=x(x^{5}-1),\quad\Omega=\frac{dx}{y}.

The locus X(C)X(\mathbb{C}) of points satisfying this equation in C2\mathbb{C}^{2} is a real 22-dimensional surface of genus 22 (with two points at infinity added to XX). The 11-form Ω\Omega from (1) is the unique (up to scale) holomorphic 11-form vanishing at the two points at infinity.

Flat geometry. There is an alternative way to describe the pair (X,Ω)(X,\Omega). Take the regular decagon in the plane and glue its opposite and parallel edges to form a surface of genus 22, with two marked points given by the vertices. If we identify the plane with C\mathbb{C}, then the 11-form Ω~=dz\tilde{\Omega}=dz will be invariant under the translations used to glue opposite edges and will descend to a 11-form Ω\Omega on the new surface. This construction gives back the same pair (X,Ω)(X,\Omega) as described in (1), although this is by no means obvious. For more on the algebraic curve in (1), see [26 C. T. McMullen, Teichmüller curves in genus two: Torsion divisors and ratios of sines. Invent. Math. 165, 651–672 (2006) , §5]. (A question for experts: what is the area of the decagon under this identification?)

This construction is quite general: starting from a pair (X,Ω)(X,\Omega) consisting of a compact Riemann surface and a holomorphic 11-form, one can associate to it a polygon in the plane by cutting the surface XX and mapping it to the plane in such a way that in local charts the 11-form Ω\Omega becomes dzdz. Equivalently, one can give charts to C\mathbb{C} near a point p0Xp_{0}\in X by pp0pΩp\mapsto\int_{p_{0}}^{p}\Omega, and the transition maps between charts are translations in C\mathbb{C}. Conversely, given a polygon in the plane (possibly disconnected), with side identifications given by translations, one can reconstruct a Riemann surface with a holomorphic 11-form using the converse to the above recipe.

Action of GL2(R)\text{{GL}}_{2}(\mathbb{R}). A polygon is determined by its sides, which are vectors in R2~C\mathbb{R}^{2}\tilde{\longrightarrow}\mathbb{C}. The group GL2(R)\text{{GL}}_{2}(\mathbb{R}) acts on polygons, keeping parallel sides parallel, so if we have a polygonal description of (X,Ω)(X,\Omega) then we obtain a new pair g(X,Ω)=(X,Ω)g\cdot(X,\Omega)=(X^{\prime},\Omega^{\prime}). One can also express the action intrinsically on the surface, by letting a matrix act on the real and imaginary parts of Ω\Omega, viewed as differential 11-forms on XX:

[ReΩImΩ]=[abcd][ReΩImΩ].\begin{bmatrix}\operatorname{Re}\Omega^{\prime}\\ \operatorname{Im}\Omega^{\prime}\end{bmatrix}=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\cdot\begin{bmatrix}\operatorname{Re}\Omega\\ \operatorname{Im}\Omega\end{bmatrix}.

Note that the holomorphic structure on XX^{\prime} is usually different from the one on XX. Furthermore, even if explicit algebraic equations are given for XX, it is typically not possible to describe the equations cutting out XX^{\prime}. The sides of the polygons describing the surfaces are computed by taking integrals of Ω\Omega on paths in XX, and the passage from algebraic equations to integrals and back is by no means explicit.

Moduli spaces of Riemann surfaces. Because algebraic equations have finitely many coefficients, one can consider full parameter spaces, or moduli spaces, of algebraic manifolds defined by the same type of equations. For Riemann surfaces we will be interested in the space H(κ)\mathcal{H}(\kappa) of pairs (X,Ω)(X,\Omega) where XX is a compact Riemann surface and Ω\Omega is a holomorphic 11-form with κ\kappa describing the multiplicities of the zeros of Ω\Omega. The process described above of obtaining a new pair (X,Ω)(X,\Omega) using a real matrix gives an action of GL2(R)\text{{GL}}_{2}(\mathbb{R}) on H(κ)\mathcal{H}(\kappa). This action, however, is not via polynomial or even holomorphic automorphisms. In Section 2 we will see, however, that there are some relations between the GL2(R)\text{{GL}}_{2}(\mathbb{R})-action and the algebraic equations defining Riemann surfaces.

For more on the GL2(R)\text{{GL}}_{2}(\mathbb{R})-action on H(κ)\mathcal{H}(\kappa), see the surveys of Zorich [36 A. Zorich, Flat surfaces. In Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 437–583 (2006) ] for an introduction as well as numerous motivations and applications, as well as the more recent surveys of Forni–Matheus [20 G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. J. Mod. Dyn. 8, 271–436 (2014) ] and Wright [33 A. Wright, From rational billiards to dynamics on moduli spaces. Bull. Amer. Math. Soc. (N.S.) 53, 41–56 (2016) ].

K3 surfaces. We now switch gears and consider algebraic surfaces, such as those given by the equation

X={(1+x2)(1+y2)(1+z2)16xyz=4},\displaystyle X=\bigl\{(1+x^{2})(1+y^{2})(1+z^{2})-16xyz=4\bigr\},
Ω=dxdyz(1+x2)(1+y2)8xy.\displaystyle\Omega=\frac{dx\wedge dy}{z(1+x^{2})(1+y^{2})-8xy}.

The 22-form Ω\Omega is nowhere-vanishing on XX and is computed via a residue construction (take the residue of dxdydzF\tfrac{dx\wedge dy\wedge dz}{F} along F=0F=0). It is interesting to consider both the complex and the real solutions of this equation, denoted X(C)X(\mathbb{C}) and X(R)X(\mathbb{R}) respectively. See Figure 1 for an example of real solutions. Algebraic curves (i.e., Riemann surfaces), such as those described in (1), have only finite automorphism groups as soon as the genus is at least two, but algebraic surfaces such as the one in (2) have dynamically interesting automorphisms such as

(x,y,z)σx(16yz(1+y2)(1+z2)x,y,z)\displaystyle(x,y,z)\xrightarrow{\sigma_{x}}\left(\frac{16yz}{(1+y^{2})(1+z^{2})}-x,y,z\right)

as well as their analogues σy,σz\sigma_{y},\sigma_{z} in which the roles of the coordinates are permuted. The formula for the automorphism σx\sigma_{x} is obtained by “freezing” the yy and zz variables, viewing (2) as a quadratic equation for xx, and exchanging the two solutions of the quadratic. In particular applying σx\sigma_{x} twice gives back the identity transformation: σxσx=1X\sigma_{x}\circ\sigma_{x}=\mathbf{1}_{X}.

Figure 1. The real solutions of (2), intersectedwith the plane x=1x=1 to obtain an elliptic curve

Kähler geometry. The complex solutions of algebraic equations such as the ones above yield projective algebraic manifolds. These admit Kähler metrics, special kinds of Riemannian metrics adapted to the complex structure. For Kähler metrics, the Riemannian volume of an algebraic submanifold is determined by its homology class alone.

Specializing further, if the algebraic manifold admits an algebraic volume form, such as Ω\Omega from (3), then Yau’s solution of the Calabi conjecture [34 S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Comm. Pure Appl. Math. 31, 339–411 (1978) ] gives canonical Kähler metrics whose Ricci curvature vanishes. To construct such metrics, one needs to solve a nonlinear PDE of Monge–Ampère type and there is no “hands-on” description of such metrics as in Figure 2.

Figure 2. Left: Real solutions of y2=x(x51)y^{2}=x(x^{5}-1). Right: Decagon with opposite sides identified,and a straight line on the surface connecting the two singularities

Moduli spaces of K3 surfaces. Moving to algebraic manifolds of higher dimensions, additional data needs to be specified in order to have well-behaved moduli spaces. For us, the most relevant will be the space KE\mathcal{K}\mathcal{E} of Ricci-flat metrics on the manifold underlying a complex K3 surface. The abbreviation KE\mathcal{K}\mathcal{E} is for Kähler–Einstein, since Ricci-flat metrics satisfy the Einstein equation Ricij=λgij\operatorname{Ric}_{ij}=\lambda g_{ij} with λ=0\lambda=0. These moduli spaces play an essential role in the study of K3 surfaces and have the remarkable feature that they are (essentially) homogeneous spaces for appropriate Lie groups. For an introduction to the geometry of K3 surfaces, see the collection of notes [2 A. Beauville, Géométrie des surfaces K⁢3: modules et périodes. Astérisque, Société Mathématique de France, Paris (1985) ] and the more recent monograph of Huybrechts [22 D. Huybrechts, Lectures on K3 surfaces. Cambridge Studies in Advanced Mathematics 158, Cambridge University Press, Cambridge (2016) ].

2 Dynamics on moduli spaces

This section describes results on the dynamics of group actions in the moduli space of Riemann and K3 surfaces equipped with appropriate flat, or Ricci-flat, metrics.

Dynamics on moduli spaces of flat surfaces. The action of the group GL2(R)\text{{GL}}_{2}\left(\mathbb{R}\right) on the moduli space H(κ)\mathcal{H}(\kappa) of Riemann surfaces with a holomorphic 11-form satisfies rigidity properties akin to those for unipotent flows in homogeneous dynamics developed by Ratner [30 M. Ratner, On Raghunathan’s measure conjecture. Ann. of Math. (2) 134, 545–607 (1991) , 31 M. Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63, 235–280 (1991) ], Margulis [24 G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogènes. C. R. Acad. Sci. Paris Sér. I Math. 304, 249–253 (1987) ] and many others. The following results were established by Eskin, Mirzakhani, and Mohammadi:

Theorem 1 ([10 A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL⁢(2,ℝ) action on moduli space. Publ. Math. Inst. Hautes Études Sci. 127, 95–324 (2018) , 11 A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL⁢(2,ℝ) action on moduli space. Ann. of Math. (2) 182, 673–721 (2015) ]).

For any pair (X,Ω)H(κ)(X,\Omega)\in\mathcal{H}(\kappa), the orbit closure M:=GL2(R)(X,Ω)\mathcal{M}:=\overline{\text{{GL}}_{2}(\mathbb{R})\cdot(X,\Omega)} is a submanifold of H(κ)\mathcal{H}(\kappa), described in local coordinates by linear relations among the sides of the polygons used to parametrize surfaces. Furthermore,1The switch from GL2\text{{GL}}_{2} to SL2\text{{SL}}_{2} is done to exclude the scaling action. any SL2(R)\text{{SL}}_{2}(\mathbb{R})-invariant ergodic probability measure must be Lebesgue supported on such a manifold.

Going back to the algebraic description of Riemann surfaces, we saw that except in special symmetric situations, it is not possible in general to relate the algebraic equations to the polygonal description of the surface. In the case of GL2(R)\text{{GL}}_{2}(\mathbb{R})-orbit closures, it is possible to give an alternative, purely algebraic description of their geometry. Specifically, recall that the Jacobian Jac(X)\text{Jac}(X) associated to a genus gg Riemann surface XX is the complex torus defined as H0(X;KX) ⁣/H1(X;Z)H^{0}(X;K_{X})^{\vee\mskip-3.0mu }/H_{1}(X;\mathbb{Z}), where H0(X;KX)H^{0}(X;K_{X}) denotes the complex gg-dimensional space of holomorphic 11-forms on XX,  ⁣{\vee\mskip-3.0mu } denotes the dual, and the first homology group H1(X;Z)H_{1}(X;\mathbb{Z}) embeds in the dual by integration along cycles. Alternatively, the Jacobian is the moduli space of holomorphic degree 00 line bundles on XX and this description provides a link between the algebraic and analytic structures on a Riemann surface. Although the automorphism group of a genus g2g\geq 2 Riemann surface is finite, the endomorphism group of its Jacobian can be much larger (real or complex multiplication give examples of such symmetries).

Theorem 2 ([14 S. Filip, Splitting mixed Hodge structures over affine invariant manifolds. Ann. of Math. (2) 183, 681–713 (2016) , 13 S. Filip, Semisimplicity and rigidity of the Kontsevich–Zorich cocycle. Invent. Math. 205, 617–670 (2016) ]).

Orbit closures M\mathcal{M} as in Theorem 1 parametrize Riemann surfaces whose Jacobians have additional endomorphisms a specific kind (such as real multiplication). Furthermore, specific combinations of the zeros of the distinguished 11-form yield torsion points on the Jacobian.

These conditions characterize M\mathcal{M} as a locus inside H(κ)\mathcal{H}(\kappa).

Additional finiteness results for orbit closures are established in [6 A. Eskin, S. Filip and A. Wright, The algebraic hull of the Kontsevich–Zorich cocycle. Ann. of Math. (2) 188, 281–313 (2018) ], jointly with Eskin and Wright.

The relation between the GL2(R)\text{{GL}}_{2}(\mathbb{R})-action and real multiplication on Jacobians was discovered by McMullen [25 C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces. J. Amer. Math. Soc. 16, 857–885 (2003) ], who also established most of the above-mentioned results in the case of genus 22 Riemann surfaces [27 C. T. McMullen, Dynamics of SL2⁢(ℝ) over moduli space in genus two. Ann. of Math. (2) 165, 397–456 (2007) ]. Möller introduced the tools of Hodge theory to the subject [29 M. Möller, Variations of Hodge structures of a Teichmüller curve. J. Amer. Math. Soc. 19, 327–344 (2006) , 28 M. Möller, Periodic points on Veech surfaces and the Mordell–Weil group over a Teichmüller curve. Invent. Math. 165, 633–649 (2006) ] which were used to connect the algebraic and combinatorial descriptions of holomorphic 11-forms on Riemann surfaces.

Billiards. Fix a polygon and consider the dynamical system consisting of a billiard ball bouncing off the sides in the customary way, with the angle of incidence equal to the angle of reflection. By studying billiards in regular nn-gons, Veech discovered the first instances of nontrivial orbit closures for the GL2(R)\text{{GL}}_{2}(\mathbb{R})-action and established along the way:

Theorem 3 ([32 W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97, 553–583 (1989) , Thm. 1.5]).

For a regular nn-gon, the number of closed billiard trajectories of length at most LL grows like cnL2c_{n}L^{2} for a constant cnc_{n}.

This is in analogy with the Gauss circle problem of counting lattice points in the plane, which corresponds to playing billiards on a square. The deeper study of the dynamics of billiards on surfaces, and polygons with rational angles, ties in with the study of the GL2(R)\text{{GL}}_{2}(\mathbb{R})-action on the moduli space H(κ)\mathcal{H}(\kappa), and this is the key to Veech’s result and many others.

An analogue for K3 surfaces. Billiard trajectories are locally given by straight lines. Besides the characterization of straight lines as giving the shortest path between points, they have the following alternative description. Take the 11-form Ω=dz=dx+1dy\Omega=dz=dx+\sqrt{-1}dy in the plane. A straight line is a curve γ\gamma such that l(γ)=γΩl(\gamma)=\left|\int_{\gamma}\Omega\right| where l(γ)l(\gamma) is the Euclidean length of γ\gamma. Note that for an arbitrary curve γ\gamma we have the inequality

l(γ)γΩl(\gamma)\geq\left|\int_{\gamma}\Omega\right|

which, in differential-geometric language, says that the 11-form Ω\Omega calibrates the straight lines.

This last point of view generalizes to K3 surfaces, where the analogue of closed billiard trajectories are special Lagrangian tori. These are real 22-dimensional tori inside a K3 surface with a Ricci-flat Kähler metric which, among many other properties, minimize volume in their homology class.

Theorem 4 ([17 S. Filip, Counting special Lagrangian fibrations in twistor families of K3 surfaces. Ann. Sci. Éc. Norm. Supér. (4) 53, 713–750 (2020) , Thm. C]).

Under appropriate assumptions on the Ricci-flat metric on a K3 surface, the number of such special Lagrangian tori, of volume bounded by VV, is asymptotic to cV20cV^{20}, for an explicit constant c>0c>0.

It is possible to make the above counting effective and give an error term of order V20εV^{20-\varepsilon}, for ε>0\varepsilon>0, which was estimated effectively by Bergeron–Matheus in the appendix to [17 S. Filip, Counting special Lagrangian fibrations in twistor families of K3 surfaces. Ann. Sci. Éc. Norm. Supér. (4) 53, 713–750 (2020) ]. Analogously to the counting result for Riemann surfaces, this one is established by studying the dynamics in the full moduli space KE\mathcal{K}\mathcal{E} of Ricci-flat metrics. Although we are asking a question about a specific one, it proves useful to study the space of all possible metrics. The idea of using dynamics on homogeneous spaces for counting results goes back to Eskin and McMullen [9 A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71, 181–209 (1993) ].

3 Dynamics on K3 surfaces

In this section we describe some results on individual automorphisms of K3 surfaces. Again, a key role in the proofs is played by Ricci-flat metrics and their moduli space on a fixed K3 surface.

Entropy. Suppose for a moment that (X,d)(X,d) is a compact metric space and f ⁣:XXf\colon X\to X is a continuous map. Define a new distance function by dn(x,y):=max0ind(fi(x),fi(y))d_{n}(x,y):=\max_{0\leq i\leq n}d(f^{i}(x),f^{i}(y)), so two points are at dnd_{n}-distance at least ε\varepsilon if along their ff-orbits, they have separated at some time at distance ε\varepsilon. Let now S(dn,ε)S(d_{n},\varepsilon) be the maximal number of ε\varepsilon-separated points in XX, i.e., any two are at dnd_{n}-distance at least ε\varepsilon. This is the number of essentially distinct trajectories, up to time nn, when observing the system with accuracy ε\varepsilon. The topological entropy htop(f)h^{top}(f) is the exponential growth rate in nn of S(dn,ε)S(d_{n},\varepsilon):

htop(f):=limε0lim supn+1nlogS(dn,ε).h^{top}(f):=\lim_{\varepsilon\to 0}\limsup_{n\to+\infty}\frac{1}{n}\log S(d_{n},\varepsilon).

There is also an associated notion of measure-theoretic entropy. Recall that an ff-invariant measure μ\mu must satisfy μ(f1(A))=μ(A)\mu(f^{-1}(A))=\mu(A) for any measurable subset AXA\subset X. To define the entropy of μ\mu, partition XX into disjoint measurable sets X1,,XkX_{1},\ldots,X_{k}. Then an orbit of a point gives rise to a sequence of elements that it visits, encoded as a sequence in {1,,k}N\{1,\ldots,k\}^{\mathbb{N}}. The number of such distinct sequences, weighted appropriately by μ\mu, grows exponentially, and the exponential growth rate is called the entropy (after taking a supremum over all finite partitions of XX).

Yomdin and Gromov theorems. Suppose now that XX is a smooth manifold and ff is a smooth diffeomorphism. Then the pullback ff^{*} acts on the cohomology groups H(X;R)H^{\bullet}(X;\mathbb{R}) and we consider its spectral radius ρ(f)\rho(f) (viewed as a linear transformation). Settling a conjecture of Shub, Yomdin proved the following:

Theorem 5 ([35 Y. Yomdin, Volume growth and entropy. Israel J. Math. 57, 285–300 (1987) ]).

The topological entropy of ff satisfies

htop(f)logρ(f).h^{top}(f)\geq\log\rho(f).

Thus topological complexity implies dynamical complexity. When XX is a complex manifold admitting a Kähler metric, and ff is a holomorphic automorphism, Gromov [21 M. Gromov, On the entropy of holomorphic maps. Enseign. Math. (2) 49, 217–235 (2003) ] established the reverse upper bound

htop(f)ρ(f)and sohtop(f)=ρ(f).h^{top}(f)\leq\rho(f)\quad\text{and so}\quad h^{top}(f)=\rho(f).

Gromov’s proof is based on the special feature of Kähler metrics that the Riemannian volume of complex submanifolds is determined by their homology class (they are calibrated submanifolds, just like the special Lagrangians in Theorem 4).

Measures on K3 surfaces. Suppose now that XX is a K3 surface and ff is an automorphism of positive topological entropy. The surface in (2) works, and a composition of automorphisms like the one in (4), one for each coordinate, gives an example. Cantat [4 S. Cantat, Dynamique des automorphismes des surfaces K⁢3. Acta Math. 187, 1–57 (2001) ] showed that there exists a unique ff-invariant measure μ\mu which maximizes entropy, i.e., hμ(f)=logρ(f)h_{\mu}(f)=\log\rho(f). But the holomorphic 22-form Ω\Omega induces another measure ΩΩ\Omega\wedge\overline{\Omega} on XX, which is canonical and invariant under the dynamics. It is then natural to ask: what is the relationship between the two measures? This was answered by Cantat–Dupont [5 S. Cantat and C. Dupont, Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy. J. Eur. Math. Soc. (JEMS) 22, 1289–1351 (2020) ] and later, using different techniques, in [19 S. Filip and V. Tosatti, Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics. arXiv:1808.08673 (August 2018). To appear in Am. J. Math.]:

Theorem 6.

With notation as above, suppose that the measure of maximal entropy μ\mu is absolutely continuous with respect to Lebesgue measure on XX. Then XX is a “Kummer surface”, i.e., obtained from a complex torus A=C2/ΛA=\mathbb{C}^{2}/\Lambda by a quotient AA/ ⁣±1A\to A/\!\pm 1 and desingularization, and the automorphism ff comes by the same construction from a linear automorphism of the torus.

The proof in [19 S. Filip and V. Tosatti, Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics. arXiv:1808.08673 (August 2018). To appear in Am. J. Math.] uses Ricci-flat metrics on K3 surfaces and their compatibility with the volume form ΩΩ\Omega\wedge\overline{\Omega}. Indeed, the Kähler form ω\omega associated to a Ricci-flat metric satisfies the identity ωω=ΩΩ\omega\wedge\omega=\Omega\wedge\overline{\Omega} (as volume forms on XX) and this poses constraints on the dynamical invariants, such as Lyapunov exponents.

Rough currents. A pseudo-Anosov homeomorphism of a real 22-dimensional surface expands/contracts a pair of measured foliations on the surface. This is a basic result of Thurston’s analysis of mapping class group elements. Analogous objects, called closed positive currents, have been constructed by Cantat on K3 surfaces [4 S. Cantat, Dynamique des automorphismes des surfaces K⁢3. Acta Math. 187, 1–57 (2001) ], and earlier for polynomial maps of the plane by Bedford–Lyubich–Smillie [3 E. Bedford, M. Lyubich and J. Smillie, Polynomial diffeomorphisms of 𝐂2. IV. The measure of maximal entropy and laminar currents. Invent. Math. 112, 77–125 (1993) ]. Theorem 6 implies that if a K3 surface is not Kummer and admits a positive entropy automorphism, then the measure of maximal entropy is singular for Lebesgue measure. General dynamical considerations imply that its Hausdorff dimension is strictly below the maximal one, and thus the closed positive currents defined above must also have less than maximal Hausdorff dimension, see [18 S. Filip and V. Tosatti, Smooth and rough positive currents. Ann. Inst. Fourier (Grenoble) 68, 2981–2999 (2018) ].

4 An overview

The geometry and topology of surfaces is a subject with a long history. Many fundamental topics have been omitted in the above discussion, yet they all play a role in motivating constructions and formulating questions in the subject. For example, although Riemann surfaces do not admit infinite-order holomorphic automorphisms in genus at least two, the study of topological automorphisms (homeomorphism and diffeomorphisms) is essential for much of low-dimensional topology in the form of the Nielsen–Thurston theory of the classification of mapping class group elements (see the monographs of Farb–Margalit [12 B. Farb and D. Margalit, A primer on mapping class groups. Princeton Mathematical Series 49, Princeton University Press, Princeton, NJ (2012) ] and the collection of articles [1 A. Fathi, F. Laudenbach, and V. Poénaru, Travaux de Thurston sur les surfaces – Séminaire Orsay. Astérisque 66, Société Mathématique de France, Paris (1979) ]). This leads to the study of measured foliations on surfaces, and their analogues on algebraic surfaces that become closed positive currents. The geometry of these last objects is far less understood than that of surface foliations.

In all instances, moduli spaces of geometric structures play a crucial role. The Teichmüller and moduli spaces of Riemann surfaces are essential for understanding the topology of surfaces, and in the case of algebraic surfaces, moduli spaces of metrics play a similar role. In the case of K3 surfaces, the moduli spaces turn out to be locally homogeneous, and this makes available all the tools of homogeneous dynamics.

Finally, understanding the dynamics in moduli spaces requires one to understand dynamical invariants called Lyapunov exponents, which play a role similar to entropy. The tools of complex geometry and Hodge theory turned out to be crucial in gaining control over these otherwise elusive dynamical invariants, and these techniques are behind many of the theorems described above. This connection was originally made by Kontsevich [23 M. Kontsevich, Lyapunov exponents and Hodge theory. In The mathematical beauty of physics (Saclay, 1996), Adv. Ser. Math. Phys. 24, World Sci. Publ., River Edge, NJ, 318–332 (1997) ], see also [8 A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. Publ. Math. Inst. Hautes Études Sci. 120, 207–333 (2014) , 7 A. Eskin, M. Kontsevich, M. Möller and A. Zorich, Lower bounds for Lyapunov exponents of flat bundles on curves. Geom. Topol. 22, 2299–2338 (2018) , 16 S. Filip, Families of K3 surfaces and Lyapunov exponents. Israel J. Math. 226, 29–69 (2018) , 15 S. Filip, Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle. Duke Math. J. 166, 657–706 (2017) ] for further developments related to Lyapunov exponents and Hodge theory.

We end with a summary of the above parallels between the geometry of Riemann and K3 surfaces in Table 1.

Table 1:

Parallels between the geometry of Riemann and K3 surfaces

Riemann surfacesK3 surfaces
Mapping classes of diffeomorphisms:pseudo-Anosov, reducible, periodicHolomorphic automorphisms:hyperbolic, parabolic, elliptic
Entropy, action on curvesEntropy, action on H2{H}^{2}
Stable and unstable foliationsStable and unstable currents
Teichmüller spacePeriod Domain(s)
Flat metricsRicci-flat metrics
Holomorphic 11-formHolomorphic 22-form
Straight lines for flat metricSpecial Lagrangians
Periodic trajectoriesSpecial Lagrangian tori
Completely Periodic FoliationsTorus Fibrations
S1\mathbb{S}^{1}: directions for straight linesS2\mathbb{S}^{2}: twistor rotation
Lyapunov exponents for families

Simion Filip is Associate Professor of Mathematics at the University of Chicago. sfilip@math.uchicago.edu

  1. 1

    The switch from GL2\text{{GL}}_{2} to SL2\text{{SL}}_{2} is done to exclude the scaling action.

References

  1. A. Fathi, F. Laudenbach, and V. Poénaru, Travaux de Thurston sur les surfaces – Séminaire Orsay. Astérisque 66, Société Mathématique de France, Paris (1979)
  2. A. Beauville, Géométrie des surfaces K⁢3: modules et périodes. Astérisque, Société Mathématique de France, Paris (1985)
  3. E. Bedford, M. Lyubich and J. Smillie, Polynomial diffeomorphisms of 𝐂2. IV. The measure of maximal entropy and laminar currents. Invent. Math. 112, 77–125 (1993)
  4. S. Cantat, Dynamique des automorphismes des surfaces K⁢3. Acta Math. 187, 1–57 (2001)
  5. S. Cantat and C. Dupont, Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy. J. Eur. Math. Soc. (JEMS) 22, 1289–1351 (2020)
  6. A. Eskin, S. Filip and A. Wright, The algebraic hull of the Kontsevich–Zorich cocycle. Ann. of Math. (2) 188, 281–313 (2018)
  7. A. Eskin, M. Kontsevich, M. Möller and A. Zorich, Lower bounds for Lyapunov exponents of flat bundles on curves. Geom. Topol. 22, 2299–2338 (2018)
  8. A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. Publ. Math. Inst. Hautes Études Sci. 120, 207–333 (2014)
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