Posted by: matheuscmss | March 16, 2016

Counting torus fibrations on a K3 surface (after S. Filip)

Last week, Simion Filip gave the talk “Counting torus fibrations on a K3 surface” at Université Paris 11 (Orsay). This post is a transcription of my notes from his lecture and, of course, all typos/mistakes are my sole responsibility.

1. Introduction

A classical problem in Dynamical Systems is the investigation of closed trajectories in billiards in polygons.

In the case of rational polygons (i.e., polygons whose angles belong to {\mathbb{Q}\pi}), it is known that closed trajectories are abundant. A popular way to establish this fact passes through the procedure of unfolding a rational polygon into a flat surface: roughly speaking, instead of letting the trajectories reflect on the boundary of the polygon, we reflect (finitely many times) the boundaries of the polygon in order to obtain straight line trajectories on a flat surface. See this excellent survey of Masur-Tabachnikov for more details.

For our purposes, we recall that a flat surface is the data {(X,\omega,\Omega)} of a Riemann surface {X}, a non-trivial holomorphic {1}-form {\Omega} and the flat metric {\omega} thought as the Kähler form {\omega=\frac{i}{2}\Omega\wedge\overline{\Omega}}. Note that {\Omega} has zeros, i.e., {\omega} has (conical) singularities, whenever {X} has genus {>1} (by Riemann-Roch theorem).

Some of the key features concerning closed trajectories in flat surfaces are:

  • closed geodesics of the flat metric come in families of parallel trajectories called cylinders in the literature;
  • such closed geodesics occur in a dense set of directions in the unit circle {S^1};
  • Eskin and Masur showed the following asymptotics for the problem of counting cylinders: there exists a constant {c>0} such that the number of (cylinders of) closed trajectories of length {\leq L} is {\sim c\cdot L^2}.

The goal of this post is to generalize this picture to higher dimensions or, more precisely, to K3 surfaces.

2. K3 surfaces

Definition 1 A compact complex {2}-dimensional manifold {X} is a K3 surface if

  • (i) {X} admits a (global) nowhere vanishing holomorphic {2}-form {\Omega};
  • (ii) the first Betti number {b_1(X)} is zero (this is equivalent to {\pi_1(X)=\{0\}} in this context).

Two basic examples of K3 surfaces are:

Example 1 Quartic surfaces in {\mathbb{P}^3(\mathbb{C})}, i.e., {\{F=0\}\subset \mathbb{P}^3(\mathbb{C})}, {F} is a polynomial of degree {\textrm{deg}(F)=4}.

Example 2 (Kummer examples) Let {A=\mathbb{C}^2/\Lambda} be a complex torus (i.e., {\Lambda} is a lattice of {\mathbb{C}^2}). Then, {A_0= A/\{\pm\textrm{Id}_A\}} has a subset \textrm{sing} of {16} singular points, and the blow-up {X=Bl_{\textrm{sing}}(A_0)} is a K3 surface.

Some basic properties of K3 surfaces include:

  • all K3’s are diffeomorphic;
  • all K3’s are Kähler (Siu);
  • {H^2(X,\mathbb{Z})} has rank {22}, the Hodge intersection form has signature {(3,19)} on {H^2(X,\mathbb{Z})}, and {H^2(X,\mathbb{Z})} is an even unimodular lattice;
  • the Hodge decomposition of {H^2(X,\mathbb{C})} is {H^2(X,\mathbb{C}) = H^{2,0}\oplus H^{1,1}\oplus H^{0,2}}, where {H^{2,0}=\mathbb{C}\cdot\Omega} has rank {1} and signature {(1,0)}, {H^{1,1}} has rank {20} and signature {(1,19)}, and {H^{0,2}=\mathbb{C}\cdot\overline{\Omega}} has rank {1} and signature {(1,0)};
  • the data in the previous two items determine the K3 surface (by Torelli theorem).

See, e.g., the lecture notes of D. Huybrechts for more details.

3. Special Lagrangian submanifolds

The natural generalization of closed trajectories on flat surfaces are special Lagrangian submanifolds (SLags).

Definition 2 Let {(X, \omega, \Omega)}, {X} is a compact complex {n}-dimensional manifold, {\omega} is a Kähler form, {\Omega} is a holomorphic {n}-form. A real {n}-dimensional submanifold {L\subset X} is a special Lagrangian submanifold (SLag) if

  • {L} is Lagrangian, i.e., {\omega|_{L}=0};
  • {L} is special, i.e., {\Omega|_{L} = d \textrm{Vol}_{\textrm{Riemannian}}}.

Remark 1 Special Lagrangians are minimal submanifolds (in the sense that they minimize the volume in their cohomology class).

The next example justifies the claim that special Lagrangian submanifolds are the analog of closed trajectories on flat surfaces.

Example 3 Consider the case {n=1}, i.e., {X} is a flat surface. In this situation, all real {1}-dimensional submanifolds are Lagrangian. On the other hand, since {\Omega} is locally {dz} (away from its divisors) in this setting, we see that a special Lagrangian is a horizontal geodesic of the flat metric. In particular, if we replace {\Omega} by {e^{i\theta}\Omega}, then the SLags become the straight line trajectories at angle {\theta} on {X}.

In a similar vein, special Lagrangian fibrations are the analog of cylinders of closed horizontal trajectories on flat surfaces.

4. Special Lagrangian fibrations

Definition 3 A fibration {X\rightarrow B} of {X} over a real {n}-dimensional base {B} is SLag if its fibers are compact {n}-dimensional SLags submanifolds of {X}. The volume {V\in\mathbb{R}_+} of such a fibration is {V=\int_L\Omega} where {L} is any fiber of {X\rightarrow B}.

Remark 2 The fibers of SLag fibrations are torii. One can compare this with the Arnold-Liouville theorem saying that the fibers of a fibration {M\rightarrow B} of a symplectic manifold {M} by compact Lagrangian submanifolds are necessarily torii. In particular, the base {B} has an integral affine structure whose structural group is the semi-direct product of {GL(\mathbb{Z}^n)} by {\mathbb{Z}^n}.

Remark 3 Similarly to the case {n=1}, a typical K3 surface doesn’t admit a SLag fibration.

K3 surfaces possess a significant amount of relevant structures. For example, a particular case of Yau’s solution to Calabi’s conjecture says that:

Theorem 4 (Yau) Let {X} be a K3 surface equipped with a Kähler form {\omega}. Then, there exists an unique {\omega'} in the same cohomology class of {\omega} in {H^2(X,\mathbb{R})} such that {\omega'} induces a Ricci-flat metric.

Moreover, K3 surfaces {X} with Ricci-flat metrics {g} are hyperKähler manifolds, i.e., they admit three complex structures {I, J, K} such that {(X,g,I,J,K)} has the following properties:

  • {g} is a Ricci-flat Riemannian metric;
  • {I, J, K} are complex structures satisfying the usual quaternionic relations: {IJ=-JI=K};
  • {I, J, K} are compatible with {g}: the forms {\omega_{\ast}=g(\dot, \ast\dot)} are closed (i.e. {d\omega_{\ast}=0}) for {\ast=I, J, K}.

Equivalently, we can write the data {(X,g,I,J,K)} of a hyperKähler manifold as {(X,\omega,\Omega)} where {\omega=\omega_I} and {\Omega = \omega_J+i\omega_K}. In this way, we obtain a presentation of K3 surfaces bearing some similarities with our definition of flat surfaces.

5. Statement of the main result

In this setting, we change direction of SLag fibrations (in analogy with the case of closed trajectories in flat surfaces) through the notion of twistor families. More concretely, we consider the sphere {\mathbb{S}^2:=\{(x,y,z)\in\mathbb{R}^3: x^2+y^2+z^2=1\}} and we denote by

\displaystyle \mathfrak{X} = X\times \mathbb{S}^2

where the fiber {X\times\{t\}} is equipped with the complex structure {I_t=xI+yJ+zK} for {t=(x,y,z)}.

At this point, we are almost ready to state the main result of this post: for technical reasons, we will give an impressionistic statement before explaining the true theorem in Remark 4 below.

Theorem 5 (Filip) Fix a (generic) twistor family {\mathfrak{X}=X\times\mathbb{S}^2}. Then, there exists {c>0} and {\delta>0} such that

\displaystyle \#\{t\in\mathbb{S}^2: X\times\{t\} \textrm{ admits a SLag fibration of volume }\leq V\} =

\displaystyle c\cdot V^{20} + O(V^{20-\delta})

as {V\rightarrow\infty}.

Remark 4 This statement is not quite true in the sense that one should count “equators in the twistor sphere” rather than counting points {t\in\mathbb{S}^2}. Indeed, this is so because if a complex structure {I_t} admits a SLag fibration at some point {x} in the equator {(\mathbb{R}t)^{\perp}\cap \mathbb{S}^2}, then one also has SLag fibrations with the same “angle” {x} as {I_t} varies along the equator {(\mathbb{R}x)^{\perp}\cap \mathbb{S}^2}.

At this point, Filip started running out of time, and, for this reason, he offered the following sketch of proof of his theorem.

The first step is to reduce to counting elliptic fibrations (i.e., holomorphic fibrations {X\rightarrow\mathbb{P}^1} whose fibers are elliptic curves).

The second step is to show that if the twistor family is not too special, then the counting problem reduces to the “linear algebra level” of the rank {22} vector space {H^2(X)} equipped with a form of signature {(3,19)}.

Finally, this last counting problem can be solved through quantitative equidistribution results on {\Gamma\backslash SO(3,19)} for the action of a certain {1}-parameter subgroup {a_s} on the quotient of the stabilizer in {SO(3,19)} of a null vector in {\mathbb{R}^{3,19}} by a lattice {\Gamma_0} (that is, the quotient of the semidirect product of {SO(3,18)} and {\mathbb{R}^{18}} by{\Gamma_0}).

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