These notes are based on the lectures taught by Dawei Chen at the Berlin Fall School on Algebraic Geometry in 2019. They are sorted and expanded by Andrei Bud
An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of algebraic geometry, dynamical systems, geometry and topology lead to the study of such objects. It comes as a natural consequence that we can employ in our study algebraic, analytic, combinatorial and dynamical perspectives. These lecture notes aim to provide an expository introduction to this subject that will emphasize the aforementioned links between different areas of mathematics. We will associate to an Abelian differential a flat surface with conical singularities such that the underlying Riemann surface is obtained from a polygon by identifying edges with one another via translation. We will focus on studying these objects in families and describe some properties of the orbit as we vary the polygon by the action of $GL_2^{+}(\mathbb{R})$ on the plane.
Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surface topology and moduli spaces of Riemann surfaces, Grothendieck calls there for a recasting of topology, in order to make it fit to the objects of semialgebraic and semianalytic geometry, and in particular to the study of the Mumford-Deligne compactifications of moduli spaces. A …
This paper focuses on the interplay between the intersection theory and the Teichmueller dynamics on the moduli space of curves. As applications, we study the cycle class of strata of the Hodge bundle, present an algebraic method to calculate the class of the divisor parameterizing abelian differentials with a non-simple zero, and verify a number of extremal effective divisors on the moduli space of pointed curves in low genus.
This is an introduction to the algebraic aspect of Teichmüller dynamics, with a focus on its interplay with the geometry of moduli spaces of curves as well as recent advances in the field.
This paper proves the Beilinson-Soul{é} vanishing conjecture for motives attached to the moduli spaces of curves of genus 0 with n marked points. As part of the proof, it is also proved that these motives are mixed Tate. As a consequence of Levine's work, one obtains then well defined categories of mixed Tate motives over the moduli spaces of curves . It is shown that morphisms between moduli spaces forgetting marked points and embedding as boundary components induce functors between those categories and how tangential bases points fit in these functorialities. Tannakian formalism attaches gro…
The aim of this paper is to study the structure of the higher-dimensional Teichmüller and Riemann moduli spaces, viewed as stacks over the category of complex manifolds. We first show that the space of complex operators on a smooth manifold admits a foliation transversely modeled on a translation groupoid, a concept that we define here. We then show how to construct explicitly a holonomy groupoid for such a structure and show that in this case its objects and morphisms form a finite-dimensional analytic space and its source and target maps are smooth morphisms. This holonomy data encodes how t…
Teichmueller curves are geodesic discs in Teichmueller space that project to an algebraic curve in the moduli space $M_g$. We show that for all $g \geq 2$ Teichmueller curves map to the locus of real multiplication in the moduli space of abelian varieties. Remark that McMullen has shown that precisely for $g=2$ the locus of real multiplication is stable under the $\SL_2(\RR)$-action on the tautological bundle $Ω\M_g$. We also show that Teichmueller curves are defined over number fields and we provide a completely algebraic description of Teichmueller curves in terms of Higgs bundles. As a cons…
We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmuller curve is proportional to the Euler characteristic of the Hilbert modular surface on which it lies. The idea is to use techniques from algebraic geometry to calculate the fundamental classes of these Teichmuller curves in certain compactifications of the Hilbert modular surfa…
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmueller curves with various geometrically defined divisors on moduli spaces of curves.
For $g,n\geq 0$ a 3-dimensional Calabi-Yau $A_\infty$-category $\mathcal C_{g,n}$ is constructed such that a component of the space of Bridgeland stability conditions, $\mathrm{Stab}(\mathcal C_{g,n})$, is a moduli space of quadratic differentials on a genus $g$ surface with simple zeros and $n$ simple poles. For a generic point in the moduli space the corresponding quantum/refined Donaldson--Thomas invariants are computed in terms of counts of finite-length geodesics on the flat surface determined by the quadratic differential. As a consequence, these counts satisfy wall-crossing formulas.
A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formula to identify eigenforms for real multiplication on genus two Jacobians. We also present evidence that Teichmüller curves admit a rich arithmetic geometry by exhibiting examples with small primes of bad reduction and notable divisors supported at their cusps.
Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such `exotic' components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group SO(p,q). Furthermore, we discuss how these exotic…
Consider the 1-dimensional Hurwitz space parameterizing covers of P^1 branched at four points. We study its intersection with divisor classes on the moduli space of curves. As an application, we calculate the slope of the Teichmuller curve parameterizing square-tiled cyclic covers and recover the sum of its Lyapunov exponents obtained by Forni, Matheus and Zorich. Motivated by the work of Eskin, Kontsevich and Zorich, we exhibit a relation among the slope of Hurwitz spaces, the sum of Lyapunov exponents and the Siegel-Veech constant for the moduli space of quadratic differentials.
This paper is a survey on the role of Higgs bundle theory in the study of higher Teichmüller spaces. Recall that the Teichmüller space of a compact surface can be identified with a certain connected component of the moduli space of representations of the fundamental group of the surface into $\mathrm{PSL}(2,{\mathbb{R}})$. Higher Teichmüller spaces correspond to special components of the moduli space of representations when one replaces $\mathrm{PSL}(2,{\mathbb{R}})$ by a real non-compact semisimple Lie group of higher rank. Examples of these spaces are provided by the Hitchin components for s…
As a higher genus version of universal mixed elliptic motives by Hain and Matsumoto, we consider mixed Teichmüller motives as certain motivic local systems on the moduli space of pointed curves. We show that the category of mixed Teichmüller motives is equivalent to a full subcategory of a certain product category of mixed Tate motives over Z and universal mixed elliptic motives. Furthermore, we show that unipotent fundamental torsors for the universal open curve give rise to a pro-object in the category of mixed Teichmüller motives. Our results can give a realization of motivic correlators pr…
This short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces, and in particular the recent breakthrough result of Eskin, Mirzakhani, and Mohammadi. We also discuss the context and applications of this result, and connections to other areas of mathematics such as algebraic geometry, Teichmuller theory, and ergodic theory on homogeneous spaces.
Prym-Teichmüller curves $W_D(4)$ constitute the main examples of known primitive Teichmüller curves in the moduli space $\mathcal{M}_3$. We determine, for each non-square discriminant $D>1$, the number and type of orbifold points in $W_D(4)$. These results, together with the formulas of Lanneau-Nguyen and Möller for the number of cusps and the Euler characteristic, complete the topological characterisation of Prym-Teichmüller curves in genus 3. Crucial for the determination of the orbifold points is the analysis of families of genus 3 cyclic covers of degree $4$ and $6$, branched over four poi…
In this paper we continue our study on the canonical metrics on the Teichmüller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carathéodory metric to the Kähler-Einstein metric, solving another old conjecture of Yau. We then prove that the Ricci curvature of the perturbed Ricci metric has negative upper and lower bounds, and it also has bounded geometry. Then we study in detail the boundary behaviors of the Kähler-Einstein metric and prove that it has bounded geometry, and all of the covariant derivatives of its curvature are uniformly bo…
We consider the Grothendieck--Teichmüller group under a new aspect. Using real algebraic geometry and web theory we show that it preserves dihedral symmetry relations, present in the fundamental groupoids of configuration spaces of marked points on $\mathbb{C}$. The motivation of this paper is to be understood in the light of Grothendieck's initial philosophy stating that throughout hidden symmetries of the moduli spaces of curves one can shed some light on the absolute Galois group. This appears as a new development of the construction of the avatar of the Grothendieck--Teichmüller group and …
In this paper, we classify three-dimensional complex Abelian varieties isogenous to a product $A_1 \times A_2$, where one of the factors admits real multiplication by a real quadratic order $\mathcal{O}_D$ of discriminant $D$. We show that the moduli space $X_D^{(3)}$ of these varieties essentially is the disjoint union of certain Hilbert modular varieties $X_{\mathfrak{a}}$, each component depending on the choice of an ideal $\mathfrak{a}$ of $\mathcal{O}_D$. We give an explicit construction of these varieties. We show that the boundary of the eigenform locus for pseudo-real multiplication by…
We construct the moduli space for equivalence classes of n-pointed tropical curves of genus g, together with its compactification given by weighted tropical curves, and establish some of its basic topological properties. We compare it to the moduli spaces of smooth and stable algebraic curves, from the combinatorial, the topological, and the Teichmüller point of view. The paper is written in an expository style, and it generalizes some results contained in sections 4-6 of arXiv:1001.2815v3.
In his 1944 paper Veränderliche Riemannsche Flächen , Teichmüller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichmüller space. In the same paper, Teichmüller introduced the so-called universal Teichmüller curve -- a space over Teichmüller space where the fiber above each point is a Riemann surface representing that point. In fact, Teichmüller proved the existence of the Teichmüller curve as a space of Riemann surfaces parametrized by an analytic space, with a…
We give a characterizaton of smooth ample Hypersurfaces in Abelian Varieties and also describe an irreducible connected component of their moduli space: it consists of the Hypersurfaces of a given polarization type, plus the iterated univariate coverings of normal type (of the same polarization type). The above manifolds yield also a connected component of the open set of Teichmüller space consisting of Kähler complex structures.
The aim of the present paper is to provide a new aspect of the $p$-adic Teichmüller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichmüller theory is the construction of the locus $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ in…
In an earlier paper [Acta Mathematica, v. 176, 1996, 145-169, alg-geom/9505024 ] the present authors and Dennis Sullivan constructed the universal direct system of the classical Teichmüller spaces of Riemann surfaces of varying genus. The direct limit, which we called the universal commensurability Teichmüller space, $T_{infty}$, was shown to carry on it a natural action of the universal commensurability mapping class group, $MC_{\infty}$. In this paper we identify an interesting cofinal sub-system corresponding to the tower of finite-sheeted characteristic coverings over any fixed base surfac…
For $2g-2+n>0$, the Teichmüller modular group $Γ_{g,n}$ of a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $Π_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\hatΠ_{g,n}$ its profinite completion. There is then a natural faithful representation $Γ_{g,n}\hookrightarrow Out(\hatΠ_{g,n})$. The procongruence completion $\checkΓ_{g,n}$ of the Teichmüller group is defined to be the closure of the Teic…
