SO(p,q)-Higgs bundles and higher Teichmüller components
Marta Aparicio-Arroyo, Steven Bradlow, Brian Collier, Oscar García-Prada, Peter Gothen, André Oliveira
备注：
Reorganized sections 2 and 3 and added appendix. Some arguments in section 5 have been improved. A new result (Proposition 7.15) has been added concerning the irreducibility of surface group representations in our exotic components, thereby strengthening the relation between our results and a conjecture of Guichard-Wienhard-Labourie
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 components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for SO(2,q), with q> 3).
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 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…
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…
In this paper we study the relation between parabolic Higgs bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a result of Hitchin, identifying those parabolic Higgs bundles that correspond to Fuchsian representations. We also study the Higgs bundles that give representations whose image is contained, after conjugation, in SL($k,\Bbb R$). We compute the real dimension of one of the components of this space of representations, which in the absence of punctures is the generalized Teichmüller space introduced by Hit…
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…
There exists on each Teichmüller space $T_g$ (comprising compact Riemann surfaces of genus $g$), a natural sequence of determinant (of cohomology) line bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''. There is a remarkable connection, (Belavin-Knizhnik), between the Mumford isomorphisms and the existence of the Polyakov string measure on the Teichmüller space. This suggests the question of finding a genus-independent formulation of these bundles and their isomorphisms. In this paper we combine a Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q…
We compute the algebraic equation of the universal family over the Kenyon-Smillie $(2,3,4)$-Teichmüller curve and give a nice geometric description of the torsion map. Moreover, we re-prove independently that the found algebraic equation describes a Teichmüller curve by computing the Picard-Fuchs equation associated to the absolute cohomology bundle.
Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichmüller spaces comprising complex structures on a certain …
For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{ü}ller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic $G$-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of…
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 this paper, we study connected components of strata of the space of quadratic differentials lying over $\T_g$. We use certain general properties of sections of line bundles to put a upper bound on the number of connected components, and a generalized version of the Gauss map as an invariant to put a lower bound on the number of such components. For strata with sufficiently many zeroes of the same order we can state precisely the number of components.
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.
We prove the existence of global sections trivializing the Hodge bundles on the Hodge metric completion space of the Torelli space of Calabi--Yau manifolds, a global splitting property of these Hodge bundles. We also prove that a compact Calabi--Yau manifold can not be deformed to its complex conjugate. These results answer certain open questions in the subject. A general result about the period map to be bi-holomorphic from the Hodge metric completion space of the Torelli space of Calabi--Yau type manifolds to their period domains is proved and applied to the cases of K$3$ surfaces, cubic fou…
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 get an upper bound of the slope of each graded quotient for the Harder-Narasimhan filtration of the Hodge bundle of a Teichmüller curve. As an application, we show that the sum of Lyapunov exponents of a Teichmüller curve does not exceed ${(g+1)}/{2}$, with equality reached if and only if the curve lies in the hyperelliptic locus induced from $\mathcal{Q}(2k_1,...,2k_n,-1^{2g+2})$ or it is a special Teichmüller curve in $Ω\mathcal{M}_g(1^{2g-2})$. It also gives an unified interpretation for many known results about the special partial sums of Lyapunov exponents on Teichmüller curves.
Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using Atiyah-Bott, Forni and Möller's works. We obtain several applications to Teichmüller dynamics conditional to the conjecture.
Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichmüller theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to construct geometric structures in some interesting cases. In this paper, w…
We consider normal covers of CP^1 with abelian deck group, branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces. We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov …
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…
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…
We study a family of Teichmüller curves T(n,m) constructed by Bouw and Möller, and previously by Veech and Ward in the cases n=2,3. We simplify the proof that T(n,m) is a Teichmüller curve, avoiding the use Möller's characterization of Teichmüller curves in terms of maximally Higgs bundles. Our key tool is a description of the period mapping of T(n,m) in terms of Schwarz triangle mappings. We prove that T(n,m) is always generated by Hooper's lattice surface with semiregular polygon decomposition. We compute Lyapunov exponents, and determine algebraic primitivity in all cases. We show that freq…
A dormant indigenous bundle is an integrable projective line bundle on a proper hyperbolic curve of positive characteristic satisfying certain conditions. Dormant indigenous bundles were introduced and studied in the $p$-adic Teichmuller theory developed by S. Mochizuki. Kirti Joshi proposed a conjecture concerning an explicit formula for the degree over the moduli stack of curves of the moduli stack classifying dormant indigenous bundles. In this paper, we give a proof for this conjecture of Joshi.
Using Hitchin's parameterization of the Hitchin-Teichmüller component of the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the $SL(n,\mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian metric solving the Higgs bundle equations. This analysis is used to estimate the asymptotics of the corresponding family of flat connections as we scale the differentials by a real parameter. We consider Higgs fields that have only one holomorphic differential $q_n$ of degree $n$ or $q_{n-1}$…
This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is related to the field of higher Teichmüller theory. Our main tool is the theory of Higgs bundles. We try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples which are our motivating…
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 first give an exposition of how the Polyakov path integral for the bosonic string produces a natural mapping class group invariant measure, $d(Poly)$, on the Teichmüller space of Riemann surfaces of each fixed genus. The description of $d(Poly)$ via the Mumford isomorphisms for determinant bundles over Teichmüller space is also explained. We then report on our recent results with I.Biswas and D.Sullivan, where we succeed in coherently fitting together this Polyakov-Mumford construction over the universal direct limit, $T_\infty$, of all the finite genus Teichmüller spaces. The virtual autom…
We consider the moduli space of polystable $L$-twisted $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a real reductive Lie group, and $L$ is a holomorphic line bundle over $X$. Evaluating the Higgs field at a basis of the ring of polynomial invariants of the isotropy representation, one defines the Hitchin map. This is a map to an affine space, whose dimension is determined by $L$ and the degrees of the polynomials in the basis. Building up on the work of Kostant-Rallis and Hitchin, in this paper, as a first step in the study of the Hitchin map, we construct a section of th…
We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree $0$ over a compact Riemann surface and the affine bundle $\mathscr{A}\to\mathscr{N}$ of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of of the quasi-Fuchsian projective connections - local holomorphic sections of $\mathscr{A}$ - that allow to pull back the Liouville symplectic form on $T^{*}\mathscr{N}$ to $\mathscr{A}$. We prove that the pullback of the Goldman form to $\mathscr{A}$ by the Riemann-Hilbert correspondence coincides with …
The tau function on the moduli space of generic holomorphic 1-differentials on complex algebraic curves is interpreted as a section of a line bundle on the projectivized Hodge bundle over the moduli space of stable curves. The asymptotics of the tau function near the boundary of the moduli space of 1-differentials is computed, and an explicit expression for the pullback of the Hodge class on the projectivized Hodge bundle in terms of the tautological class and the classes of boundary divisors is derived. This expression is used to clarify the geometric meaning of the Kontsevich-Zorich formula …
Let $k$ be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve $X_1$ defined over $k$, there exists a lifting $X$ of the curve to the ring $W(k)$ of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over $X/W$. As a consequence, it gives rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group $π_1(X_K)$ of the generic fiber of $X$. This curve $X$ and its associated representation is in close relation wit…
We study closures of GL_2(R)-orbits on the total space of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that, in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to strata. For non-generic strata similar results can be shown by a case-by-case inspection. We also propose to study a notion of 'linear manifold' that comprises Teichmueller curves, Hilbert modular s…
We consider the action of a finite subgroup of the mapping class group $Mod(S)$ of an oriented compact surface $S$ of genus $g \geq 2$ on the moduli space $\mathcal{R}(S,G)$ of representations of $π_1(S)$ in a connected semisimple real Lie group $G$. Kerckhoff's solution of the Nielsen realization problem ensures the existence of an element $J$ in the Teichmüller space of $S$ for which $Γ$ can be realised as a subgroup of the group of automorphisms of $X=(S,J)$ which are holomorphic or antiholomorphic. We identify the fixed points of the action of $Γ$ on $\mathcal{R}(S,G)$ in terms of $G$-Higg…
This work is motivated by two central questions in the birational geometry of moduli spaces of curves -- Fulton's conjecture and the effective cone of $\bar M_g$. We study the algebro-geometric aspect of Teichmuller curves parameterizing square-tiled surfaces with two applications: (a) there exist infinitely many rigid curves on the moduli space of hyperelliptic curves, they span the same extremal ray of the cone of moving curves and their union is Zariski dense, hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for e…
