Connected components of the moduli spaces of Abelian differentials with prescribed singularities

Abstract

Consider the moduli space of pairs (C,w) where C is a smooth compact complex curve of a given genus and w is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.

42 pages, 12 figures, LaTeX

Cited by in corpus (164)

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• The moduli space of twisted canonical divisors
• Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials
• A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces
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• Spin Hurwitz numbers and topological quantum field theory
• Branched projective structures with quasi-Fuchsian holonomy
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• Full rank affine invariant submanifolds
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• The Deligne-Mumford and the Incidence Variety Compactifications of the Strata of $\Omega {\mathcal{M}}_g$
• Non-ergodic Z-periodic billiards and infinite translation surfaces
• The Liouville-type theorem for integrable Hamiltonian systems with incomplete flows
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• Volumes of strata of Abelian differentials and Siegel-Veech constants in large genera
• GL(2,R)-Invariant Measures in Marked Strata: Generic Marked Points, Earle-Kra for Strata, and Illumination
• Lower bounds for Lyapunov exponents of flat bundles on curves
• Classification of Rauzy-Veech groups: proof of the Zorich conjecture
• Ergodic infinite group extensions of geodesic flows on translation surfaces
• Rel leaves of the Arnoux-Yoccoz surfaces
• Sobolev regularity of solutions of the cohomological equation
• Enumeration of meanders and Masur-Veech volumes
• GL(2,R) Orbit Closures in Hyperelliptic Components of Strata
• Quotient groups of the fundamental groups of certain strata of the moduli space of quadratic differentials
• Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
• Affine geometry of strata of differentials
• Square-tiled surfaces and rigid curves on moduli spaces
• Cluster exchange groupoids and framed quadratic differentials
• Drawing cone spherical metrics via Strebel differentials
• Translation surfaces and the curve graph in genus two
• Rank two affine submanifolds in $\mathcal{H}(2,2)$ and $\mathcal{H}(3,1)$
• Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations
• Quasimodularity and large genus limits of Siegel-Veech constants
• Teichmüller dynamics in the eyes of an algebraic geometer
• Self-Inverses, Lagrangian Permutations and Minimal Interval Exchange Transformations with Many Ergodic Measures
• Two-Dimensional Yang-Mills Theory and Moduli Spaces of Holomorphic Differentials
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• Degenerations of Abelian Differentials
• Degenerations of quadratic differentials on CP1
• Moduli spaces for Lamé functions and Abelian integrals of the second kind
• The Hausdorff Dimension of Non-Uniquely Ergodic directions in $\mathcal{H}(2)$ is almost everywhere $1/2$
• The Pillowcase Distribution and Near-Involutions
• Exponential Mixing for the Teichmuller flow in the Space of Quadratic Differentials
• Teichmueller curves in genus three and just likely intersections in $G_m^{n}x{G}_a^n$
• A short introduction to translation surfaces, Veech surfaces, and Teichmueller dynamics
• Equivariant cohomology of moduli spaces of genus three curves with level two structure
• Horocycle dynamics: new invariants and eigenform loci in the stratum H(1,1)
• Baker-Akhiezer spinor kernel and tau-functions on moduli spaces of meromorphic differentials
• Parallelogram decompositions and generic surfaces in H^{hyp}(4)
• Parity of the spin structure defined by a quadratic differential
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• On the wall-crossing formula for quadratic differentials
• On the cohomological dimension of the moduli space of Riemann surfaces
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• Non-varying sums of Lyapunov exponents of Abelian differentials in low genus
• Pseudo-Anosov homeomorphisms on translation surfaces in hyperelliptic components have large entropy
• A large deviations bound for the Teichmuller flow
• Limit Theorems for Translation Flows
• Counting pairs of saddle connections
• On non-primitive Weierstrass points
• Weak-mixing polygonal billiards
• Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow
• GL^+(2,R)-orbits in Prym eigenform loci
• Reconstructing orbit closures from their boundaries
• A non-varying phenomenon with an application to the wind-tree model
• Geometry of periodic regions on flat surfaces and associated Siegel-Veech constants
• Strata of abelian differentials and the Teichmueller dynamics
• Rank One Orbit Closures in H^{hyp}(g-1,g-1)
• Geodesics on Flat Surfaces
• Explicit Teichmüller curves with complemetary series
• Self-inverses in Rauzy Classes
• Classification of Rauzy classes in the moduli space of quadratic differentials
• On the locus of curves with an odd subcanonical marked point
• On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials
• Hodge and Teichmüller
• Hyperelliptic Gorenstein curves and logarithmic differentials
• Existence and Uniqueness of the Measure of Maximal Entropy for the Teichmueller Flow on the Moduli Space of Abelian Differentials
• Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow
• Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms
• Decay of Correlations for the Rauzy-Veech-Zorich Induction Map on the Space of Interval Exchange Transformations and the Central Limit Theorem for the Teichmueller Flow on the Moduli Space of Abelian Differentials
• The image in the moduli space of curves of strata of meromorphic and quadratic differentials
• Moduli spaces of branched covers of Veech surfaces I: d-symmetric differentials
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• Double Scaling String Theory of QCD in Two Dimensions
• Non arithmetic affine invariant orbifolds in Hodd(2, 2) and H(3, 1)
• Moduli space of meromorphic differentials with marked horizontal separatrices
• Integrable Hamiltonian systems with incomplete flows and Newton's polygons
• Tau functions, Hodge classes and discriminant loci on moduli spaces of Hitchin's spectral covers
• Moduli spaces of meromorphic functions and determinant of Laplacian
• Loci of curves with subcanonical points in low genus
• The Arnoux-Yoccoz Teichmueller disc
• Diagonal changes for surfaces in hyperelliptic components
• On the asymptotics of determinant of Laplacian at the principal boundary of the principal stratum of the moduli space of Abelian differentials
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• Systolic geometry of translation surfaces
• Period realization of meromorphic differentials with prescribed invariants
• Variational formulas on spaces of SL(2) Hitchin's spectral covers
• Bowen's construction for the Teichmueller flow
• The cohomological equation for Roth type interval exchange maps
• Crossing the transcendental divide: from translation surfaces to algebraic curves
• Semiclassical Entropy of BPS States in 4d $\mathcal{N}=2$ Theories and Counts of Geodesics
• On the Non-Uniform Hyperbolicity of the Kontsevich-Zorich Cocycle for Quadratic Differentials
• A combinatorial move on the set of Jenkins-Strebel differentials
• Counting saddle connections in a homology class modulo $q$
• Rotational component spaces for infinite-type translation surfaces
• Logarithmic asymptotics for the number of periodic orbits of the Teichmueller flow on Veech's space of zippered rectangles
• Positivity of divisor classes on the strata of differentials
• Exponential mixing for the Teichmuller flow
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• Continuity of the orthogeodesic foliation and ergodic theory of the earthquake flow
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• A priori bounds for GIETs, affine shadows and rigidity of foliations in genus 2
• Billiards in right triangles and orbit closures in genus zero strata
• Finite translation surfaces with maximal number of translations
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• Diagonal flow detects topology of strata
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• Shrinking rates of horizontal gaps for generic translation surfaces
• Meanders, hyperelliptic pillowcase covers, and the Johnson filtration
• Existence of closed geodesics through a regular point on translation surfaces
• Quadratic differentials as stability conditions: collapsing subsurfaces
• Monodromy kernels for strata of translation surfaces
• Minimal non uniquely ergodic flipped IETs
• Horizontal strips and spaces of quadratic differentials
• On the class of caustic on the moduli space of odd spin curves
• Horocycle flow orbits and lattice surface characterizations
• Tau function and moduli of spin curves
• Residual generic ergodicity of periodic group extensions over translation surfaces
• The asymptotic behavior of the monodromy representations of the associated families of compact CMC surfaces
• Subcanonical points on projective curves and triply periodic minimal surfaces in the Euclidean space
• Deviation of ergodic averages for area-preserving flows on surfaces of higher genus
• Finiteness results for Teichmueller curves
• The fine structure of Kontsevich-Zorich strata for genus 3
• The distribution of gaps for saddle connection directions
• Deviation of ergodic averages for rational polygonal billiards
• Linear manifolds in the moduli space of one-forms
• Irreducibility and stable rationality of the loci of Weierstrass points on curves of genus at most six
• Krein formula and S-matrix for Euclidean Surfaces with Conical Singularities
• Dynamics of Non-Classical Interval Exchanges
• Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds
• On Convergence of Random Walks on Moduli Space
• Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture