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

By M. Kontsevich, A. Zorich. 2002

589 citations recorded by the source corpus.

Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c703954be982bc00428c369b5000a61d21c05866

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In Geometry and Topology

1. Martin Möller
2. A. Zorich
3. Martin Moeller
4. A. Eskin
5. Dawei Chen
6. G. Forni
7. Carlos Matheus
8. Artur Avila

In Mathematical Physics

1. A. Zorich
2. Martin Möller
3. Martin Moeller
4. Dawei Chen
5. A. Eskin
6. Artur Avila
7. Carlos Matheus
8. G. Forni

Citations in the corpus

• SQUARE-TILED CYCLIC COVERS (2010, background): “The ambient stratum H (2,2,2) for the square-tiled surface corresponding to the cyclic cover M6(1,1,1,3) contains two connected components representing even and odd parity of the spinstructure of the corresponding holomorphic 1-form, see [23].”
• Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials (2007, methodology): “ture of this differential, see [16] and Appendix C.”
• Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow (2010, background): “Each stratum has a natural complex affine structure, though it is not necessarily connected, the (finitely many) connected components having been classified by Kontsevich-Zorich [KZ03] and Lanneau [Lan08].”
• CONNECTED COMPONENTS OF THE STRATA OF THE MODULI SPACE OF MEROMORPHIC DIFFERENTIALS (2012, methodology): “1, which is the analogous version of a key argument in [14].”
• Quantitative recurrence and large deviations for Teichmüller geodesic flow (2005, background): “Strata are not always connected, however, they have at most finitely many components [16], and are invariant under SL(2,R).”
• Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials (2019, background): “For special μ the strata ΩMg,n(μ) can be disconnected, with up to three spin and hyperelliptic connected components, classified by Kontsevich and Zorich ([KZ03]).”
• Billiards and Teichmüller curves on Hilbert modular surfaces (2003, background): “For more on spaces of holomorphic 1-forms and quadratic differentials, see for example [Mas1], [V2], [Ko], [EO] and [KZ].”
• Limit theorems for translation flows (2008, background): “To a connected component H of the space Mκ one can assign a Rauzy class R in such a way that the following is true [37], [30].”
• Exponential mixing for the Teichmüller flow (2005, background): “The strata Mg; are not necessarily connected (a classication of connected components is given in [ KZ ]).”
• Quasimodularity and large genus limits of Siegel-Veech constants (2016, background): “if the profile consists of cycles of odd length only, the parity of the spin structure of [31] distinguishes two components.”
• A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle (2010, background): “The strata are in general not connected and can have up to three connected components [28].”
• Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants (2002, methodology): “Moduli spaces of these structures have been studied in [Ve4], [MS], [Ko], [ KonZo ], and [EMa].”