Euler characteristics of Teichmüller curves in genus two

By Matt Bainbridge. 2006

115 citations recorded by the source corpus.

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 are naturally 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 Teichmiiller curves in certain compactifications of the Hilbert modular surfaces. This is done by defining meromorphic sections of line bundles over Hilbert modular surfaces which vanish along these Teichmuller curves. We apply these results to calculate the Siegel-Veech constants for counting closed billiards paths in certain L-shaped polygons. We also calculate the Lyapunov exponents of the Kontsevich-Zorich cocycle for any ergodic, SL 2 (R)-invariant measure on the moduli space of Abelian differentials in genus two (previously calculated in unpublished work of Kontsevich and Zorich).

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d05da672fec9d8f8b3f798691a8ddc78a8a2b65f

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In Computational Theory and Mathematics

1. G. Forni
2. Carlos Matheus
3. Martin Möller
4. C. Ulcigrai
5. Krzysztof Frkaczek
6. A. Zorich
7. Matt Bainbridge

In Geometry and Topology

1. Carlos Matheus
2. G. Forni
3. A. Zorich
4. Martin Möller
5. J. Yoccoz
6. Matt Bainbridge
7. C. Ulcigrai
8. Krzysztof Frkaczek

In Mathematical Physics

1. Carlos Matheus
2. A. Zorich
3. G. Forni
4. Martin Möller
5. J. Yoccoz
6. C. Ulcigrai
7. Krzysztof Frkaczek
8. Matt Bainbridge

Citations in the corpus

• Non-ergodic Z-periodic billiards and infinite translation surfaces (2011, background): “2In [4] Bainbridge actually computes the explicit value of λ2 for any μ probability measure invariant for the Teichmüller flow in the genus two strata H(2) and H(1, 1).”
• The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3 (2012, background): “In fact, it is sufficient also in genus two, but we ignore this case as the boundary of the eigenform locus was previously calculated in the genus two case in [Bai07].”
• A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle (2010, background): “Bainbridge result, based on C. McMullen’s classification [33] theorem for SL(2,R)-invariant probability measures in genus 2, implies the simplicity of the Kontsevich–Zorich spectrum.”
• A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces (2013, background): “[6]): the nontrivial exponents are ±3 if ω has a double zero,±1 2 if ω has two simple zeros.”
• SQUARE-TILED CYCLIC COVERS (2010, background): “Bainbridge [4] has proved that the second exponent λ2 equals to 1/2 for all ergodic SL(2,R)-invariant measures supported in the stratum H (1,1), corresponding to two simple zeros of the holomorphic differential, andλ2 equals to 1/3 for all ergodic SL(2,R)-invariant measures supported in the stratum ”