Lectures on the Sl(2,r) Action on Moduli Space
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چکیده
Suppose g ≥ 1, and let α = (α1, . . . , αn) be a partition of 2g− 2, and let H(α) be a stratum of Abelian differentials, i.e. the space of pairs (M,ω) where M is a Riemann surface and ω is a holomorphic 1-form on M whose zeroes have multiplicities α1 . . . αn. The form ω defines a canonical flat metric on M with conical singularities at the zeros of ω. Thus we refer to points of H(α) as flat surfaces or translation surfaces. For an introduction to this subject, see the survey [Zo]. The spaceH(α) admits an action of the group SL(2,R) which generalizes the action of SL(2,R) on the space GL(2,R)/SL(2,Z) of flat tori. Period Coordinates. Let Σ ⊂ M denote the set of zeroes of ω. Let {γ1, . . . , γk} denote a symplectic Z-basis for the relative homology group H1(M,Σ,Z). We can define a map Φ : H(α)→ C by
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Lectures on the Sl(2,r) Action on Moduli Space
Suppose g ≥ 1, and let α = (α1, . . . , αn) be a partition of 2g− 2, and let H(α) be a stratum of Abelian differentials, i.e. the space of pairs (M,ω) where M is a Riemann surface and ω is a holomorphic 1-form onM whose zeroes have multiplicities α1 . . . αn. The form ω defines a canonical flat metric on M with conical singularities at the zeros of ω. Thus we refer to points of H(α) as flat sur...
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