On the Rational Homotopy Type of a Moduli Space of Vector Bundles over a Curve
نویسنده
چکیده
We study the rational homotopy of the moduli space NX of stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g ≥ 2. The symplectic group Aut(H1(X,Z)) ∼= Sp(2g,Z) has a natural action on the rational homotopy groups πn(NX)⊗ZQ. We prove that this action extends to an action of Sp(2g,C) on πn(NX)⊗ZC. We also show that πn(NX)⊗ZC is a non–trivial Sp(2g,C)–representation for each n ≥ 2g−1. In particular, NX is a rationally hyperbolic space. In the special case where g = 2, for each n ∈ N, we compute the leading Sp(2g,C)–representation occurring in πn(NX)⊗Z C.
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