On the rational homotopy type of a moduli space of vector bundles over a curve

Abstract

We study the rational homotopy of the moduli space N X that parametrizes the isomorphism classes of all stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g, with g ≥ 2. The symplectic group Aut(H 1 (X, Z)) ∼ = Sp(2g, Z) has a natural action on the rational homotopy groups π n (N X )⊗ Z Q. We prove that this action extends to an action of Sp(2g, C) on π n (N X )⊗ Z C. We also show that π n (N X )⊗ Z C is a non-trivial representation of Sp(2g, C) ∼ = Aut (H 1 (X, C)) for all n ≥ 2g − 1. In particular, N X 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 π n (N X ) ⊗ Z C.

Rationally elliptic toric varieties

“…Elliptic complex manifolds of complex dimension less than or equal to 2 have been classified in [1]. Also, the rational homotopy type of moduli spaces of certain vector bundles over complex curves has been analysed in [3]. Our results here complement these references and show, in particular, the existence of a very large number of (non isomorphic) hyperbolic varieties.…”

“…On the other hand, Example 3.5( 6) exhibits a hyperbolic toric variety X of complex dimension 3 with P X (t) = (1 + t 2 ) 3 . Such examples do not exist for the other possible choice of the Poincaré polynomial in the same dimension given by Theorem 3.3, namely the polynomial (1 + t 2 )(1 + t 2 + t 4 ).…”

“…The 3-dimensional elliptic smooth toric varieties are, up to isomorphism, CP 3 , P(O CP 2 ⊕O CP 2 (c)), P(O CP 1 ⊕O CP 1 (a)⊕O CP 1 (b)), and CP 1 -bundles over Hirzebruch surfaces. These varieties have the rational homotopy type of CP 3 #CP 3 , CP 1 ×CP 2 and a quotient (S 3 × S 3 × S 3 )/T 3 respectively.…”

“…If b 2 = 1, then X = CP 3 . (If b 2 = 2, then X is either P(O CP 2 ⊕ O CP 2 (c)) or P(O CP 1 ⊕ O CP 1 (a) ⊕ O CP 1 (b)) and thus, it has the rational homotopy type of CP 3 #CP3 or CP 1 × CP 2 respectively. (3) If b 2 = 3, then X is a CP 1 -bundle over a Hirzebruch surface and has the rational homotopy type of a quotient (S 3 × S 3 × S 3 )/T 3 .…”