Consider a simply connected, smooth, projective, complex surface X . Let Mkf ( X ) be the moduli space of framed irreducible anti-self-dual connections on a principal SU(2)-bundle over X with second Chern class k > 0, and let Ckf (X) be the corresponding space of all framed connections, modulo gauge equivalence. A famous conjecture by M. Atiyah and J. Jones says that the inclusion map Mkf ( X ) → Ckf ( X ) induces isomorphisms in homology and homotopy through a range that grows with k. In this paper, we focus on the fundamental group, π1. When this group is finite or polycyclic-by-finite, we prove that if the π1-part of the conjecture holds for a surface X, then it also holds for the surface obtained by blowing up X at n points. As a corollary, we get that the π1-part of the conjecture is true for any surface obtained by blowing up n times the complex projective plane at arbitrary points. Moreover, for such a surface, the fundamental group π1 (M f ( X )) is either trivial or isomorphic to Z2 .
On the geometry of moduli spaces of anti-self-dual connections
Ballico, Edoardo;
2012-01-01
Abstract
Consider a simply connected, smooth, projective, complex surface X . Let Mkf ( X ) be the moduli space of framed irreducible anti-self-dual connections on a principal SU(2)-bundle over X with second Chern class k > 0, and let Ckf (X) be the corresponding space of all framed connections, modulo gauge equivalence. A famous conjecture by M. Atiyah and J. Jones says that the inclusion map Mkf ( X ) → Ckf ( X ) induces isomorphisms in homology and homotopy through a range that grows with k. In this paper, we focus on the fundamental group, π1. When this group is finite or polycyclic-by-finite, we prove that if the π1-part of the conjecture holds for a surface X, then it also holds for the surface obtained by blowing up X at n points. As a corollary, we get that the π1-part of the conjecture is true for any surface obtained by blowing up n times the complex projective plane at arbitrary points. Moreover, for such a surface, the fundamental group π1 (M f ( X )) is either trivial or isomorphic to Z2 .I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione