ar X iv : m at h / 98 04 13 5 v 2 [ m at h . D G ] 2 4 A ug 1 99 8 DEGENERATE CHERN - WEIL THEORY AND EQUIVARIANT COHOMOLOGY

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our work [5] on reproving wall crossing formulas in Seiberg-Witten theory, where the Lie group is the circle. As applications, we derive two localization formulas of Kalkman type for G = SU(2) or SO(3)actions on compact manifolds with boundary. One of the formulas is then used to yield a very simple proof of a localization formula due to Jeffrey-Kirwan [15] in the case of G = SU(2) or SO(3). Throughout this paper, G will be a compact connected Lie group, with g as its Lie algebra. Assume that G acts freely on a smooth manifold P . Then the quotient map P → P/G = M gives P a structure of principal G-bundle. The celebrated Chern-Weil theory gives us a homomorphism cw : S(g) → H(M), (1) called the Chern-Weil homomorphism. Here S(g) is the algebra of polynomials on g which is invariant under the adjoint representation of G on g. The Chern-Weil construction uses a connection 1-form ω ∈ (Ω(P ) × g) and its curvature 2-form Ω = dω + 12 [ω, ω]. The equation dΩ = [Ω, ω] can be used to show that for any invariant polynomial F ∈ S(g), F (Ω) is the pullback of a closed form on M . This defines the homomorphism (1). Furthermore, for two connections ω and ω with curvatures Ω and Ω respectively, there is a canonically defined differential form T(ω0,ω1)F on M , called the transgression form, such that dT(ω0,ω1)F = F (Ω )− F (Ω). Therefore, the Chern-Weil homomorphism is independent of the choice of ω. We call this Chern’s formulation. Cartan [7] presented Weil’s formulation, which we shall review in §1. Through Weil’s formulation, Cartan (§5 in [8]) discovered that the Chern-Weil homomorphism can be factored as S(g) φ → H G(P ) (r)∗ → H(M), whereH G(P ) is the equivariant cohomology of P , and φ is the homomorphism which gives H G(P ) the structure of an H (BG) ∼= S(g)-module. The homomorphism (r)∗ is induced from a homomorphism on the chain level obtained by a similar Chern-Weil construction. 1991 Mathematics Subject Classification: Primary 55N91, 57R20, 57S15, 58F05. The authors are supported in part by NSF 1 2 HUAI-DONG CAO & JIAN ZHOU In this paper, we shall generalize the above picture to the case that the G-action on a smooth manifold W is only locally free on a dense open set W 0 ⊂ W . Using a connection ω on W , and a cut-off function f , we shall construct homomorphisms cw f : S(g ) → H G(W ), and (r f )∗ : H ∗ G(W ) → H G(W ), such that cw f = (r G f )∗ ◦φ. Here (r f )∗ is induced from a homomorphism r f G at the chain level in Cartan model for equivariant cohomology. We shall also construct transgression operator to show that cw f and (r G f )∗ are independent of the choices of connection ω and the cut-off function f . An important observation, pointed out to us by Professor Michèle Verge, is that when one takes f ≡ 0, then our calculation shows that the homomorphism (r f )∗ is the identity map. The main results of this paper are stated in Theorem 2.1-2.6. We call these results the degenerate ChernWeil theory. We remark that our approach corresponds to Chern’s formulation. It depends on calculations by brute force. It is interesting to find a Weil’s formulation, which might make the argument simpler. Even though the results of this paper provide an invariant for non-free group actions (which is interesting in its own respect), the main motivation is to give a method of choosing a nice representative for an equivariant cohomological class to obtain localization results. At the chain level, for suitable choice of ω and f , r f gives us a nice way to change an equaivraint closed form α within its equivariant cohomological class to r f (α), with the following property: in a neighborhood of the sigular set of the group action, r f (α) = α, outside a larger neighborhood, r G f (α) is the pullback of an ordinary differential form from the quotient. This provides a simple explanation for the localization phenomenon in equivariant cohomology. When deg(α) = dim(X), one often considers integral ∫ X α. But we have ∫