Localization for $K$-contact manifolds

Abstract

We prove an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology of K-contact manifolds. As a consequence, we deduce analogues of Witten's nonabelian localization and the Jeffrey-Kirwan residue formula, which relate equivariant basic integrals on a contact manifold M to basic integrals on the contact quotient M 0 := µ −1 (0)/G, where µ denotes the contact moment map for the action of a torus G. In the special case that M → N is an equivariant Boothby-Wang fibration, our formulae reduce to the usual ones for the symplectic manifold N . ContentsRemark 1.5. In §5.1 we explain in detail how Theorems 1.1 and 1.4 may be used to deduce the analogous theorems for symplectic manifolds that occur as M/F in the case that R induces a free S 1 -action. In this sense, these theorems provide a strict generalization of their symplectic analogues, at least in the case of an integral symplectic form and a Hamiltonian group action that lifts to the S 1 -bundle in the Boothby-Wang fibration [BW58].Remark 1.6. The first named author has obtained a surjectivity result for the basic contact Kirwan map [Cas17]. Since basic cohomology satisfies Poincaré duality (see Lemma 2.3), Theorem 1.4 in principle provides a method to compute the kernel of theProposition 4.3. The distribution Q η (y) may be represented by a piecewise polynomial function.As in the proof of Lemma 4.2, the previous equation becomesRecall that the local normal form of the moment map is given by µ(p, z) = z. Then, for sufficiently small y, δ(−µ − y) is supported away from S h and it follows that ∆ = 0. This means that, for sufficiently small y, Q η (y) = 1 (2π) s/2 g µ −1 (0)×B h α ∧ π * q * η 0 ∧ e idα+i −µ−y,φ dφ = (2π) s/2 µ −1 (0)×B h α ∧ π * q * η 0 ∧ e idα δ(−µ − y)

Basic Kirwan surjectivity for K-contact manifolds

“…The proof of [GS99, Theorem 11.1.1] is equally applicable in the basic (see also [CF17]) and boundary setting so that we obtain…”

“…A basic equivariant Thom form is a closed form τ ∈ C d G,c (νA, F ) satisfying p * τ = 1, with p * : C k G,c (νA, F ) → C k−d G (A, F ) denoting fibrewise integration. A basic equivariant Thom form can be constructed analogously to [GS99, Chapter 10] with an invariant basic connection form (see also [CF17]). Note that a G-invariant basic connection form θ has to exist: By [Mol88, Proposition 2.8], there always exists a connection that is adapted to the lifted foliation, i.e., such that the tangent spaces to the leaves are horizontal.…”