2.1 Equivariant cohomology

Let M be a smooth manifold equipped with a smooth action of a compact Lie group G. Write $S^\omega (\mathbf {\mathfrak g}^*)$ for the complex vector space of analytic series on $\mathbf {\mathfrak g}$ with complex coefficients converging in a neighborhood of $0\in \mathbf {\mathfrak g}$ . The coadjoint action of G on $\mathbf {\mathfrak g}^*$ induces a natural action on $S^\omega (\mathbf {\mathfrak g}^*)$ .

Definition 2.1. The complex of analytic G-equivariant differential forms is the complex vector space of G-invariants

$$ \begin{align*} \Omega^*_G(M):=\left(\Omega^*(M,{\mathbb C})\otimes S^\omega(\mathbf{\mathfrak g}^*)\right)^G\, \end{align*} $$

for the natural G-action on the tensor product, equipped with the equivariant differential

(2.1) $$ \begin{align} d_{\mathbf{\mathfrak g}}:\Omega_G^*(M)&\longrightarrow\Omega_G^*(M),\nonumber\\ \sigma(X)&\longmapsto d\sigma(X)+2\pi i\,\widetilde{X}\mathbin{\lrcorner}\sigma(X)\,. \end{align} $$

The cohomology of the complex $(\Omega ^*_G(M),d_{\mathbf {\mathfrak g}})$ is denoted by $H^*_G(M)$ and called equivariant cohomology of M with analytic coefficients.

In the above definition and in the sequel, we will often write equivariant differential forms $\sigma (X)\in \Omega ^*_G(M)$ with explicit dependence on the variable $X\in \mathbf {\mathfrak g}$ . We point out that there are several competing conventions for the definition of the equivariant differential in the literature, with the factor $2\pi i$ in (2.1) replaced by other constants. Similarly, the sign in (2.10) is a convention. We follow here Meinrenken’s conventions in [Reference Meinrenken28]. The following version of the Stokes Lemma for equivariant differential forms can be found for instance in [Reference Bott and Tu5, §4].

Lemma 2.2 (Equivariant Stokes’ lemma)

Let $U\subset M$ be an open set such that its closure $\overline U\subset M$ is a compact submanifold with boundary. Then, for any $\sigma \in \Omega ^*_G(M)$ we have

$$ \begin{align*} \int_U d_{\mathbf{\mathfrak g}}\sigma(X)=\int_{\partial\overline U}\sigma(X)\,. \end{align*} $$

For a submanifold $N\subset M$ we write $\nu _{\Sigma _N}:\Sigma _N\to N$ for its normal bundle inside $TM$ and identify N canonically with the zero section in $\Sigma _N$ . The following version of the classical homotopy lemma for equivariant differential forms follows for instance from [Reference Goertsches and Zoller10, Theorem 6.1].

Lemma 2.3 (Equivariant homotopy lemma)

Let $N\subset M$ be a submanifold preserved by the action of G, and let $\Phi _N:V_N\to U_N$ be a G-equivariant diffeomorphism between a tubular neighborhood $V_N\subset \Sigma _N$ of the zero section of $\Sigma _N$ and a tubular neighborhood $U_N\subset M$ of N. Then, for any equivariant cohomology class $[\varrho ] \in H^*_G(M)$ there exists an equivariant form $\beta _N \in \Omega _{G}^\ast (V_N)$ such that

(2.2) $$ \begin{align} \Phi_{N}^\ast(\varrho|_{U_N}) = \nu_{\Sigma_N}^\ast \mathrm{inc}_N^*\varrho|_{V_N}+d_{\mathbf{\mathfrak g}} \beta_N. \end{align} $$

Furthermore, the equivariant form $\beta _N \in \Omega _{G}^\ast (V_N)$ can be chosen such that $\operatorname {{\mathrm {inc}}}_N^*\beta _N\equiv 0$ and such that for any open set $S\subset V_N$ on which we already have $\Phi _{N}^\ast (\varrho |_{U_N})|_{S} = \nu _{\Sigma _N}^\ast \mathrm {inc}_N^*\varrho |_{S}$ , one has $\beta _N|_{S}\equiv 0$ .

The second part of Lemma 2.3, which will be used in Section 5 to establish Lemma 5.3, follows from the fact that N is a strong deformation retract of $U_N$ and the explicit form of the homotopy operator used for instance in [Reference Goertsches and Zoller10, Theorem 6.1] to construct the equivariant form $\beta _N \in \Omega _{G}^\ast (V_N)$ of Formula (2.2).

Consider now the important special case when the G-action is locally free, so that the quotient map

(2.3) $$ \begin{align} \pi:M\longrightarrow M/G \end{align} $$

is a G-principal bundle over the orbifold $M/G$ . We then have the following basic result.

Proposition 2.4. The pullback by the quotient map (2.3) induces an isomorphism of complexes

(2.4) $$ \begin{align} \pi^*:(\Omega(M/G,{\mathbb R}),d)\xrightarrow{~\sim~}(\Omega(M)^G_{\mathrm{bas}},d)\,, \end{align} $$

where the subcomplex of basic differential forms $\Omega (M)^G_{\mathrm {bas}}\subset \Omega (M,{\mathbb R})$ is defined by

(2.5) $$ \begin{align} \Omega(M)^G_{\mathrm{bas}}:=\{\sigma\in\Omega^*(M,{\mathbb R})^G~|~\widetilde{X}\mathbin{\lrcorner}\sigma=0\}\,, \end{align} $$

The cohomology of the complex (2.5) is called the basic cohomology of M, and Proposition 2.4 shows that it is isomorphic to the de Rham cohomology $H^*(M/G)$ of the orbifold $M/G$ . On the other hand, one has the following fundamental notion in Chern-Weil theory, which will be of crucial importance in this paper.

Definition 2.5. A connection for a locally free G-action on M is a $\mathbf {\mathfrak g}$ -valued $1$ -form $\Theta \in \Omega ^1(M,\mathbf {\mathfrak g})$ satisfying for all $X\in \mathbf {\mathfrak g}$

$$ \begin{align*} \tilde X \mathbin{\lrcorner} \Theta=X\,. \end{align*} $$

In what follows, we will mainly be interested in the case where $G=S^1$ is the circle group, which we realize as the subgroup of complex numbers of modulus $1$ , inducing an identification

(2.6) $$ \begin{align} \mathbf{\mathfrak g}&\xrightarrow{\,\sim\,}{\mathbb R}\nonumber\\ X&\longmapsto x\,, \end{align} $$

in such a way that $X\in \mathbf {\mathfrak g}$ exponentiates to $e^{2\pi i x}\in S^1\subset {\mathbb C}$ . This induces in turn an identification $\mathbf {\mathfrak g}^*\simeq {\mathbb R}$ of the dual of the Lie algebra with ${\mathbb R}$ and of the Lebesgue measure on the interval $[0,1]$ with the normalized Haar measure on $S^1$ . Under this identification, Definition 2.1 becomes

(2.7) $$ \begin{align} \Omega^*_{S^1}(M)=\Omega(M,S^\omega({\mathbb R}))^{S^1}\,, \end{align} $$

the complex vector space of $S^1$ -invariant differential forms with values in entire analytic series of the variable $x\in {\mathbb R}$ . We will write $S^1$ -equivariant differential forms $\sigma (x)\in \Omega ^*_{S^1}(M)$ with explicit dependence in the variable $x\in {\mathbb R}$ , when they are understood in the identification (2.7), so that they can actually be seen as functions of $x\in {\mathbb R}$ with values in $S^1$ -invariant differential forms. Furthermore, Definition 2.5 of a connection for the $S^1$ -action on M becomes under this identification a $1$ -form $\Theta \in \Omega ^1(M,{\mathbb R})$ such that for any $X\in \mathbf {\mathfrak g}$ with image $x\in {\mathbb R}$ by (2.6), we have

(2.8) $$ \begin{align} \tilde X \mathbin{\lrcorner} \Theta=x\,. \end{align} $$

The following basic lemma is then a straightforward consequence of Proposition 2.4 and Definition 2.5 via the identification (2.6).

Lemma 2.6. Assume that the $S^1$ -action on M is locally free, let $\Theta \in \Omega ^1(M,{\mathbb R})$ be an associated connection in the sense of (2.8), and let $\sigma \in \Omega (M/S^1)$ be closed. Then we have

(2.9) $$ \begin{align} \int_M\pi^*\sigma\wedge\Theta=\int_{M/S^1}\sigma \end{align} $$

in terms of the isomorphism of complexes (2.4). In particular, the integral on the left-hand side of (2.9) only depends on the basic cohomology class of $\sigma $ .

2.2 Dirac operators and invariant Riemann-Roch numbers

To introduce our proper setup, let us now focus on the case of a Hamiltonian G-action on a compact connected symplectic manifold $(M,\omega )$ of dimension $2n$ . Recall that a G-equivariant map ${ \mathcal J}: M \to \mathbf {\mathfrak g}^\ast $ is called a moment map for such an action if for all $X\in \mathbf {\mathfrak g}$ , the function ${ \mathcal J}(X)\in \mathrm {C^{\infty }}(M,{\mathbb R})$ satisfies

(2.10) $$ \begin{align} d{ \mathcal J}(X) =-\widetilde X \mathbin{\lrcorner} \omega\,, \end{align} $$

where d is the de Rham differential and $\mathbin {\lrcorner }$ denotes the contraction. By definition, a Hamiltonian G-action on a symplectic manifold $(M,\omega )$ induces a locally free action on the level set ${ \mathcal J}^{-1}(\{0\})$ if and only if $0$ is a regular value of ${ \mathcal J}:M\to \mathbf {\mathfrak g}^*$ . For $G=S^1$ , under the identification (2.6) the moment map ${ \mathcal J}:M\to \mathbf {\mathfrak g}^*$ corresponds to a function ${ \mathcal J}\in \mathrm {C^{\infty }}(M,{\mathbb R})$ which we still call moment map, such that for any $X\in \mathbf {\mathfrak g}$ with image $x\in {\mathbb R}$ by (2.6), we have

(2.11) $$ \begin{align} { \mathcal J}(X)=x{ \mathcal J}\in\mathrm{C^{\infty}}(M,{\mathbb R})\,. \end{align} $$

Next, let $(M,\omega )$ be endowed with a so-called prequantizing line bundle $(L,h^L,\nabla ^L)$ , so that $(L,h^L)$ is a Hermitian line bundle over M with a Hermitian connection $\nabla ^L$ whose curvature $R^L\in \Omega ^2(M,{\mathbb C})$ satisfies the prequantization condition (1.2). We denote by $\mathrm {C^{\infty }}(M,L)$ the space of smooth sections of L. Let further G be a connected compact Lie group such that G acts on L over M, preserving the Hermitian metric $h^L$ and the connection $\nabla ^L$ . Such an action is called prequantized, and the induced action of G on $(M,\omega )$ then preserves the symplectic form $\omega $ . Following for instance [Reference Meinrenken28, (25)] adapted to our conventions, there is a canonical choice of moment map ${ \mathcal J}: M \to \mathbf {\mathfrak g}^\ast $ for a prequantized action of G on $(M,\omega )$ defined by the Kostant formula

(2.12) $$ \begin{align} { \mathcal J}(X)s:=\frac{i}{2\pi}\left(L_Xs-\nabla^{L}_{\widetilde X}s\right) \end{align} $$

for all $s\in \mathrm {C^{\infty }}(M,L)$ and $X\in \mathbf {\mathfrak g}$ , where $\widetilde X\in \mathrm {C^{\infty }}(M,TM)$ denotes the fundamental vector field on M associated with X and $L_X$ denotes the Lie derivative with respect to X induced by the G-action. This is the moment map which will underlie all our considerations from now on.

Next, choose a G-invariant compatible almost complex structure $J\in \operatorname {\mathrm {End}}(TM)$ over $(M,\omega )$ , inducing a splitting

(2.13) $$ \begin{align} TM\otimes{\mathbb C}=T^{(1,0)}M\oplus T^{(0,1)}M \end{align} $$

on the complexification $TM\otimes {\mathbb C}$ of $TM$ into the eigenspaces of J corresponding to the eigenvalues i and $-i$ , respectively. Consider the total exterior product

(2.14) $$ \begin{align} \Lambda( T^{*(0,1)}M):=\bigoplus_{j=0}^n \Lambda^j( T^{*(0,1)}M)\,, \end{align} $$

where $T^{*(0,1)}M$ denotes the dual bundle of $T^{(0,1)}M$ . For any $v\in TM$ with decomposition $v=v^{1,0}+v^{0,1}$ according to (2.13), we define its Clifford action on $\alpha \in \Lambda ( T^{*(0,1)}M)$ by

(2.15) $$ \begin{align} c(v)\alpha:=\sqrt{2}\,(v^{1,0})^*\wedge\alpha-v^{0,1}\mathbin{\lrcorner}\alpha\,, \end{align} $$

where $(v^{1,0})^*$ denotes the metric dual of $v^{1,0}$ in $T^{*(0,1)}M$ with respect to the induced Hermitian metric $g^{TM}$ defined by (1.3). As explained for instance in [Reference Lawson and Michelsohn23, Appendix D], the Clifford action (2.15) on $\Lambda ( T^{*(0,1)}M)$ is associated with the canonical $\mathrm {Spin}^c$ structure of the almost Hermitian manifold $(M,J,g^{TM})$ , and there is an induced connection $\nabla ^{\Lambda ( T^{*(0,1)}M)}$ on $\Lambda ( T^{*(0,1)}M)$ , which we call the Clifford connection. Following [Reference Ma and Marinescu26, §1.3.1], the Clifford connection $\nabla ^{\Lambda ( T^{*(0,1)}M)}$ is given in any local orthonormal frame $\{e_j\}_{j=1}^n$ of $(TM,g^{TM})$ by the formula

$$ \begin{align*} \nabla^{\Lambda( T^{*(0,1)}M)} =d+\frac{1}{4}\sum_{j,k=1}^{2n}\langle\Gamma^{TM} e_j,e_k\rangle_{g^{TM}}c(e_j)c(e_k)+\frac{1}{2}\Gamma^{\mathrm{det}\,}\,, \end{align*} $$

where $\Gamma ^{TM}$ is the local connection form of the Levi-Civita connection $\nabla ^{TM}$ associated with $g^{TM}$ and $\Gamma ^{\mathrm {det}\,}$ is the local connection form associated with the connection on $\mathrm {det}\,(T^{(1,0)}M):=\Lambda ^n( T^{(1,0)}M)$ induced by $\nabla ^{TM}$ . Finally, for any $m\in {\mathbb N}$ , we write $\Omega ^{0,*}(M,L^m)$ for the space of smooth sections of $\Lambda ( T^{*(0,1)}M)\otimes L^m$ , where $L^m$ denotes the m-th tensor power of L. We then have the following

Definition 2.7. For any $m\in {\mathbb N}$ , the Spin $^c$ -Dirac operator $D_m$ is defined in any local orthonormal frame $\{e_j\}_{j=1}^n$ of $(TM,g^{TM})$ by the formula

$$ \begin{align*} D_m=\sum\limits_{j=1}^{2n} c(e_j)\nabla^{\Lambda( T^{*(0,1)}M)\otimes L^m}_{e_j}:\Omega^{0,*}(M,L^m)\longrightarrow\Omega^{0,*}(M,L^m)\,, \end{align*} $$

where $\nabla ^{\Lambda ( T^{*(0,1)}M)\otimes L^m}$ is the connection on $\Lambda ( T^{*(0,1)}M)\otimes L^m$ induced by $\nabla ^{\Lambda ( T^{*(0,1)}M)}$ and $\nabla ^{L^m}$ .

By definition of the Clifford action (2.15), the $\mathrm {Spin}^c$ -Dirac operator $D_m$ interchanges the space $\Omega ^{0,+}(M,L^m) \subset \Omega ^{0,*}(M,L^m)$ of even-degree forms with the space $\Omega ^{0,-}(M,L^m)\subset \Omega ^{0,*}(M,L^m)$ of odd-degree forms in the decomposition (2.14) of $\Lambda ( T^{*(0,1)}M)$ . We write $D_m^\pm $ for the restriction of $D_m$ to $\Omega ^{0,\pm }(M,L^m)$ . On the other hand, as shown for instance in [Reference Ma and Marinescu26, Lemma 1.3.4], $D_m$ is a formally self-adjoint elliptic operator on $\Omega ^{0,*}(M,L^m)$ with respect to the $\mathrm {L}^2$ -Hermitian product induced by $g^{TM}$ and $h^L$ . In particular, it has finite-dimensional kernel inside $\Omega ^{0,*}(M,L^m)$ , which allows us to state the following definition.

Definition 2.8. For any $m\in {\mathbb N}$ , the Riemann-Roch number $\text {RR}(M,L^m)\in {\mathbb N}$ of the bundle $L^m$ over M is defined by the formula

$$ \begin{align*} \text{RR}(M,L^m):=\dim\operatorname{\mathrm{Ker}} D_m^+-\dim\operatorname{\mathrm{Ker}} D_m^-\,. \end{align*} $$

The standard invariance property of indices of elliptic operators with respect to deformations shows that these Riemann-Roch numbers do not depend on the choice of an almost complex structure J nor on $h^L$ and $\nabla ^L$ satisfying the prequantization condition (1.2).

On the other hand, recall that the action of G preserves all additional data on M and L by construction, so that there is an induced action of G on $\Omega ^{0,*}(M,L^m)$ commuting with $D_m$ . In particular, this induces unitary representations of G on the finite dimensional Hermitian vector spaces $\operatorname {\mathrm {Ker}} D_m^+$ and $\operatorname {\mathrm {Ker}} D_m^-$ . For any $g\in G$ , we write

$$ \begin{align*} \chi^{(m)}(g):=\operatorname{\mathrm{Tr}}_{\operatorname{\mathrm{Ker}} D_m^+}[g]-\operatorname{\mathrm{Tr}}_{\operatorname{\mathrm{Ker}} D_m^-}[g]\,. \end{align*} $$

We write $(\operatorname {\mathrm {Ker}} D_m^+)^G$ and $(\operatorname {\mathrm {Ker}} D_m^-)^G$ for the subspaces of G-invariant vectors inside $\operatorname {\mathrm {Ker}} D_m^+$ and $\operatorname {\mathrm {Ker}} D_m^-$ respectively.

Definition 2.9. For any $m\in {\mathbb N}$ , the G-invariant Riemann-Roch number ${\text {RR}^G(M,L^m)\in {\mathbb N}}$ of the bundle $L^m$ over M is defined by the formula

$$ \begin{align*} \text{RR}^G(M,L^m):=\dim(\operatorname{\mathrm{Ker}} D_m^+)^G-\dim(\operatorname{\mathrm{Ker}} D_m^-)^G\,. \end{align*} $$

Again by invariance of the index of elliptic operators, the G-invariant Riemann-Roch number does not depend on the choice of a G-invariant almost complex structure J nor on the choice of G-invariant $h^L$ and $\nabla ^L$ satisfying the prequantization condition (1.2). Note also that we have

(2.16) $$ \begin{align} \text{RR}^G(M,L^m)&=\int_G\operatorname{\mathrm{Tr}}_{\operatorname{\mathrm{Ker}} D_m^+}[g]\,dg-\int_G\operatorname{\mathrm{Tr}}_{\operatorname{\mathrm{Ker}} D_m^-}[g]\,dg=\int_G\,\chi^{(m)}(g)\,dg\,, \end{align} $$

where $dg$ is the normalized Haar volume form on G.

2.3 Equivariant characteristic forms and Riemann-Roch formulas

In the setting of Section 2.2, and following [Reference Berline, Getzler and Vergne2, Definition 7.5], we define for any Hermitian vector bundle $(E,h^E)\to M$ to which the G-action lifts and which is equipped with a G-equivariant Hermitian connection $\nabla ^E$ the associated moment map ${ \mathcal J}^{E}:M\to \operatorname {\mathrm {End}}(E)\otimes \mathbf {\mathfrak g}^*$ via the Kostant formula

$$ \begin{align*} { \mathcal J}^{E}(X):=L_X-\nabla^{E}_{\widetilde X}\,, \qquad X \in \mathbf{\mathfrak g}, \end{align*} $$

where $L_X$ denotes the Lie derivative with respect to X induced by the action of G on E. Its equivariant curvature $R^{E}_{\mathbf {\mathfrak g}}(X)\in \Omega ^*(M,\operatorname {\mathrm {End}}(E))$ evaluated at $X\in \mathbf {\mathfrak g}$ is then given by the formula

$$ \begin{align*} R^{E}_{\mathbf{\mathfrak g}}(X):=R^{E}+2\pi i{ \mathcal J}^{E}(X)\,, \end{align*} $$

where $R^{E}$ denotes the curvature of $\nabla ^E$ . The associated equivariant Todd form is the equivariantly closed form $\operatorname {\mathrm {Td}}_{\mathbf {\mathfrak g}}(E)\in \Omega ^*_G(M)$ defined for all $X\in \mathbf {\mathfrak g}$ by the formula

(2.17) $$ \begin{align} \operatorname{\mathrm{Td}}_{\mathbf{\mathfrak g}}(E,X):=\mathrm{det}\,_{E}\left(\frac{R^{E}_{\mathbf{\mathfrak g}}(X)/2\pi i} {\exp(R^{E}_{\mathbf{\mathfrak g}}(X)/2\pi i)-{\mathrm{Id}_{E}}}\right). \end{align} $$

One readily checks that its cohomology class in $H^*_G(M)$ does not depend on the choice of a Hermitian metric $h^E$ and connection $\nabla ^E$ . In the important case of the tangent bundle $TM\to M$ equipped with the chosen G-invariant compatible almost complex structure $J\in \operatorname {\mathrm {End}}(TM)$ over $(M,\omega )$ , one can consider the induced Hermitian metric $g^{TM}$ given by Formula (1.3) and the associated Levi-Civita connection $\nabla ^{TM}$ , which are both G-equivariant. This induces a Hermitian metric and connection on $T^{(1,0)}M$ via the splitting (2.13), and the associated equivariant Todd form $\operatorname {\mathrm {Td}}_{\mathbf {\mathfrak g}}(M)\in \Omega ^*_G(M)$ given by $\operatorname {\mathrm {Td}}_{\mathbf {\mathfrak g}}(M,X):=\operatorname {\mathrm {Td}}_{\mathbf {\mathfrak g}}(T^{(1,0)}M,X)$ does not depend on the choice of the compatible almost complex structure, and hence is an invariant of the Hamiltonian action of G on $(M,\omega )$ . The closed form $\operatorname {\mathrm {Td}}(M):=\operatorname {\mathrm {Td}}_{\mathbf {\mathfrak g}}(M,0)\in \Omega ^*(M)$ is called the Todd form of M, and its cohomology class is a symplectic invariant of $(M,\omega )$ .

Recalling the prequantization condition (1.2), the equivariant Chern character of $L^m$ is the equivariantly closed form $\operatorname {\mathrm {ch}}_{\mathbf {\mathfrak g}}(L^m)\in \Omega ^*_G(M)$ defined for all $m\in {\mathbb N}$ and $X\in \mathbf {\mathfrak g}$ by the formula

(2.18) $$ \begin{align} \operatorname{\mathrm{ch}}_{\mathbf{\mathfrak g}}(L^m,X):= \exp(m\omega +2\pi im{ \mathcal J}(X))\,. \end{align} $$

The closed form $\operatorname {\mathrm {ch}}(L^m):=\operatorname {\mathrm {ch}}_{\mathbf {\mathfrak g}}(L^m,0)=e^{m\omega }\in \Omega ^*(M)$ is called the Chern character of  $L^m$ .

Consider now for any $g\in G$ the fixed point set

(2.19) $$ \begin{align} M^g:=\left\{ p \in M~|~g\cdot p =p \right\}\subset M, \end{align} $$

which is a smooth submanifold of M, and write $\Sigma _{M^g}\to M^g$ for its normal bundle inside $TM$ . It is naturally endowed with an action of the centralizer $Z_g\subset G$ of g, and we write $\mathfrak {z}_g$ for its Lie algebra. Then all of the equivariant characteristic forms considered above make sense for the action of $Z_g$ on the vector bundles $\Sigma _{M^g}$ , $TM^g$ , and $L^m|_{M^g}$ over $M^g$ . In particular, we write $R^{\Sigma _{M^g}}_{\mathfrak {z}_g}(X)\in \Omega ^2(M^g,\operatorname {\mathrm {End}}(\Sigma _{M^g}))$ for the equivariant curvature associated with the connection on $\Sigma _{M^g}$ induced by the Levi-Civita connection of $g^{TM}$ .

We can now state the following fundamental Kirillov formula, which we present in a guise that can be found in [Reference Meinrenken28, Theorem 3.1 and Remark (4) after Theorem 2.1].

Theorem 2.10. [Reference Berline and Vergne3, Th. 3.23]

For any $m\in {\mathbb N}$ , $g\in G$ , and $X\in \mathfrak {z}_g$ sufficiently close to $0$ we have

$$ \begin{align*} \chi^{(m)}(g\exp(X))=\int_{M^g}\, \frac{\operatorname{\mathrm{Td}}_{\mathfrak{z}_g}(M^g,X)\,\operatorname{\mathrm{Tr}}_{L^m}[g^{-1}]\operatorname{\mathrm{ch}}_{\mathfrak{z}_g}(L^m|_{M^g},X)} {\mathrm{det}\,_{\Sigma_{M^g}}(1-g\exp(R^{\Sigma_{M^g}}_{\mathfrak{z}_g}(X)/2\pi i))}. \end{align*} $$

Note that $g^{-1}\in G$ acts fiberwise on the complex line bundle $L^m|_{M^g}$ by multiplication with a scalar denoted by $\operatorname {\mathrm {Tr}}_{L^m}[g^{-1}]$ in Theorem 2.10. On the one hand, applied to $g=e$ , the Kirillov formula takes the elegant form

(2.20) $$ \begin{align} \chi^{(m)}(\exp(X))=\int_M\,\operatorname{\mathrm{Td}}_{\mathbf{\mathfrak g}}(M,X)\,\operatorname{\mathrm{ch}}_{\mathbf{\mathfrak g}}(L^m,X) \end{align} $$

for any $m\in {\mathbb N}$ and $X\in \mathbf {\mathfrak g}$ sufficiently close to $0$ , compare [Reference Berline, Getzler and Vergne2, Theorem 8.2]. On the other hand, Theorem 2.10 applied to $X=0$ recovers the equivariant Atiyah-Segal-Singer index theorem [Reference Atiyah and Segal1]

(2.21) $$ \begin{align} \chi^{(m)}(g)=\int_{M^g}\, \frac{\operatorname{\mathrm{Td}}(M^g)\,\operatorname{\mathrm{Tr}}_{L^m}[g^{-1}]\exp(-m R^{L}/2\pi i)} {\mathrm{det}\,_{\Sigma_{M^g}}(1-g\exp(R^{\Sigma_{M^g}}/2\pi i))}, \end{align} $$

which is valid for any $m\in {\mathbb N}$ and $g\in G$ . In particular, Theorem 2.10 applied to $g=e$ and $X=0$ recovers the classical Hirzebruch-Riemann-Roch formula (1.5) for the usual Riemann-Roch numbers of Definition 2.8.

3.1 Local normal form theorem

Let $F\in {\mathcal F}$ and note that F is a symplectic submanifold of M. When considering a fiber bundle over F with total space E, we will use the notation $\nu _E:E\to F$ for its bundle projection. Let $\nu _{\Sigma _F}:\Sigma _F\to F$ be the symplectic normal bundle of F inside $TM$ , so that we have a decomposition

(3.1) $$ \begin{align} (TM,\omega)|_{F}=(TF,\omega_{F})\oplus (\Sigma_F,\omega^{\perp}) \end{align} $$

into symplectic vector bundles, where $\omega _F:=\mathrm {inc}_F^\ast \,\omega \in \Omega ^2(F)$ and $\omega ^\perp $ is the fiberwise restriction of $\omega $ to $\Sigma _F$ . Note that the action of $S^1$ on $(M,\omega )$ induces a fiberwise action on $\Sigma _F$ , and choose an $S^1$ -invariant compatible complex structure $J_{\Sigma _F}\in \operatorname {\mathrm {End}}(\Sigma _F)$ on $(\Sigma _F,\omega ^{\perp })$ . The Formula (1.3) then induces a Hermitian metric $g_{\Sigma _F}:=\omega ^{\perp }(\cdot ,J_{\Sigma _F}\cdot )$ on the complex bundle $(\Sigma _F,J_{\Sigma _F})$ , making $\Sigma _F$ into a Hermitian vector bundle over F. For any $k\in {\mathbb N}$ and under the identification (2.6) we write

$$ \begin{align*} \Sigma_F^{(k)}:=\left\{v\in\Sigma_F~\big|~\exp(X)\cdot v=e^{2\pi ikx}v\,\text{ for all } X\in\mathbf{\mathfrak g}\right\} \end{align*} $$

for the associated isotypic component, as well as $W:=\{k\in {\mathbb Z}~|~\Sigma _F^{(k)}\neq 0\}\subset {\mathbb Z}$ for the set of weights, which is finite and does not contain $0$ by definition of $\Sigma _F$ , and let $\ell _F^{(k)}:=\mathrm {rk}_{\mathbb C}\,\Sigma _F^{(k)}$ be the complex rank of the vector bundle $\Sigma _F^{(k)}$ . We thus have a finite decomposition

(3.2) $$ \begin{align} \Sigma_F=:\bigoplus_{k\in W}\Sigma_F^{(k)} \end{align} $$

into hermitian subbundles, with associated structure group

$$ \begin{align*} K_F:=\prod_{k\in W}\,\mathrm{U}\big(\ell_F^{(k)}\big)\,, \end{align*} $$

so that there is a principal $K_F$ -bundle $\nu _{P_F}:P_F \rightarrow F$ satisfying

(3.3) $$ \begin{align} \Sigma_F\cong P_F\times_{K_F}\bigoplus_{k\in W}{\mathbb C}^{\ell_F^{(k)}}\,, \end{align} $$

and inducing the decomposition (3.2), where $K_F$ acts linearly on $\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ . The diagonal action of $S^1$ on $\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ of weight $k\in W$ on each summand ${\mathbb C}^{\ell _F^{(k)}}$ commutes with the action of $K_F$ , inducing an $S^1$ -action on the right-hand side of (3.3), which makes it an $S^1$ -equivariant identification. Note also that the symplectic structure on $\Sigma _F$ is induced by the standard complex structure on each ${\mathbb C}^{\ell _F^{(k)}}$ via this decomposition. In what follows, we will write $W_\pm :=\{w\in W\,|\,\pm w\in {\mathbb N}\}$ for the sets of positive and negative weights, respectively, so that by (3.2) there is a decomposition

(3.4) $$ \begin{align} \Sigma_F=\Sigma_F^+\oplus\Sigma_F^-, \end{align} $$

where $\Sigma _F^\pm :=\bigoplus _{k\in W_\pm }\Sigma _F^{(k)}$ . Setting $ \ell ^\pm _F:=\mathrm {rk}_{\mathbb C}\,\Sigma _F^\pm =\sum _{k\in W_\pm }\ell _F^{(k)} $ , the decomposition (3.4) is induced by the natural embedding $K_F\subset U(\ell ^+_F)\times U(\ell ^-_F)$ . We write ${\mathbb C}^{\ell ^\pm _F}=\bigoplus _{k\in W_\pm }{\mathbb C}^{\ell _F^{(k)}}$ and set $\ell _F:=\mathrm {rk}_{\mathbb C}\,\Sigma _F$ .

Let $\omega _{\Sigma _F}$ be a symplectic form on a neighborhood $V_F$ of the zero section in $\Sigma _F$ such that its restriction to the fibers of $\Sigma _F$ coincides with the standard symplectic form induced by $\omega ^{\perp }$ , while the restriction $(T\Sigma _F,\omega _{\Sigma _F})|_F$ to the zero section $F\subset \Sigma _F$ coincides with $(TM,\omega )|_F$ via the natural identification of $TM$ with $T\Sigma _F$ over F. The next proposition gives a canonical description of the moment map ${ \mathcal J}:M\to {\mathbb R}$ in a neighborhood of each $F \in {\mathcal F}$ , and constitutes the basis for the computations of the next sections. It is a straightforward consequence of the equivariant Darboux lemma, as explained for instance in [Reference Guillemin, Lerman and Sternberg13, Section 2.2].

Proposition 3.2 (Local normal form theorem)

For each $F\in {\mathcal F}$ there is an $S^1$ -equivariant symplectomorphism $\Phi _F:V_F\rightarrow U_F$ from the open tubular neighborhood $(V_F,\omega _{\Sigma _F})$ of the zero section in $\Sigma _F$ onto an open neighborhood $U_F\subset (M,\omega )$ of F, such that

$$ \begin{align*} { \mathcal J} \circ \Phi_F([\wp,w])= \frac 12 Q_F(w) + { \mathcal J}(F),\quad\text{for all}\quad [\wp,w]\in \Sigma_F\cong P_F\times_{K_F}\bigoplus_{k\in W}{\mathbb C}^{\ell_F^{(k)}}, \end{align*} $$

where $Q_F$ is the $K_F$ -invariant quadratic form on $\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ defined by

(3.5) $$ \begin{align} Q_F(w):=\sum_{k\in W}\,k\,\left\| \pi_k(w) \right\|^2. \end{align} $$

We write

(3.6) $$ \begin{align} Z_F:=\{[\wp,w]\in \Sigma_F\,|\,w\neq 0\text{ and } Q_F(w)=0\}\subset\Sigma_F \end{align} $$

for the submanifold formed fiberwise by the smooth points of the $0$ -level set of the quadratic form $Q_F$ . Recalling the stratification (1.7), the set $Z_F$ then corresponds to the regular stratum of $J^{-1}(\{0\})$ in the local normal coordinates of Proposition 3.2, since

(3.7) $$ \begin{align} \Phi_{F}(V_F\cap Z_F)=U_F\cap { \mathcal J}^{-1}(\{0\})^{\mathrm{reg}}. \end{align} $$

3.2 Local model

In this section, we introduce for each $F\in {\mathcal F}$ coordinates on $\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ with respect to the zero level set of the quadric $Q_F$ , which will be extended in Section 3.3 to the symplectic normal bundle using the framework of the previous section. We will consider $\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ as equipped with the diagonal $S^1$ -action of weight $k\in W$ on each summand ${\mathbb C}^{\ell _F^{(k)}}$ . To begin, we define the sets

(3.8) $$ \begin{align} {\mathbb C}^{\ell_F^\pm}_{\bullet}&:= {\mathbb C}^{\ell_F^\pm}\setminus \{0\}, \qquad\qquad\qquad\qquad\text{in case}\quad \ell_F^\pm>0,\ell_F^\mp=0, \nonumber\\ {\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}&:=({\mathbb C}^{\ell_F^+}\setminus \{0\})\times ({\mathbb C}^{\ell_F^-}\setminus \{0\}),\qquad\!\!\kern-1.3pt\!\text{in case}\quad \ell_F^+,\ell_F^->0. \end{align} $$

Further, for any $k_0\in W$ we write

$$ \begin{align*} \pi_{k_0}:\bigoplus_{k\in W}{\mathbb C}^{\ell_F^{(k)}}\to{\mathbb C}^{\ell_F^{(k_0)}} \end{align*} $$

for the canonical projection, and introduce the weighted Euclidean norms

(3.9) $$ \begin{align} \left\| w \right\|_{F\pm}^2:=\pm \sum_{k\in W^\pm}\,k\,\left\| \pi_k(w) \right\|^2, \quad\text{for all}\quad w\in {\mathbb C}^{\ell_F^\pm}=\bigoplus_{k\in W^\pm}{\mathbb C}^{\ell_F^{(k)}}. \end{align} $$

We then introduce the ellipsoids

(3.10) $$ \begin{align} S^{2\ell_F^\pm-1}_\pm:=\big\{w \in{\mathbb C}^{\ell_F^\pm} \mid\; \left\| w \right\|_{F\pm}=1 \big\},\quad\text{when}\quad\ell_F^\pm>0\,, \end{align} $$

Note that the norms $\left \| \cdot \right \|_{F\pm }$ are both $S^1$ - and $K_F$ -invariant. The quadratic form $Q_F$ from (3.5) can be written in norm notation as

$$ \begin{align*} Q_F(w)=\left\| w^+ \right\|_{F+}^2-\left\| w^- \right\|_{F-}^2,\quad\text{for all}\quad w=(w^+,w^-)\in {\mathbb C}^{\ell_F^++\ell_F^-}. \end{align*} $$

Later, it will also be convenient to consider the weighted Hermitian norm

(3.11) $$ \begin{align} \left\| w \right\|_F^2:=\left\| w^+ \right\|_{F+}^2+\left\| w^- \right\|_{F-}^2,\quad\text{for all}\quad w=(w^+,w^-)\in {\mathbb C}^{\ell_F^++\ell_F^-}. \end{align} $$

Next, for each weight $k\in W$ , we introduce the following standard symplectic and angular forms over ${\mathbb C}^{\ell _F^{(k)}}$ for any $w=(w_1'+iw_1",\cdots , w_{\ell _F^{(k)}}'+iw_{\ell _F^{(k)}}") \in {\mathbb C}^{\ell _F^{(k)}}$ by setting

(3.12) $$ \begin{align} \omega_{(k)}|_w:=\sum_{j=1}^{\ell_F^{(k)}}\,dw^{\prime}_j\wedge dw^{\prime\prime}_j,\qquad \alpha_{(k)}|_w:=\sum_{j=1}^{\ell_F^{(k)}}\,w^{\prime}_j \,d w^{\prime\prime}_j-w^{\prime\prime}_j \,d w^{\prime}_j\,. \end{align} $$

They are related by $\omega _{(k)}=\frac {1}{2}d\alpha _{(k)}$ . For any $X\in \mathbf {\mathfrak g}$ inducing the fundamental vector field $\tilde X\in \mathrm {C^{\infty }}({\mathbb C}^{\ell _F^{(k)}},T{\mathbb C}^{\ell _F^{(k)}})$ associated with the $S^1$ -action on ${\mathbb C}^{\ell _F^{(k)}}$ of weight $k\in W$ we then get

(3.13) $$ \begin{align} (\tilde X \mathbin{\lrcorner} \alpha_{(k)})|_w = k\, x\, \left\| w \right\|^2,\quad\text{for all}\quad w\in {\mathbb C}^{\ell_F^{(k)}}\,, \end{align} $$

where $x\in {\mathbb R}$ is the image of $X\in \mathbf {\mathfrak g}$ via the identification (2.6). If we now introduce the $1$ -forms

(3.14) $$ \begin{align} \alpha^{\pm}:=\pm\sum_{k\in W^{\pm}} \pi_k^*\alpha_{(k)} \in\Omega^1({\mathbb C}^{\ell_F^\pm},{\mathbb R})\,, \end{align} $$

then (3.13) and (3.9) imply that the $1$ -forms defined by

(3.15) $$ \begin{align} \theta^\pm|_w:=\frac{\alpha^{\pm} |_w}{ \left\| w \right\|_{F\pm}^2}, \qquad \text{if } w \in {\mathbb C}^{\ell_F^\pm}_{\bullet}, \end{align} $$

(3.16) $$ \begin{align} \theta|_w:=\frac{1}{2}\left(\frac{\alpha^{+}|_w}{ \left\| w \right\|_{F+}^2}+\frac{\alpha^{-}|_w}{ \left\| w \right\|_{F-}^2}\right), \qquad \text{if } w \in {\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}, \end{align} $$

are connection forms in the sense of Formula (2.8) for the diagonal $S^1$ -action of weight $k\in W$ on the k-th summand on ${\mathbb C}^{\ell _F^\pm }_{\bullet }$ and ${\mathbb C}^{\ell _F^+,\ell _F^-}_{\bullet \bullet }$ respectively defined in (3.8).

The following simple lemma will be crucial for all our local and global considerations. Recall the general notation for inclusions and projections introduced at the beginning of Section 2.

Lemma 3.3. Let $S\subset \bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ be an embedded smooth submanifold, $f:(0,\infty )\to (0,\infty )$ a smooth function, and consider the smooth map $\Psi _{S,f}:S\times (0,\infty )\to \bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ , $(w,r)\mapsto f(r)w$ . Then one has for each $k\in W$

$$ \begin{align*} \Psi_{S,f}^\ast (\pi_k^*\alpha_{(k)}) =f(r)^2\, \mathrm{pr}_S^\ast \mathrm{inc}^\ast_S(\pi_k^*\alpha_{(k)})\,. \end{align*} $$

Proof. Let $v\in T_{y}(S \times (0,\infty ))$ be a tangent vector at $y:=(w, r)\in S \times (0,\infty )$ and $\gamma _v(t)=(w(t), r(t))$ a smooth curve with $\dot \gamma _v(0)=v$ , $\gamma _v(0)=y$ . Identifying $T_{y}(S \times (0,\infty ))$ with a subspace of $\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}\times {\mathbb R}$ we get

$$ \begin{align*} (\Psi_{S,f})_{\ast}(v)=\frac d {dt} (f(r(t)) w(t))|_{t=0}=\frac d {dt} f(r(t))|_{t=0} w + f(r)\dot w(0), \end{align*} $$

so that with the identification $w_j\equiv w_j' \operatorname {\mathrm {\partial }}_{w_j'}+w_j" \operatorname {\mathrm {\partial }}_{w_j"}$ in the notation of (3.12) we obtain

$$ \begin{align*} \Psi_{S,f}^\ast(\pi_k^*\alpha_{(k)})|_y (v)&= \frac d {dt} f(r(t))|_{t=0} \underbrace{(w_j' dw_j"- w_j" dw_j')|_{\Psi_{S,f}(y)} (w)}_{=0}\\ &\quad+ f(r) (w_j' dw_j"- w_j" dw_j')|_{\Psi_{S,f}(y)}(\dot w(0))\\ &= f(r)^2\, \mathrm{pr}_S^\ast \mathrm{inc}_S^\ast \pi_k^*\alpha_{(k)}|_y (v). \\[-32pt]\end{align*} $$

We will now proceed with separate treatments of the cases $0\in \partial \,{ \mathcal J}(M)$ and $0\in \operatorname {\mathrm {Int}}{ \mathcal J}(M)$ , respectively. As explained in the introduction, in the first case ${\mathcal F}_0$ consists of only one component $F={ \mathcal J}^{-1}(\left \{ 0 \right \})$ and $Q_F$ is definite, while in the second case, $Q_F$ is indefinite for all $F\in {\mathcal F}^0$ .

3.2.1 Definite case

Let us begin with the easier definite case, so that either $\ell _F^-=0$ or $\ell _F^+=0$ , and write $\ell _F:=\ell _F^+>0$ if $F\in {\mathcal F}_+$ and $\ell _F:=\ell _F^->0$ if $F\in {\mathcal F}_-$ . The diagonal action of $S^1$ on ${\mathbb C}^{\ell _F}=\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ considered in Section 3.1 restricts to a locally free action on $S^{2\ell _F-1}$ , and we introduce the connection form

(3.17) $$ \begin{align} \Theta:=\mathrm{inc}^\ast_{S^{2\ell_F-1}}\theta \in\Omega^1(S^{2\ell_F-1},{\mathbb R}) \end{align} $$

as the restriction of (3.15) to $S^{2\ell _F-1}$ .

We now introduce spherical polar coordinates in ${\mathbb C}^{\ell _F}_{\bullet }$ via the diffeomorphism

(3.18) $$ \begin{align} \Psi:S^{2\ell_F-1} \times(0,\infty)&\longrightarrow {\mathbb C}^{\ell_F}_{\bullet}\nonumber\\ \qquad (w,r)&\longmapsto r w\,, \end{align} $$

which is equivariant for the action of $S^1$ introduced above on the first factor and the trivial action on the second factor of $S^{2\ell _F-1}\times (0,\infty )$ . It defines on ${\mathbb C}^{\ell _F}_{\bullet }$ the radial coordinate

(3.19) $$ \begin{align} r=\left\| w \right\|_F=\sqrt{|Q_F(w)|},\quad\text{for all}\quad w\in {\mathbb C}^{\ell_F}_{\bullet}, \end{align} $$

which is related to the quadratic form $Q_F$ by

(3.20) $$ \begin{align} Q_F|_{{\mathbb C}^{\ell_F}_{\bullet}}=\pm r^2\quad \text{if }F\in {\mathcal F}_\pm. \end{align} $$

Applying Lemma 3.3 to $S=S^{2\ell _F-1}$ and $f=\mathrm {id}$ one immediately sees that

$$ \begin{align*} \Psi^\ast (\alpha^{\pm}) =r^2\, \mathrm{pr}_{S^{2\ell_F-1}}^\ast \mathrm{inc}^\ast_{S^{2\ell_F-1}}\alpha^{\pm} \,. \end{align*} $$

Since $\omega _{\mathrm {std}}=\pm \frac {1}{2}d\theta $ and $\Theta =\mathrm {inc}^\ast _{S^{2\ell _F-1}}\theta $ , we deduce from this the useful identity

(3.21) $$ \begin{align} \Psi^\ast (\omega_{\mathrm{std}}|_{{\mathbb C}^{\ell_F}_{\bullet}})&=\pm\frac{1}{2}d(r^2\, \mathrm{pr}_{S^{2\ell_F-1}}^\ast \Theta)=\pm r\,d r \wedge\mathrm{pr}_{S^{2\ell_F-1}}^\ast \Theta \pm\frac{1}{2}r^2\, \mathrm{pr}_{S^{2\ell_F-1}}^\ast d\Theta \quad \text{if }F\in {\mathcal F}_\pm. \end{align} $$

3.2.2 Indefinite case

Let us now turn to the indefinite case, so that $\ell _F^+,\ell _F^->0$ . Departing from the quadric formed by the zero level set of $Q_F$ we define the slit quadric

(3.22) $$ \begin{align} \mathcal{Q}_\times:=Q_F^{-1}(\{0\})\setminus\{0\}\subset {\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}\,, \end{align} $$

which is a smooth submanifold of the set ${\mathbb C}^{\ell _F^+,\ell _F^-}_{\bullet \bullet }$ defined in (3.8). The diagonal action of $S^1$ on ${\mathbb C}^{\ell _F}=\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ considered in Section 3.1 restricts to locally free $S^1$ -actions on $\mathcal {Q}_\times \subset {\mathbb C}^{\ell _F^+,\ell _F^-}_{\bullet \bullet }$ and the product of ellipsoids

(3.23) $$ \begin{align} S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-\subset {\mathbb C}^{\ell_F^++\ell_F^-}= {\mathbb C}^{\ell_F^+}\oplus {\mathbb C}^{\ell_F^-} \end{align} $$

of the ellipsoids (3.10). Besides the connection form (3.16) we introduce now also the $1$ -form

(3.24) $$ \begin{align} \bar \theta|_w:=\frac{1}{2}\left(\frac{\alpha^+|_w}{ \left\| w \right\|_{F+}^2}-\frac{\alpha^-|_w}{ \left\| w \right\|_{F-}^2}\right) \end{align} $$

on ${\mathbb C}^{\ell _F^+,\ell _F^-}_{\bullet \bullet }$ , and introduce the corresponding restrictions

(3.25) $$ \begin{align} \Theta:=\mathrm{inc}^\ast_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}\theta, \qquad \overline \Theta:=\mathrm{inc}^\ast_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}\bar\theta \end{align} $$

to $S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-$ . Clearly, $\Theta \in \Omega ^1(S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-,{\mathbb R})$ is a connection for the $S^1$ -action on $S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-$ . On the other hand, the $1$ -form $\overline \Theta \in \Omega ^1(S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-,{\mathbb R})$ is basic in the sense of Proposition 2.4.

Let us now introduce polar moment coordinates in ${\mathbb C}^{\ell _F^+,\ell _F^-}_{\bullet \bullet }$ via the diffeomorphism

(3.26) $$ \begin{align} \Psi:S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-\times(0,\infty)\times {\mathbb R} &\longrightarrow {\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}, \nonumber\\ (w^+,w^-,r,q)&\longmapsto \Big (\sqrt{\sqrt{r^4+q^2}+q}\; w^+, \sqrt{\sqrt{r^4+q^2}- q}\; w^-\Big)\,. \end{align} $$

The map $\Psi $ is at the basis of our local model. It is equivariant under the action of $S^1$ introduced above on $S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-$ and the trivial action on $(0,\infty )\times {\mathbb R} $ , and defines for $w \in {\mathbb C}^{\ell _F^+,\ell _F^-}_{\bullet \bullet }$ the coordinates

(3.27) $$ \begin{align} r=\sqrt{\left\| w^+ \right\|_{F+} \, \left\| w^- \right\|_{F-}}, \qquad q=\frac{1}{2}Q_F(w). \end{align} $$

The slit quadric $\mathcal {Q}_\times \subset {\mathbb C}^{\ell _F^+,\ell _F^-}_{\bullet \bullet }$ defined in (3.22) corresponds to the set $\{q=0\}$ , while the coordinate r is radial on ${\mathcal {Q}_\times }$ . Moreover, composing the retraction

(3.28) $$ \begin{align} \mathrm{ret}_{\mathcal{Q}_\times}:{\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}&\longrightarrow \mathcal{Q}_\times,\nonumber\\ w=(w^+,w^-)&\longmapsto \sqrt{\left\| w^+ \right\|_{F+} \, \left\| w^- \right\|_{F-}}\Big(\frac{w^+}{\left\| w^+ \right\|_{F+}},\frac{w^-}{\left\| w^- \right\|_{F-}}\Big) \end{align} $$

with $\Psi $ gives us an $S^1$ -equivariant surjection

(3.29) $$ \begin{align} \pi_{\mathcal{Q}_\times}:S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-\times(0,\infty)\times {\mathbb R}&\longrightarrow \mathcal{Q}_\times,\nonumber\\ (w^+,w^-,r,q)&\longmapsto \Psi(w^+,w^-,r,0)=(rw^+,rw^-). \end{align} $$

Let $\mathrm {pr}_{S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-}:S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-\times (0,\infty )\times {\mathbb R}\longrightarrow S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-$ be the canonical projection. We then have

Lemma 3.4. Write $\alpha :=\frac 12 (\alpha ^++\alpha ^-)$ and $\overline \alpha :=\frac 12 (\alpha ^+-\alpha ^-)$ . In the coordinates (3.27), one has

$$ \begin{align*} \Psi^\ast (\alpha|_{{\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}}) &=\sqrt{r^4+q^2} \;\mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast \Theta + q \,\mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast \overline \Theta,\\ \Psi^\ast (\overline\alpha|_{{\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}}) &=\sqrt{r^4+q^2}\; \mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast \overline \Theta + q\, \mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast \Theta\,. \end{align*} $$

Proof. Consider $\Psi $ and $\mathrm {pr}_{S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-}$ as families of maps defined on $S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-\times (0,\infty )$ and parametrized by $q\in {\mathbb R}$ . Defining $f_q^\pm :(0,\infty )\to (0,\infty )$ by $f_q^\pm (r):=\sqrt {\sqrt {r^4+q^2}\pm q}$ , we can apply Lemma 3.3 with $S=S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-$ and $f=f_q^\pm $ to get

$$ \begin{align*}\Psi^\ast (\alpha^\pm|_{{\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}})=f^\pm_q(r)^2\mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast\mathrm{inc}^\ast_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-} \alpha^\pm\,,\end{align*} $$

concluding the proof.

A direct consequence of Lemma 3.4 is the following indefinite version of (3.21).

Corollary 3.5. In the coordinates (3.26) one has

$$ \begin{align*} \Psi^\ast (\omega_{\mathrm{std}}|_{{\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}}) = \pi_{\mathcal{Q}_\times}^\ast\mathrm{inc}_{\mathcal{Q}_\times}^\ast\omega_{\mathrm{std}}+ d\Big ( q\, \mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast \Theta + \big ( \sqrt{r^4+q^2} -r^2\big ) \mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast \overline \Theta\Big ). \end{align*} $$

Proof. As $\omega _{\mathrm {std}}=d\bar \alpha $ , we apply d on both sides of the equation in Lemma 3.4 involving $\bar \alpha $ , yielding

$$ \begin{align*} \Psi^\ast (\omega_{\mathrm{std}}|_{{\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}}) &= d\big ( q\, \mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast \Theta + \sqrt{r^4+q^2} \mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast \overline \Theta\big ). \end{align*} $$

We now assert that

(3.30) $$ \begin{align} \pi_{\mathcal{Q}_\times}^\ast\mathrm{inc}_{\mathcal{Q}_\times}^\ast\omega_{\mathrm{std}}=d\big (r^2 \mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast \overline \Theta \big ). \end{align} $$

Indeed, since $\overline \Theta =\mathrm {inc}^\ast _{S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-}\bar \alpha $ , this is a consequence of the relation

$$ \begin{align*} \pi_{\mathcal{Q}_\times}^\ast\mathrm{inc}_{\mathcal{Q}_\times}^\ast\bar \alpha= r^2\, \mathrm{pr}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast\mathrm{inc}_{S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-}^\ast\bar \alpha \end{align*} $$

which follows from Lemma 3.3 applied with $S=S^{2\ell _F^+-1}_+\times S^{2\ell _F^--1}_-$ , $f=\mathrm {id}$ , and $q\in {\mathbb R}$ as an additional parameter.

3.3 Application to the symplectic normal bundle

We now translate the fiberwise considerations from Section 3.2 into global statements on the symplectic normal bundle $\Sigma _F$ , identified with the associated bundle $P_F\times _{K_F}\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ as in (3.3).

Choose a $K_F$ -connection on the principal bundle $\nu _{P_F}:P_F \rightarrow F$ , inducing a Hermitian connection $\nabla ^{\Sigma _F}$ on $\Sigma _F$ that preserves the decomposition (3.2). This provides us with a splitting of the short exact sequence

$$ \begin{align*} 0 \longrightarrow \nu_{\Sigma_F}^*\Sigma_F \longrightarrow T\Sigma_F \longrightarrow \nu_{\Sigma_F}^*TF \longrightarrow 0 \end{align*} $$

of vector bundles over $\Sigma _F$ , yielding the global decomposition

(3.31) $$ \begin{align} T\Sigma_F\cong T^{\mathrm{hor}}F \oplus \nu_{\Sigma_F}^*\Sigma_F\,, \end{align} $$

where $T^{\mathrm {hor}}F\simeq \nu _{\Sigma _F}^*TF$ is the horizontal distribution defined by $\nabla ^{\Sigma _F}$ and $\nu _{\Sigma _F}^*\Sigma _F\subset T\Sigma _F$ is the vertical tangent bundle to the fibers. Choose a compatible almost complex structure $J_F\in \operatorname {\mathrm {End}}(TF)$ over $(F,\omega _F)$ . Together with the natural complex structure $J_{\Sigma _F}\in \operatorname {\mathrm {End}}(\Sigma _F)$ , this induces via the decomposition (3.31) an $S^1$ -invariant almost complex structure $J\in \operatorname {\mathrm {End}}(T\Sigma _F)$ over the total space of $\Sigma _F$ .

For any $k_0\in W$ , we define $f_{k_0}\in \mathrm {C^{\infty }}(\Sigma _F,{\mathbb R})$ by setting

(3.32) $$ \begin{align} f_{k_0}([\wp,w]):=\frac{1}{2}\left\| \pi_{k_0}(w) \right\|^2,\quad \text{for all}\quad [\wp,w]\in P_F\times_{K_F}\bigoplus_{k\in W}{\mathbb C}^{\ell_F^{(k)}}, \end{align} $$

where $\left \| \cdot \right \|$ denotes the standard Hermitian norm. Note that $f_{k_0}$ is well-defined since for each $k\in W$ the structure group $K_F$ acts by $\mathrm {U}(\ell _F^{(k)})$ on ${\mathbb C}^{\ell _F^{(k)}}$ and thus preserves the Hermitian norm. Using this, we define the $1$ -form $\alpha _{k_0}\in \Omega ^1(\Sigma _F,{\mathbb R})$ by

(3.33) $$ \begin{align} \alpha_{k_0}(v):=-df_{k_0}(Jv),\quad\text{for all}\quad v\in T\Sigma_F\,. \end{align} $$

Comparing with (3.12), one readily checks that for any $p\in F$ the restriction of $\alpha _{k_0}\in \Omega ^1(\Sigma _F,{\mathbb R})$ to the fiber $(\Sigma _F)_p\subset \Sigma _F$ , seen as a submanifold in the total space of $\Sigma _F$ , satisfies

(3.34) $$ \begin{align} \alpha_{k_0}|_{(\Sigma_F)_p}=\pi_{k_0}^*\alpha_{(k_0)}\,, \end{align} $$

in any trivialization $(\Sigma _F)_p\simeq \bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ compatible with the $K_F$ -principal bundle structure (3.3). Following (3.14), (3.15) and (3.16), we write

(3.35) $$ \begin{align} \alpha^\pm_F:=\pm \sum_{k\in W^\pm}\alpha_k\,, \end{align} $$

so that the $1$ -forms given by

(3.36) $$ \begin{align} \theta_F^\pm|_{[\wp,w]}:=\frac{\alpha^\pm_F|_{[\wp,w]}}{\left\| w \right\|_{F\pm}^2}\quad \text{for}\quad [\wp,w]\in P_F\times_{K_F}{\mathbb C}^{\ell_F^\pm}_{\bullet}, \end{align} $$

(3.37) $$ \begin{align} \theta_F|_{[\wp,w]}:=\frac{1}{2}\left(\frac{\alpha^+_F|_{[\wp,w]}}{\left\| w \right\|_{F+}^2}+ \frac{\alpha^-_F|_{[\wp,w]}}{\left\| w \right\|_{F-}^2}\right)\quad \text{for}\quad [\wp,w]\in P_F\times_{K_F}{\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}, \end{align} $$

define connections for the $S^1$ -action on $P_F\times _{K_F}{\mathbb C}^{\ell _F^\pm }_{\bullet }$ and $P_F\times _{K_F}{\mathbb C}^{\ell _F^+,\ell _F^-}_{\bullet \bullet }$ , respectively, such that for all $p\in F$ the restriction of $\theta _F^\pm ,\,\theta _F$ to the fiber over p coincides with $\theta ^\pm ,\,\theta $ in any trivialization $(\Sigma _F)_p\simeq \bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ compatible with the $K_F$ -principal bundle structure (3.3). We also define the vertical form $\omega _{\mathrm {vert}}\in \Omega ^2(\Sigma _F,{\mathbb R})$ by

(3.38) $$ \begin{align} \omega_{\mathrm{vert}}:=\frac{1}{2}\sum_{k\in W}\,d\alpha_k\,. \end{align} $$

Comparing with (3.12) and (3.34), for all $p\in F$ the restriction of $\omega _{\mathrm {vert}}\in \Omega ^2(\Sigma _F,{\mathbb R})$ to the fiber $(\Sigma _F)_p\subset \Sigma _F$ satisfies

(3.39) $$ \begin{align} \omega_{\mathrm{vert}}|_{(\Sigma_F)_p}=\omega_{\mathrm{std}}\,, \end{align} $$

in any trivialization $(\Sigma _F)_p\simeq \bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ compatible with the $K_F$ -principal bundle structure (3.3), where $\omega _{\mathrm {std}}$ is the standard symplectic form on $\bigoplus _{k\in W}{\mathbb C}^{\ell _F^{(k)}}$ induced by (3.12) for each $k\in W$ . Furthermore, note that the restriction $T\Sigma _F|_F$ over the zero section $F\subset \Sigma _F$ naturally identifies with the restriction $TM|_F$ over $F\subset M$ , and the restriction of $\omega _{\mathrm {vert}}$ to $T\Sigma _F|_F$ coincides with $\omega ^{\perp }$ in the decomposition (3.1). Consequently, we can explicitly choose the symplectic form $\omega _{\Sigma _F}$ from Proposition 3.2 to be given by

(3.40) $$ \begin{align} \omega_{\Sigma_F}:=\omega_{\mathrm{vert}}+\nu_{\Sigma_F}^*\,\omega_F \in\Omega^2(\Sigma_F,{\mathbb R}). \end{align} $$

By construction, the almost complex structure $J\in \operatorname {\mathrm {End}}(T\Sigma _F)$ over $\Sigma _F$ is compatible with $\omega _{\Sigma _F}$ , and we write $g^{T\Sigma _F}:=\omega _{\Sigma _F}(\cdot ,J\cdot )$ for the induced $S^1$ -invariant Riemannian metric over $V_F$ .

3.3.1 Definite case

Let $F\in {\mathcal F}$ be such that $Q_F$ is definite. Recall the definition of $\ell _F$ from the beginning of Section 3.2.1. As the linear action of $K_F$ on ${\mathbb C}^{\ell _F}$ preserves $S^{2\ell _F-1}$ , we can consider the associated sphere bundle $ S_F:= P_F\times _{K_F}S^{2\ell _F-1} $ inside $\Sigma _F=P_F\times _{K_F}{\mathbb C}^{\ell _F}$ , with bundle projection $\nu _{S_F}:S_F\to F$ . The $S^1$ -action on $\Sigma _F$ restricts to a locally free $S^1$ -action on $S_F$ for which we have the connection

(3.41) $$ \begin{align} \Theta_{S_F}:=\mathrm{inc}^\ast_{S_F}\theta_F \end{align} $$

given by restriction of the connection $\theta _F$ from (3.36). Using (3.8) to define the subbundle

$$ \begin{align*} \Sigma_{F\bullet}:=P_F\times_{K_F}{\mathbb C}^{\ell_F}_{\bullet} = \Sigma_F\setminus F, \end{align*} $$

the $S^1$ -equivariant diffeomorphism $\Psi $ from (3.18) globalizes to an $S^1$ -equivariant diffeomorphism

(3.42) $$ \begin{align} \Psi_{F}: S_F\times(0,\infty)&\longrightarrow \Sigma_{F\bullet}\nonumber\\ ([\wp,w], r)&\longmapsto [\wp,\Psi(w,r)]\,, \end{align} $$

where $S^1$ acts on $S_F\times (0,\infty )$ by the product of the $S^1$ -action on $S_F$ and the trivial action on $(0,\infty )$ . $\Psi _{F}$ promotes the coordinate r from (3.19) to a global fiber-radial coordinate r on the bundle $\Sigma _{F\bullet }$ . Its relation to $Q_F$ is given by (3.20) with ${\mathbb C}^{\ell _F}_{\bullet }$ replaced by $\Sigma _{F\bullet }$ . Note that $S_F\times (0,\infty )$ is a fiber bundle over F with the projection $\nu _{S_F\times (0,\infty )}:=\nu _{S_F}\circ \mathrm {pr}_{S_F}$ . We state the global version of (3.21) in

Corollary 3.6. In the coordinate (3.19) provided by the diffeomorphism (3.42) we have

$$ \begin{align*} \Psi_{F}^\ast (\omega_{\Sigma_F}|_{\Sigma_{F\bullet}})=\nu_{S_F\times(0,\infty)}^\ast\omega_F\pm d\Big(\frac{r^2}{2} \mathrm{pr}_{S_F}^\ast \Theta_{S_F}\Big)\quad \text{if }F\in {\mathcal F}_\pm. \end{align*} $$

Proof. In view of the definition (3.41) of $\Theta _{S_F}$ and the definition (3.42) of $\Psi _{F}$ the assertion follows from (3.40), (3.21), and (3.39).

For a better overview, we illustrate the maps appearing in Corollary 3.6 in the diagram

(3.43)

in which everything commutes except the triangle formed by $\mathrm {pr}_{S_F}$ , $\mathrm {inc}_{S_F}$ , and $\Psi _{F}$ , which commutes only up to $S^1$ -equivariant homotopy.

3.3.2 Indefinite case

Let $F\in {\mathcal F}$ be such that $Q_F$ is indefinite. As the linear action of $K_F$ on ${\mathbb C}^{\ell _F^++\ell _F^-}$ preserves the bi-ellipsoid (3.23), we can consider the associated fiber bundle

(3.44) $$ \begin{align} S_F:= P_F\times_{K_F}(S^{2\ell_F^+-1}_+\times S^{2\ell_F^--1}_-) \end{align} $$

inside $\Sigma _F\cong P_F\times _{K_F}{\mathbb C}^{\ell _F^++\ell _F^-}$ . Furthermore, the $0$ -level set $Z_F\subset \Sigma _F\setminus F$ defined in (3.6) has the fiber bundle structure

(3.45) $$ \begin{align} Z_F\cong P_F\times_{K_F}\mathcal{Q}_\times\, \end{align} $$

associated with the slit quadric (3.22). Recall that we write $\nu _{S_F}:S_F\to F$ and $\nu _{Z_F}:Z_F\to F$ for the fiber bundle projections. The action of $S^1$ on $\Sigma _F$ restricts to locally free $S^1$ -actions on $S_F$ and $Z_F$ , respectively, and we set

(3.46) $$ \begin{align} \Theta_{S_F}&:=\mathrm{inc}_{S_F}^\ast\theta_F=\frac{1}{2}(\Theta^+_{S_F}+\Theta^-_{S_F}),\quad\overline \Theta_{S_F}:=\frac{1}{2}(\Theta^+_{S_F}-\Theta^-_{S_F})\quad\text{with}\quad \Theta^\pm_{S_F}:=\mathrm{inc}_{S_F}^\ast \theta^\pm_F,\nonumber\\ \Theta_{Z_F}&:=\mathrm{inc}_{Z_F}^\ast\theta_F=\frac{1}{2}(\Theta^+_{Z_F}+\Theta^-_{Z_F}),\quad \overline \Theta_{Z_F}:=\frac{1}{2}(\Theta^+_{Z_F}-\Theta^-_{Z_F})\quad \text{with}\quad \Theta^\pm_{Z_F}:=\mathrm{inc}_{Z_F}^\ast\theta_F^\pm\,, \end{align} $$

where $\theta _F$ is the connection form on $\Sigma _F\setminus F$ from (3.37) and the $1$ -forms $\alpha _k$ were defined in (3.33). Thus, $\Theta _{S_F}$ and $\Theta _{Z_F}$ are connections for the $S^1$ -actions on $S_F$ and $Z_F$ , respectively. On the other hand, $\overline \Theta _{S_F}$ and $\overline \Theta _{Z_F}$ are basic differential forms in the sense of Proposition 2.4. Using (3.8) to define the subbundle

$$ \begin{align*}\Sigma_{F\bullet\bullet}:=P_F\times_{K_F}{\mathbb C}^{\ell_F^+,\ell_F^-}_{\bullet\bullet}, \end{align*} $$

the $S^1$ -equivariant diffeomorphism $\Psi $ from (3.26) globalizes to an $S^1$ -equivariant diffeomorphism

(3.47) $$ \begin{align} \Psi_{F}: S_F\times(0,\infty)\times {\mathbb R} &\longrightarrow \Sigma_{F\bullet\bullet}\nonumber\\ ([\wp,w], r,q)&\longmapsto [\wp,\Psi(w,r,q)], \end{align} $$

where $S^1$ acts on $S_F\times (0,\infty )\times {\mathbb R}$ by the product of the $S^1$ -action on $S_F$ and the trivial action on $(0,\infty )\times {\mathbb R}$ . The map $\Psi _{F}$ defines coordinates (3.27) on $\Sigma _{F\bullet \bullet }$ , in which $Z_F$ coincides with $\{q=0\}$ . The retraction $\mathrm {ret}_{\mathcal {Q}_\times }$ from (3.28) and the surjection $\pi _{\mathcal {Q}_\times }$ from (3.29) globalize to an $S^1$ -equivariant retraction and an $S^1$ -equivariant surjection onto $Z_F$ , respectively, by defining

(3.48) $$ \begin{align} \mathrm{ret}_{Z_F}: \Sigma_{F\bullet\bullet} &\longrightarrow Z_F\nonumber\\ [\wp,w]&\longmapsto [\wp,\mathrm{ret}_{\mathcal{Q}_\times}(w)], \end{align} $$

as well as

$$ \begin{align*} \pi_{Z_F}:=\mathrm{ret}_{Z_F}\circ \Psi_{F}: S_F\times(0,\infty)\times {\mathbb R} &\longrightarrow Z_F,\\ ([\wp,w], r,q)&\longmapsto [\wp,\pi_{\mathcal{Q}_\times}(w,r,q)]=\Psi_{F}([\wp,w], r,0)=[\wp,rw]. \end{align*} $$

The restricted diffeomorphism

(3.49) $$ \begin{align} \Psi_{F}:S_F\times(0,\infty)\times\{0\}\longrightarrow Z_F, \end{align} $$

denoted again just by $\Psi _F$ for simplicity, provides the radial coordinate r on $Z_F$ . Note that the space $S_F\times (0,\infty )$ , which we identify canonically with $S_F\times (0,\infty )\times \{0\}$ , as well as the space $S_F\times (0,\infty )\times {\mathbb R}$ , are fiber bundles over F with the corresponding projections $\nu _{S_F\times (0,\infty )}$ , $\nu _{S_F\times (0,\infty )\times {\mathbb R}}$ given by the composition of $\nu _{S_F}$ with the canonical projections onto $S_F$ , respectively. These constructions lead us to the following

Corollary 3.7. In the coordinates (3.27) defined by the diffeomorphism (3.47) one has

$$ \begin{align*} \Psi_{F}^\ast (\omega_{\Sigma_F}|_{\Sigma_{F\bullet\bullet}})= \pi_{Z_F}^\ast \mathrm{inc}_{Z_F}^\ast\omega_{\Sigma_F}+ d\Big ( q\, \mathrm{pr}_{S_F}^\ast \Theta_{S_F} + \big ( \sqrt{r^4+q^2} -r^2\big ) \mathrm{pr}_{S_F}^\ast \overline \Theta_{S_F}\Big ), \end{align*} $$

as well as

$$ \begin{align*} \Psi_{F}^\ast(\Theta_{Z_F})&=\mathrm{pr}_{S_F}^\ast \Theta_{S_F},\\ \Psi_F^\ast\mathrm{inc}_{Z_F}^\ast\omega_{\Sigma_F}&=r^2\mathrm{pr}_{S_F}^\ast\mathrm{inc}_{S_F}^\ast\omega_{\mathrm{vert}} + r \,d r\wedge\mathrm{pr}_{S_F}^\ast \overline \Theta_{S_F} +\nu_{S_F\times(0,\infty)}^*\,\omega_F. \end{align*} $$

Proof. By the definition of $\omega _{\Sigma _F}$ in (3.40) and (3.39), of $\Theta _{Z_F}$ in (3.46), and of $\omega _{\Sigma _F}$ , $\omega _{\mathrm {vert}}$ in (3.40) and (3.38), respectively, the claims are direct consequences of Lemma 3.4 and Corollary 3.5.

For an overview, the maps introduced above, and in particular those involved in Corollary 3.7, are illustrated in the diagram

(3.50)

This is the indefinite version of (3.43). In the diagram, everything commutes except the triangles involving an inclusion followed by the retraction $\mathrm {ret}_{Z_F}$ , a projection, or the surjection $\pi _{Z_F}$ , which commute only up to $S^1$ -equivariant homotopy. Note that the two canonical projections onto $S_F$ are both denoted by $\mathrm {pr}_{S_F}$ , whereas each fiber bundle projection onto F has its own name.

4.1 Regular Kirwan map of a singular symplectic quotient

Let now $(M,\omega ,{ \mathcal J})$ be a general Hamiltonian $S^1$ -manifold, so that the symplectic quotient (1.1) is only a stratified space, and assume that $0\in \partial \,{ \mathcal J}(M)$ . To introduce the regular Kirwan maps appearing in Theorem 1.1, we will need the following definition.

Definition 4.2. A connection $\Theta \in \Omega ^1({ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}},{\mathbb R})$ for the $S^1$ -action on ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ is said to have normal form near the singularities if for each $F\in {\mathcal F}^0$ one has

$$ \begin{align*} \Phi_F^\ast (\Theta|_{U_F\cap { \mathcal J}^{-1}(\{0\})^{\mathrm{reg}}}) = \Theta_{Z_F}|_{V_F\cap Z_F}, \end{align*} $$

where $U_F\subset M$ is a neighborhood of F as in Proposition 3.2 and $\Theta _{Z_F}\in \Omega ^1(Z_F,{\mathbb R})$ denotes the connection form (3.46) over $Z_F\subset \Sigma _F$ via (3.7).

Using a partition of unity, one readily constructs a connection with normal form near all $F\in {\mathcal F}^0$ .

Definition 4.3. The regular Kirwan map is the map of complexes

(4.3) $$ \begin{align} \kappa:(\Omega_{S^1}^*(M),d_{\mathbf{\mathfrak g}})\longrightarrow (\Omega({\mathscr M}_0^{\mathrm{reg}}),d) \end{align} $$

defined by the restriction of the formula (4.1) to the regular stratum ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ of the stratification (1.7) and using a connection $\Theta \in \Omega ^1({ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}},{\mathbb R})$ with normal form near the singularities in the sense of Definition 4.2.

The regular Kirwan map of Definition 4.3 appears in the regular term of the Hirzebruch-Riemann-Roch type formula of Theorem 1.1 inside an integral over the non-compact manifold ${\mathscr M}_0^{\mathrm {reg}}$ , which is naturally oriented by its symplectic volume form. The next Lemma shows that this integral in fact converges.

Lemma 4.4. For all $\varrho \in \Omega _{S^1}^*(M)$ and $m\in {\mathbb N}$ , the integral

(4.4) $$ \begin{align} \int_{{\mathscr M}_0^{\mathrm{reg}}}e^{m\omega_0}\kappa(\varrho):=\lim_{j\to \infty}\int_{U_j}e^{m\omega_0}\kappa(\varrho) \end{align} $$

converges, where $\{U_j\}_{j\in {\mathbb N}}$ denotes an arbitrary exhaustion of ${\mathscr M}_0^{\mathrm {reg}}$ by relatively compact open sets, and does not depend on the choice of the exhaustion.

Proof. For any $F\in {\mathcal F}^0$ , recall from (3.7) that the diffeomorphism $\Phi _F$ of the local normal form of Proposition 3.2 sends $Z_F\cap V_F$ to ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}\cap U_F$ , where $Z_F=Q_F^{-1}(\{0\})\setminus F$ has been introduced in Equation (3.6). Keeping in mind the coordinates (3.27), for any small enough $\varepsilon>0$ let $B_{F,\varepsilon }\subset { \mathcal J}^{-1}(\{0\})\cap U_F$ be the neighborhood of F defined by

(4.5) $$ \begin{align} B_{F,\varepsilon}:=\left\{x\in{ \mathcal J}^{-1}(\{0\})~\bigg|~ \Phi_F^{-1}(x)=[\wp,w]\text{ with } \sqrt{\|w^+\|_{F+}\,\|w^-\|_{F-}}<\varepsilon\right\}\,. \end{align} $$

Let $\Theta \in \Omega ^1({ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}},{\mathbb R})$ be a connection of normal form near the singularities as in Definition 4.2, so that its restriction to ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}\cap U_F$ pulls back along $\Phi _F$ to the connection $\Theta _{Z_F}\in \Omega ^1(Z_F,{\mathbb R})$ introduced in (3.46). Let $\{U_j\}_{j\in {\mathbb N}}$ be an exhaustion of ${\mathscr M}_0^{\mathrm {reg}}$ by relatively compact open sets. Using Lemma 2.6 and the fact that the Kirwan map (4.3) is a map of complexes we now get for any $\varrho \in \Omega _{S^1}^*(M)$ , $j,\,m\in {\mathbb N}$ , and $\varepsilon>0$ small enough the equality

$$ \begin{align*} \int_{U_j}e^{m\omega_0}\kappa(\varrho)&= \int_{\pi_0^{-1}(U_j)} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\kappa(\varrho)\wedge\Theta\\ &=\int_{\pi_0^{-1}(U_j)\setminus \bigcup_{F\in{\mathcal F}^0}B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\kappa(\varrho)\wedge\Theta+\sum_{F\in{\mathcal F}^0}\int_{\pi_0^{-1}(U_j)\cap B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\kappa(\varrho)\wedge\Theta. \end{align*} $$

Since ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}\setminus \bigcup _{F\in {\mathcal F}^0}B_{F,\varepsilon }$ is compact, we have a well-defined limit

$$\begin{align*}\lim_{j\to \infty}\int_{\pi_0^{-1}(U_j)\setminus \bigcup_{F\in{\mathcal F}^0}B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\kappa(\varrho)\wedge\Theta=\int_{{ \mathcal J}^{-1}(\{0\})^{\mathrm{reg}}\setminus \bigcup_{F\in{\mathcal F}^0}B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\kappa(\varrho)\wedge\Theta, \end{align*}$$

which is clearly independent of the exhaustion. Let us now fix $F\in {\mathcal F}^0$ and focus on the corresponding summand

$$ \begin{align*} \int_{\pi_0^{-1}(U_j)\cap B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\kappa(\varrho)\wedge\Theta&=\int_{\pi_0^{-1}(U_j)\cap B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\varrho\Big(\frac{i}{2\pi} d \Theta\Big)\wedge\Theta\\ &=\int_{\Phi_F^{-1}(\pi_0^{-1}(U_j)\cap B_{F,\varepsilon})} e^{m\, \mathrm{inc}_{Z_F}^\ast \omega_{\Sigma_F} } \mathrm{inc}_{Z_F}^\ast\Phi_F^\ast \varrho\Big(\frac{i}{2\pi}d\Theta_{Z_F}\Big)\,\Theta_{Z_F}. \end{align*} $$

Since $\{U_j\}_{j\in {\mathbb N}}$ is an exhaustion of ${\mathscr M}_0^{\mathrm {reg}}$ , for all $j\in {\mathbb N}$ large enough, there exist $R_j,r_j\in (0,\varepsilon )$ with $R_j>r_j$ satisfying $R_j,r_j\to 0$ as $j\to \infty $ such that the set $\mathscr U_{j,\varepsilon }:=\Psi _F^{-1}(\Phi _F^{-1}(\pi _0^{-1}(U_j)\cap B_{F,\varepsilon }))\subset S_F\times (0,\infty )$ satisfies $S_F\times (R_j,\varepsilon )\subset \mathscr U_{j,\varepsilon }\subset S_F\times (r_j,\varepsilon )$ . Pulling back along $\Psi _F$ and using Corollary 3.7, we get

(4.6) $$ \begin{align} &\int_{\pi_0^{-1}(U_j)\cap B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\kappa(\varrho)\wedge\Theta= \\ &\nonumber \int_{\mathscr U_{j,\varepsilon}} e^{m (r^2\mathrm{pr}_{S_F}^\ast\mathrm{inc}_{S_F}^\ast\omega_{\mathrm{vert}} + r \,d r\wedge\mathrm{pr}_{S_F}^\ast \overline \Theta_{S_F} +\nu_{S_F\times(0,\infty)}^*\,\omega_F)} \Psi_{F}^\ast\mathrm{inc}_{Z_F}^\ast\Phi_F^\ast \varrho\Big(\frac{i}{2\pi} \mathrm{pr}_{S_F}^\ast d \Theta_{S_F}\Big)\, \mathrm{pr}_{S_F}^\ast \Theta_{S_F}. \end{align} $$

Now, by definition (3.46) of $\Theta _{S_F}\in \Omega ^1(Z_F,{\mathbb R})$ one readily sees that all forms in the integrand of the right-hand side of (4.7) are bounded on the whole space $S_F\times (0,\varepsilon )$ , except possibly the form $\Psi _{F}^\ast \mathrm {inc}_{Z_F}^\ast \Phi _F^\ast \varrho $ . However, from (3.49) and (3.26) we have

(4.7) $$ \begin{align} \Psi_{F}:S_F\times(0,\infty)&\longrightarrow Z_F\subset\Sigma_F,\nonumber\\ ([\wp,w],r)&\longmapsto [\wp,rw], \end{align} $$

so that $\Psi _{F}$ is simply the restriction to $Z_F\subset \Sigma _F$ of polar coordinates over $\Sigma _F$ . Since $\Phi _F^\ast \varrho $ is defined on a neighborhood of the $0$ -section $V_F\subset \Sigma _F$ rather than just on $Z_F\cap V_F$ , one readily computes that it stays bounded in polar coordinates in the neighborhood $V_F\subset \Sigma _F$ of the $0$ -section, hence its restriction $\Psi _{F}^\ast \mathrm {inc}_{Z_F}^\ast \Phi _F^\ast \varrho $ to $Z_F$ stays bounded over $S_F\times (0,\varepsilon )$ as well. Therefore, the integral over the whole $S_F\times (0,\varepsilon )$ of the integrand of the right-hand side of (4.7) converges and we have a well-defined limit

(4.8) $$ \begin{align} & \lim_{j\to+\infty}\int_{\pi_0^{-1}(U_j)\cap B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\kappa(\varrho)\wedge\Theta= \\ &\nonumber \int_{S_F\times (0,\varepsilon)} e^{m (r^2\mathrm{pr}_{S_F}^\ast\mathrm{inc}_{S_F}^\ast\omega_{\mathrm{vert}} + r \,d r\wedge\mathrm{pr}_{S_F}^\ast \overline \Theta_{S_F} +\nu_{S_F\times(0,\infty)}^*\,\omega_F)} \Psi_{F}^\ast\mathrm{inc}_{Z_F}^\ast\Phi_F^\ast \varrho\Big(\frac{i}{2\pi} \mathrm{pr}_{S_F}^\ast d \Theta_{S_F}\Big)\, \mathrm{pr}_{S_F}^\ast \Theta_{S_F}, \end{align} $$

which is independent of the exhaustion. This finishes the proof.

Note from Equation (4.1) that for any $\varrho \in \Omega ^*_{S^1}(M)$ with real values, the integrand of formula (4.4) defines a signed measure on ${\mathscr M}_0^{\mathrm {reg}}$ up to a multiplicative constant, so that the fact that the limit (4.4) exists and is independent of the exhaustion actually shows that the integral converges in the usual sense. This is also evident from the proof of Lemma 4.4.

The following result will be used in a crucial way in Section 5.4 to establish Theorem 5.7 on the topological interpretation of the asymptotics of the Witten integral, which forms the core of the proof of Theorem 1.1.

Lemma 4.5. Let $\sigma \in \Omega ^*_{S^1}(M)$ be such that for every $F\in {\mathcal F}^0$ we have $\operatorname {{\mathrm {inc}}}_F^*\sigma \equiv 0$ . Then for all $m\in {\mathbb N}$ , we have

(4.9) $$ \begin{align} \int_{{\mathscr M}_0^{\mathrm{reg}}}e^{m\omega_0}\kappa(d_{\mathbf{\mathfrak g}}\sigma)=0\,, \end{align} $$

where $\kappa :\Omega _{S^1}^*(M)\to \Omega ({\mathscr M}_0^{\mathrm {reg}})$ denotes the Kirwan map (4.3).

Proof. For any $F\in {\mathcal F}^0$ , recall the definition of the neighborhood (4.5) of F. Using Lemma 2.6 and the fact that the Kirwan map (4.3) is a map of complexes we can apply Stokes’ theorem to get

$$ \begin{align*} \int_{{\mathscr M}_0^{\mathrm{reg}}}e^{m\omega_0}\kappa(d_{\mathbf{\mathfrak g}}\sigma)&= \int_{{ \mathcal J}^{-1}(\{0\})^{\mathrm{reg}}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}d\kappa(\sigma)\wedge\Theta\\ &=\lim_{\varepsilon\to 0} \int_{{ \mathcal J}^{-1}(\{0\})\setminus \bigcup_{F\in{\mathcal F}^0}B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}d\kappa(\sigma)\wedge\Theta\\ &=\sum_{F\in{\mathcal F}^0}\lim_{\varepsilon\to 0} \int_{\partial B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega}\kappa(\sigma)\wedge\Theta\\ &\qquad\qquad\qquad\qquad-\lim_{\varepsilon\to 0} \int_{{ \mathcal J}^{-1}(\{0\})\setminus \bigcup_{F\in{\mathcal F}^0}B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^\ast\omega}\kappa(\sigma)\wedge d\Theta\\ &=\sum_{F\in{\mathcal F}^0}\lim_{\varepsilon\to 0} \int_{\partial B_{F,\varepsilon}} e^{m\operatorname{{\mathrm{inc}}}_0^*\omega_0} \operatorname{{\mathrm{inc}}}_0^\ast\sigma\big(\frac{i}{2\pi}d\Theta\big)\wedge\Theta, \end{align*} $$

where the second term of the third line vanishes since the integrand is basic for the $S^1$ -action and therefore its top form component is zero. Next, we define for all small enough $\varepsilon>0$ the diffeomorphism

(4.10) $$ \begin{align} \tilde b_{F,\varepsilon}:S_F&\longrightarrow \Phi_F^{-1}(\partial B_{F,\varepsilon})=\Psi_F(S_F\times\{\varepsilon\}\times\{0\})\subset Z_F\subset \Sigma_F,\nonumber\\ p=[\wp,w]&\longmapsto\Psi_F(p,\varepsilon,0)=[\wp,\varepsilon w] \end{align} $$

using the diffeomorphism $\Psi _F$ from (3.47), and we pull back along $\Phi _F\circ \tilde b_{F,\varepsilon }$ to get

(4.11) $$ \begin{align} \int_{{\mathscr M}_0^{\mathrm{reg}}}e^{m\omega_0}\kappa(d_{\mathbf{\mathfrak g}}\sigma) =\sum_{F\in{\mathcal F}^0}\lim_{\varepsilon\to 0} \int_{S_F}e^{m\tilde b_{F,\varepsilon}^*\omega_{\Sigma_F}} \tilde b_{F,\varepsilon}^*\Phi_F^*{\sigma_{|U_F}}\big(\frac{i}{2\pi}d\tilde b_{F,\varepsilon}^*\Theta_{Z_F}\big)\wedge\tilde b_{F,\varepsilon}^*\Theta_{Z_F}. \end{align} $$

We now investigate each of the three pullbacks along $\tilde b_{F,\varepsilon }$ on the right-hand side. From Corollary 3.7, we get

(4.12) $$ \begin{align} \tilde b_{F,\varepsilon}^*\omega_{\Sigma_F}&=\varepsilon^2\mathrm{inc}_{S_F}^\ast\omega_{\mathrm{vert}}+\nu_{S_F}^*\,\omega_F,\nonumber\\ \tilde b_{F,\varepsilon}^*\Theta_{Z_F}&=\Theta_{S_F}. \end{align} $$

Furthermore, we claim that for any differential form $\alpha \in \Omega ^*(M)$ we have

(4.13) $$ \begin{align} \lim_{\varepsilon\to 0}\tilde b_{F,\varepsilon}^*\Phi_F^*{\alpha_{|U_F}}= \nu_{S_F}^*\alpha_F\quad \text{in } \Omega^*(S_F), \end{align} $$

where we write $\alpha _F:=\operatorname {{\mathrm {inc}}}_F^*\alpha \in \Omega ^*(F)$ and we use on $\Omega ^*(S_F)$ the standard Fréchet topology of uniform convergence of all derivatives, recalling that $S_F$ is compact. To prove the claim, we first note that it is enough to prove (4.13) when $\alpha \in \Omega ^0(M)=\mathrm {C^{\infty }}(M)$ is a smooth function, since (4.13) is a local formula involving pullbacks which commute with wedge products and the exterior differential d, which is a continuous operator $\Omega ^*(S_F)\to \Omega ^*(S_F)$ , and any differential form of positive degree can locally be written as a sum of wedge products of differentials of smooth functions. Now, by passing to a local trivialization of the fiber bundle $\Sigma _F$ over a local chart of F, the claim (4.13) for $\alpha \in \mathrm {C^{\infty }}(M)$ reduces to the claim that for $n,m\in {\mathbb N}_0$ , $m>0$ , the operator

$$ \begin{align*} b_\varepsilon: \mathrm{C^{\infty}_c}({\mathbb R}^n\times {\mathbb R}^m)&\longrightarrow\mathrm{C^{\infty}_c}({\mathbb R}^n\times S^{m-1}), \qquad f\longmapsto b_\varepsilon(f)(x,y):=f(x,\varepsilon y)\,, \end{align*} $$

satisfies for each $f\in \mathrm {C^{\infty }_c}({\mathbb R}^n\times {\mathbb R}^m)$ that

$$ \begin{align*} b_\varepsilon(f)\stackrel{\varepsilon\to 0}{\longrightarrow}f_0 \quad \text{in } \mathrm{C^{\infty}_c}({\mathbb R}^n\times S^{m-1}), \end{align*} $$

where $f_0(x,y):=f(x,0)$ . One easily verifies that this holds, finishing the proof of the claim.

Combining (4.12) and (4.13) allows us to evaluate the limit in (4.11), yielding

$$ \begin{align*} \int_{{\mathscr M}_0^{\mathrm{reg}}}e^{m\omega_0}\kappa(d_{\mathbf{\mathfrak g}}\sigma)& =\sum_{F\in{\mathcal F}^0} \int_{S_{F}}e^{m\nu_{S_F}^*\omega_F} \nu_{S_F}^*\sigma_F\big(\frac{i}{2\pi}d\Theta_{S_F}\big)\wedge\Theta_{S_F} =0\,, \end{align*} $$

since $\sigma _F:=\operatorname {{\mathrm {inc}}}_F^*\sigma \equiv 0$ by assumption. This concludes the proof.

Lemma 4.5 gives a partial cohomological interpretation for the regular Kirwan map (4.3), since it shows that the integral of the left-hand side of (4.9) does not depend on the choice of a connection with normal form near the singularities. Theorem 5.7 will actually show a posteriori that the condition $\operatorname {{\mathrm {inc}}}_F^*\sigma \equiv 0$ is not necessary for (4.9) to hold.

4.2 Kirwan map of a partial resolution

In order to compare our results with those in [Reference Meinrenken and Sjamaar30], let us now discuss the Kirwan map of a partial resolution. Assume that $0\in \operatorname {\mathrm {Int}}{ \mathcal J}(M)$ , and recall the notations of Section 3. For any connected component $F\in {\mathcal F}^0$ of the set of fixed points $M^{S^1}$ contained in ${ \mathcal J}^{-1}(\{0\})$ we consider the complex blow-up

(4.14) $$ \begin{align} \beta_{\widetilde{\Sigma}_F}: \widetilde{\Sigma}_F&\longrightarrow\Sigma_F \end{align} $$

of $\Sigma _F$ along its zero section $F\subset \Sigma _F$ in the sense of [Reference Guillemin and Sternberg12, Section 8], which is equivariant with respect to the $S^1$ -action considered in Section 3. The strict transform of the submanifold $Z_F\subset \Sigma _F$ introduced in (3.6) is defined as the closure ${\widetilde {Z}_F:= \overline {\beta _{\widetilde {\Sigma }_F}^{-1}(Z_F)}\subset \widetilde {\Sigma }_F}$ and inherits a natural structure of a smooth $S^1$ -submanifold of $\widetilde {\Sigma }_F$ . Setting $\mathbb {F}^0:=\sqcup _{F\in {\mathcal F}^0} F\subset M^{S^1}$ we then define the complex blow-up of M along $\mathbb {F}^0\subset M$ as the unique $S^1$ -equivariant map

(4.15) $$ \begin{align} \beta:\widetilde M\longrightarrow M \end{align} $$

which restricts to an $S^1$ -equivariant diffeomorphism over $M\setminus \mathbb {F}^0$ and which, over each open set $U_F\subset M$ as in Proposition 3.2 is given by the map $\beta _{\widetilde {\Sigma }_F}$ over $V_F$ . Like $\widetilde Z_F$ , the strict transform $\widetilde Z$ of the subset ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}\subset M$ is defined as the closure ${\widetilde Z:=\overline {\beta ^{-1} ({ \mathcal J}^{-1}(\{0\})^{{\mathrm {reg}}})}\subset \widetilde M}$ and inherits a natural structure of a smooth $S^1$ -submanifold of $\widetilde M$ . Since the $S^1$ -action on $\widetilde Z$ is locally free, one is led to the following definition.

Definition 4.6. The partial resolution of the symplectic quotient ${\mathscr M}_0$ is defined as the orbifold

$$ \begin{align*} \widetilde{\mathscr M}_0:=\widetilde Z/S^1\,, \end{align*} $$

together with the map $\beta _0:\widetilde {\mathscr M}_0\longrightarrow {\mathscr M}_0$ induced by the map (4.15) after taking quotients by the $S^1$ -action. The unique form $\widetilde \omega _0\in \Omega ^2(\widetilde {\mathscr M}_0,{\mathbb R})$ satisfying

$$ \begin{align*}\widetilde\pi_0^*\widetilde\omega_0=\operatorname{{\mathrm{inc}}}_{\widetilde Z}^\ast\beta^*\omega\,,\end{align*} $$

where $\widetilde \pi _0:\widetilde Z\longrightarrow \widetilde {\mathscr M}_0$ denotes the quotient map, is called the degenerate symplectic form of $\widetilde {\mathscr M}_0$ .

Partial resolutions were introduced for arbitrary compact Lie group actions by Kirwan in [Reference Kirwan22] in the algebraic case and by Meinrenken and Sjamaar in [Reference Meinrenken and Sjamaar30, Section 4.1.2] in the symplectic case. As it is explained in [Reference Delarue, Ramacher and Schmitt7, Corollary 3.14], our definition agrees with this definition in the case $G=S^1$ .

We now define the following alternative notion of a Kirwan map for singular symplectic quotients by an $S^1$ -action, even though $\widetilde M$ does not admit a canonical structure of a Hamiltonian $S^1$ -manifold.

Definition 4.7. The resolution Kirwan map is the map

(4.16) $$ \begin{align} \widetilde\kappa:=\kappa\circ\beta^*:H^*_{S^1}(M)\longrightarrow H^*(\widetilde{\mathscr M}_0) \end{align} $$

where $\beta ^*:H^*_{S^1}(M)\to H^*_{S^1}(\widetilde M)$ is induced by the map (4.15) and $\kappa :H^*_{S^1}(\widetilde {M})\longrightarrow H^*(\widetilde {\mathscr M}_0)$ is defined at the level of complexes by the formula (4.1) with $\operatorname {{\mathrm {inc}}}_0:{ \mathcal J}^{-1}(\{0\})\to M$ replaced by the inclusion map $\operatorname {{\mathrm {inc}}}_{\widetilde Z}:\widetilde Z\to \widetilde M$ .

The following result gives another topological interpretation of the regular Kirwan map (4.3), strengthening Lemma 4.5 under an appropriate combinatorial condition on the $S^1$ -action around the fixed points.

Lemma 4.8. With the notation of Section 3.1, assume that the weights of the $S^1$ -action satisfy the condition

(4.17) $$ \begin{align} \frac{\# W^+}{\sum_{k\in W^+}|k|}=\frac{\# W^-}{\sum_{k\in W^-}|k|} \end{align} $$

for each $F\in {\mathcal F}^0$ . Then, for any connection $\Theta \in \Omega ^1({ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}},{\mathbb R})$ with normal form near the singularities in the sense of Definition 4.2 there exists a connection $\widetilde \Theta \in \Omega ^1(\widetilde Z,{\mathbb R})$ for the $S^1$ -action on the strict transform $\widetilde Z$ satisfying

(4.18) $$ \begin{align} {\widetilde\Theta}|_{\beta^{-1}({ \mathcal J}^{-1}(\{0\})^{\mathrm{reg}})}= (\beta|_{\beta^{-1}({ \mathcal J}^{-1}(\{0\})^{\mathrm{reg}})})^*\Theta\,. \end{align} $$

In particular, for any $\varrho \in \Omega _{S^1}^*(M)$ we have

(4.19) $$ \begin{align} \int_{{\mathscr M}_0^{\text{reg}}}\,e^{\omega_0}\kappa(\varrho) =\int_{\widetilde{\mathscr M}_0}\,e^{\widetilde\omega_0}\widetilde\kappa(\varrho)\,. \end{align} $$

Proof. Let $F\in {\mathcal F}^0$ and equip the complex vector bundle $\nu _F:\Sigma _F\to F$ with the Hermitian norm $\frac {1}{\sqrt {2}}\|\cdot \|_F$ defined by Equation (3.11). Write $\nu _{\mathscr {S}_F}:\mathscr {S}_F\to F$ for the associated unit sphere bundle and consider the natural diffeomorphism

(4.20) $$ \begin{align} \Psi_{\mathscr{S}_F}:\mathscr{S}_F\times(0,\infty)& \xrightarrow{~\sim~}\Sigma_F\backslash F\nonumber\\ (w,t)&\longmapsto tw\,. \end{align} $$

Then, the bi-ellipsoid bundle $S_F\to F$ defined in (3.44) satisfies $S_F\subset \mathscr {S}_F$ . Next, using the notation (3.35), let us set

(4.21) $$ \begin{align} \Theta_{\mathscr{S}_F}:=\frac{1}{2}\operatorname{{\mathrm{inc}}}_{\mathscr{S}_F}^*(\alpha^+_F+\alpha^-_F) \in\Omega^1(\mathscr{S}_F,{\mathbb R})\,. \end{align} $$

Comparing with Equations (3.37) and (3.41) one gets the identity $\Theta _{S_F}=\operatorname {{\mathrm {inc}}}_{S_F}^*\Theta _{\mathscr {S}_F}$ , while via the restricted diffeomorphism (3.49) one has $\Psi _F^*\Theta _{Z_F}=\mathrm {pr}_{S_F}^*\Theta _{S_F}$ . On the other hand, by Definitions (3.13) and (3.14) one readily checks that (4.21) is basic for the $S^1$ -action over $\mathscr {S}_F$ induced by multiplication with $e^{2\pi i\theta }\in {\mathbb C}^*$ for all $\theta \in {\mathbb R}/{\mathbb Z}$ if and only if the condition (4.17) is satisfied. Write $\vartheta _F\in \Omega ^1(\Sigma _F\backslash F,{\mathbb R})$ for the unique form over the complement of the zero section inside $\Sigma _F$ satisfying $\Psi _{\mathscr {S}_F}^*\vartheta _F =\mathrm {pr}_{\mathscr {S}_F}^*\Theta _{\mathscr {S}_F}$ . Then, according to [Reference Guillemin and Sternberg12, Section 11] there exists a unique form $\widetilde \vartheta _F\in \Omega ^1(\widetilde {\Sigma }_F,{\mathbb R})$ satisfying

$$ \begin{align*} \widetilde\vartheta_F|_{\beta_{\widetilde \Sigma_F}^{-1}(\Sigma_F\backslash F)} =(\beta_{\widetilde \Sigma_F}|_{\beta_{\widetilde \Sigma_F}^{-1}(\Sigma_F\backslash F)})^*\vartheta_{F}\,. \end{align*} $$

By restricting $\widetilde \vartheta _F$ to $\widetilde {Z}_F\subset \widetilde {\Sigma }_F$ one gets a form ${\Theta }_{\widetilde Z_F}\in \Omega ^1(\widetilde {Z}_F,{\mathbb R})$ satisfying

$$ \begin{align*}\Theta_{\widetilde Z_F}|_{\beta_{\widetilde \Sigma_F}^{-1}(Z_F)} =(\beta_{\widetilde \Sigma_F}|_{\beta_{\widetilde \Sigma_F}^{-1}(Z_F)})^*\Theta_{Z_F}. \end{align*} $$

In view of Definition 4.2 of a connection $\Theta \in \Omega ^1({ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}},{\mathbb R})$ with normal form near the singularities this concludes the proof of (4.18). Formula (4.19) is then a straightforward consequence of the Definitions 4.3 and 4.7 of the regular and resolution Kirwan maps, using the fact that $ \beta ^{-1}_0({\mathscr M}_0^{\text {reg}})$ is of full measure inside $\widetilde {\mathscr M}_0$ .

Lemma 4.8 shows that, under the combinatorial condition (4.17), the first term in (1.10) can be interpreted topologically in terms of the partial resolution $\widetilde {\mathscr M}_0$ of ${\mathscr M}$ introduced in Definition 4.6. We will show in Section 6.3 that this condition is actually realized on a large class of examples.

Remark 4.9. It is instructive to compare the resolution Kirwan map of Definition 4.7 with the Kirwan map considered by Jeffrey, Kiem, Kirwan, and Woolf in [Reference Jeffrey, Kiem, Kirwan and Woolf16] in the case $G=S^1$ . They work in the complex case, so that ${\mathscr M}_0$ is obtained as a GIT quotient by a ${\mathbb C}^*$ -action of a smooth projective variety M, and work instead with the so-called intersection cohomology $IH^{*}({\mathscr M}_{0})$ of the singular quotient. Relying on the fact that the intersection cohomology naturally occurs as a summand inside the usual cohomology $H^{*}(\widetilde {\mathscr M}_{0})$ of the partial resolution $\widetilde {\mathscr M}_0$ of Definition 4.6, they consider the canonical map

(4.22) $$ \begin{align} \kappa_{IH}\colon H^{*}_{G}(M)\longrightarrow H^{*}(\widetilde{\mathscr M}_{0})\longrightarrow IH^{*}({\mathscr M}_{0}), \end{align} $$

obtained by composition of the resolution Kirwan map $\widetilde \kappa :H^*_{S^1}(M)\longrightarrow H^*(\widetilde {\mathscr M}_0)$ of Definition 4.7 with the canonical projection from $H^{*}(\widetilde {\mathscr M}_{0})$ onto the summand $IH^{*}({\mathscr M}_{0})$ . Nevertheless, in the general symplectic setting, the relation between $IH^{*}({\mathscr M}_{0})$ and $H^{*}(\widetilde {\mathscr M}_{0})$ is not as clear as in the complex setting considered in [Reference Jeffrey, Kiem, Kirwan and Woolf16], and there might not be a canonical choice of an intersection Kirwan map in general.

5.1 Preliminaries

We begin now with the study of the asymptotics of the Witten integral 5.1. Conceptually, we will follow the method of Meinrenken in [Reference Meinrenken28, Proof of Theorem 3.3], the crucial idea being a retraction onto the zero level set of the moment map and the use of the equivariant homotopy Lemma 2.3 and equivariant Stokes’ Lemma 2.2, combined with the classical stationary phase principle. But since we do not assume zero to be a regular value of the moment map, the situation is much more involved. In fact, we will combine a retraction onto the regular stratum of ${ \mathcal J}^{-1}(\left \{ 0 \right \})$ with retractions onto the several components of the singular stratum of ${ \mathcal J}^{-1}(\left \{ 0 \right \})$ .

As a first step, we choose – once and for all – for each $F\in {\mathcal F}$ open sets $U_F\subset M$ and $V_F\subset \Sigma _F$ as in Proposition 3.2. For technical purposes, we choose them small enough such that the symplectic form (3.40) is actually non-degenerate on a slightly larger tubular neighborhood $\widetilde V_F\subset \Sigma _F$ of the zero section of $\Sigma _F$ containing $\overline V_F$ , so that Proposition 3.2 applies to a corresponding neighborhood $\widetilde U_F\subset M$ containing $\overline U_F$ with a symplectomorphism $\widetilde \Phi _F:\widetilde V_F\to \widetilde U_F$ extending $\Phi _F$ , while keeping the sets $\widetilde U_F$ obtained for the different F disjoint. This setup will be kept fixed in all of the following.

Lemma 5.2. There exists $\delta>0$ and an $S^1$ -invariant open neighborhood $W\subset M\setminus M^{S^1}$ of ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ together with a retraction $\mathrm {ret}_0:W\to { \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ and an $S^1$ -equivariant diffeomorphism

(5.3) $$ \begin{align} \Phi_W:W\xrightarrow{\,\sim\,}{ \mathcal J}^{-1}(\{0\})^{\mathrm{reg}}\times(-\delta,\delta),\qquad p\longmapsto(\mathrm{ret}_0(p),{ \mathcal J}(p)), \end{align} $$

such that, in case $0\in \operatorname {\mathrm {Int}}\,J(M)$ , we have $\Phi _F^{-1}(U_F\cap W)\subset \Sigma _{F\bullet \bullet }$ for all $F\in {\mathcal F}^0$ and

(5.4) $$ \begin{align} \Phi_F^{-1}\circ( \mathrm{ret}_{0}|_{U_F\cap W} ) =\mathrm{ret}_{Z_F}\circ\Phi_F^{-1}|_{U_F\cap W}, \end{align} $$

while $W\cap \widetilde {U}_F=\emptyset $ in case $0\in \partial \,{ \mathcal J}(M)$ .

Proof. Let $g^{TM}$ be an $S^1$ -invariant Riemannian metric on M satisfying $\Phi _F^*(g^{TM}|_{\widetilde {U}_F})=g^{T\Sigma _F}$ in the local normal form coordinates of Proposition 3.2 for all $F\in {\mathcal F}$ . Write $\mathrm {grad}\,{ \mathcal J}\in \mathrm {C^{\infty }}(M,TM)$ for the associated Riemannian gradient of ${ \mathcal J}:M\to {\mathbb R}$ , and let $\xi \in \mathrm {C^{\infty }}(M\backslash M^{S^1},TM)$ be the vector field

(5.5) $$ \begin{align} \xi:=\frac{\mathrm{grad}\,{ \mathcal J}}{\|\mathrm{grad}\,{ \mathcal J}\|_{g^{TM}}^2}, \end{align} $$

where we use that the Hamiltonian $S^1$ -action is locally free on $M\backslash M^{S^1}$ , so that $\mathrm {grad}\,{ \mathcal J}$ is nowhere vanishing on $M\backslash M^{S^1}$ . In case $0\in \operatorname {\mathrm {Int}}{ \mathcal J}(M)$ , the vector field (5.5) is transverse to ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ , satisfies $d{ \mathcal J}\mathbin {\lrcorner } \xi =1$ over $M\backslash M^{S^1}$ , and in view of (3.27) is mapped to the vector field $\frac {\partial }{\partial q}$ by the diffeomorphism of Proposition 3.2 over $\widetilde U_F\subset M$ for each ${F\in {\mathcal F}^0}$ . From this explicit description near each $F\in {\mathcal F}^0$ and the fact that the set ${K:={ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}\cap (M\setminus \bigcup _{F\in {\mathcal F}^0} U_F)}$ is compact, it follows that there is a $\delta>0$ such that the flow $\Phi _t^{\xi }$ of $\xi $ is defined on ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ for all $t\in (-\delta ,\delta )$ . Consequently, the smooth map

(5.6) $$ \begin{align} \Phi^{\xi}:{ \mathcal J}^{-1}(\{0\})^{\mathrm{reg}}\times(-\delta,\delta)& \longrightarrow M \nonumber\\ (p,t)&\longmapsto\Phi_t^{\xi}(p)\,, \end{align} $$

is a diffeomorphism onto its image satisfying ${ \mathcal J}(\Phi ^{\xi }(p,t))=t$ for all $p\in { \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ and $t\in (-\delta ,\delta )$ . Setting $W:=\mathrm {im}(\Phi ^{\xi })\subset M$ , we define the diffeomorphism (5.3) to be the inverse of (5.6). Recalling the definition of $\mathrm {ret}_{Z_F}$ in (3.48), all claimed properties except the last one are satisfied by construction. Finally, in the case $0\in \partial \,{ \mathcal J}(M)$ , then K is disjoint from $\bigcup _{F\in {\mathcal F}^0} \widetilde U_F$ simply because ${ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ is disjoint from $\widetilde U_F$ . Hence in that case, it suffices to take $\delta $ small enough to get that $W\cap \widetilde {U}_F=\emptyset $ for all $F\in {\mathcal F}^0$ .

The following simple consequence of the equivariant Stokes’ Lemma 2.2 and the equivariant homotopy Lemma 2.3 will allow us to greatly simplify the computations of the next section.

Lemma 5.3. The Witten integral of Definition 5.1 only depends on the equivariant cohomology class of $\varrho \in \Omega _{S^1}^\ast (M)$ inside $H_{S^1}^*(M)$ . Moreover, each equivariant cohomology class in $H_{S^1}^*(M)$ has a representative $\varrho \in \Omega _{S^1}^\ast (M)$ satisfying

(5.7) $$ \begin{align} \Phi_{F}^\ast(\varrho|_{U_F}) = (\nu_{\Sigma_F}^\ast\mathrm{inc}_F^\ast \varrho)|_{V_F}\quad\text{for all}\;F\in {\mathcal F}^0 \end{align} $$

and

(5.8) $$ \begin{align} \varrho|_W=\operatorname{{\mathrm{ret}}}_{0}^*\operatorname{{\mathrm{inc}}}_0^*\varrho, \end{align} $$

where $\operatorname {{\mathrm {ret}}}_{0}:W\to { \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ is as in Lemma 5.2 with a suitable $\delta>0$ and $\text {inc}_0:{ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}\to M$ denotes the inclusion map.

Proof. The first claim is standard. Indeed, let $\varrho \in \Omega _{S^1}^\ast (M)$ be equivariantly closed. In view of the fact that $ e^{2\pi im{ \mathcal J}(x)} e^{m \omega }\in \Omega _{S^1}^\ast (M)$ is equivariantly closed, Lemma 2.2 implies for any equivariant form $\gamma \in \Omega _{S^1}^\ast (M)$ that

$$ \begin{align*} \langle{\mathcal W}_m(\varrho+d_{\mathbf{\mathfrak g}}\gamma),\phi\rangle&= \int_{{\mathbb R}} \left[\int_{M} e^{2\pi im x { \mathcal J}} e^{m \omega} \varrho(x) +\int_{M} e^{2\pi imx{ \mathcal J}} e^{m \omega} d_{\mathbf{\mathfrak g}}\gamma(x) \right] \phi(x) \,d x\\ &=\int_{{\mathbb R}} \left[\int_{M} e^{2\pi imx{ \mathcal J}} e^{m \omega} \varrho(x) +\int_{M} d_{\mathbf{\mathfrak g}}\big(e^{2\pi im x { \mathcal J}} e^{m \omega} \gamma(x)\big) \right] \phi(x) \,d x\\ &=\int_{{\mathbb R}} \left[\int_{M} e^{2\pi imx{ \mathcal J}} e^{m \omega} \varrho(x) \right] \phi(x) \,d x=\langle{\mathcal W}_m(\varrho),\phi\rangle, \end{align*} $$

proving the first claim of the lemma.

Let us consider now an arbitrary equivariantly closed $\widetilde \varrho \in \Omega _{S^1}^\ast (M)$ . To prove the remaining assertions of the lemma, we need to construct an equivariantly closed ${\varrho \in \Omega _{S^1}^\ast (M)}$ in the cohomology class $[\widetilde \varrho ]\in H_{S^1}^*(M)$ satisfying (5.7) and (5.8). Using Lemma 2.3 with $N=F$ and $U_N=\widetilde U_F$ we get for each $F\in {\mathcal F}^0$ an equivariant form $\gamma _F\in \Omega _{S^1}^\ast (\widetilde V_F)$ such that

(5.9) $$ \begin{align} \widetilde \Phi_{F}^\ast(\widetilde\varrho|_{\widetilde U_F})=(\nu_{\Sigma_F}^\ast\operatorname{{\mathrm{inc}}}_F^*\widetilde\varrho)|_{\widetilde V_F}+d_{\mathbf{\mathfrak g}}\gamma_F\,. \end{align} $$

Let $\chi _F\in \mathrm {C^{\infty }}(M)$ have compact support in $\widetilde U_F$ and be identically equal to $1$ on $U_F$ . Setting

$$ \begin{align*} \gamma:=\sum_{F\in {\mathcal F}^0}\chi_F (\widetilde\Phi_F^{-1})^\ast\gamma_F\in\Omega_{S^1}^\ast(M)\,, \end{align*} $$

the equivariantly closed form $\widehat \varrho :=\widetilde \varrho -d_{\mathbf {\mathfrak g}}\gamma \in \Omega _{S^1}^\ast (M)$ satisfies $\widetilde \Phi _{F}^\ast ( \widehat \varrho |_{U_F}) = (\nu _{\Sigma _F}^\ast \operatorname {{\mathrm {inc}}}_F^\ast \widetilde \varrho )|_{V_F}$ . But $\nu _{\Sigma _F}\circ \widetilde \Phi _F^{-1}\circ \mathrm {inc}_F=\mathrm {inc}_F$ , so that (5.9) implies for each $F\in {\mathcal F}^0$

$$ \begin{align*}\mathrm{inc}_F^\ast\, d_{\mathbf{\mathfrak g}}\gamma=\mathrm{inc}_F^*\widetilde\varrho-\mathrm{inc}_F^\ast(\widetilde \Phi_F^{-1})^\ast((\nu_{\Sigma_F}^\ast\mathrm{inc}_F^*\widetilde\varrho)|_{\widetilde V_F})=0, \end{align*} $$

yielding $\mathrm {inc}_F^*\widehat \varrho =\mathrm {inc}_F^*\widetilde \varrho $ and consequently

(5.10) $$ \begin{align} \widetilde\Phi_{F}^\ast(\widehat\varrho|_{U_F})=(\nu_{\Sigma_F}^\ast\operatorname{{\mathrm{inc}}}_F^*\widehat\varrho)|_{V_F}\,. \end{align} $$

Next, choose $\delta>0$ in Lemma 5.2 such that it applies also with a slightly larger $\widehat \delta>\delta $ and gives us two corresponding retractions $\widehat {\operatorname {{\mathrm {ret}}}}_{0}:\widehat W\to { \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ and $\operatorname {{\mathrm {ret}}}_{0}:W\to { \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ from open subsets $\widehat W,W\subset M$ onto $J^{-1}(\{0\})^{\mathrm {reg}}$ such that $\widehat {\operatorname {{\mathrm {ret}}}}_{0}|_{W}=\operatorname {{\mathrm {ret}}}_{0}$ . Let $\overline {W}$ be the closure of W in M. Then the compact set $\overline {W}\setminus \bigcup _{F\in {\mathcal F}^0}U_F$ lies in $M\setminus M^{S^1}$ and its intersection with $J^{-1}(\{0\})^{\mathrm {reg}}$ is compact. By making $\delta $ , and hence W, smaller while keeping $\widehat \delta $ fixed, we can therefore achieve that $\widehat W$ contains $\overline {W}\setminus \bigcup _{F\in {\mathcal F}^0}U_F$ . Using Lemma 2.3 again, now with $N={ \mathcal J}^{-1}(\{0\})^{\mathrm {reg}}$ and $U_N=\widehat W$ , there is a form $\sigma \in \Omega _{S^1}^\ast (\widehat W)$ such that

(5.11) $$ \begin{align} \widehat\varrho|_{\widehat W}=\widehat {{\mathrm{ret}}}_{0}^*{{\mathrm{inc}}}_{0}^*\widehat\varrho+d_{\mathbf{\mathfrak g}}\sigma\,. \end{align} $$

On the other hand, in case $0\in \operatorname {\mathrm {Int}}{ \mathcal J}(M)$ , for any $F\in {\mathcal F}^0$ one has $\nu _{\Sigma _F}\circ \operatorname {{\mathrm {inc}}}_{Z_F}\circ \operatorname {{\mathrm {ret}}}_{Z_F}=\nu _{\Sigma _F|\Sigma _{F\bullet \bullet }}$ so that from (5.10) we deduce that $\widetilde \Phi _{F}^\ast (\widehat \varrho |_{\widehat W \cap U_F}) =\operatorname {{\mathrm {ret}}}_{Z_F}^*\operatorname {{\mathrm {inc}}}_{Z_F}^*\widetilde \Phi _{F}^\ast (\widehat \varrho |_{\widehat W \cap U_F})$ for such an F. Thanks to Lemma 5.2 we therefore already know that

$$ \begin{align*} \widehat\varrho|_{\widehat W\cap U_F}=(\widehat {{\mathrm{ret}}}_{0}|_{\widehat W\cap U_F})^*{{\mathrm{inc}}}_{0}^*(\widehat\varrho|_{\widehat W \cap U_F})\,. \end{align*} $$

Invoking the second part of the equivariant homotopy Lemma 2.3 and recalling that $\widehat W$ is disjoint from $\widetilde U_F$ for each $F\in {\mathcal F}^0$ in case $0\in \partial \,{ \mathcal J}(M)$ , we can thus choose $\sigma $ in Equation (5.11) such that $\sigma |_{\widehat W\cap U_F}\equiv 0$ for all $F\in {\mathcal F}^0$ . Let $\chi \in \mathrm {C^{\infty }_c}(\widehat W)$ be identically equal to $1$ on $\overline W\setminus \bigcup _{F\in {\mathcal F}^0}U_F$ . Then we can extend $\sigma $ by $0$ by setting $\widehat \sigma :=\chi \sigma \in \Omega _{S^1}^\ast (M)$ , and $\varrho :=\widehat \varrho -d_{\mathbf {\mathfrak g}}\widehat \sigma \in \Omega _{S^1}^\ast (M)$ satisfies the properties (5.7) and (5.8). Since it differs from $\widehat \varrho \in \Omega _{S^1}^\ast (M)$ , and hence from $\widetilde \varrho \in \Omega _{S^1}^\ast (M)$ , by an equivariantly exact form, $[\varrho ]=[\widetilde \varrho ]\in H_{S^1}^*(M)$ , which concludes the proof.