On a Hitchin–Thorpe inequality for manifolds with foliated boundaries

Abstract

We prove a Hitchin–Thorpe inequality for noncompact 4-manifolds with foliated geometry at infinity by extending on previous work by Dai and Wei. After introducing the objects at hand, we recall some preliminary results regarding the G-signature formula and the rho invariant, which are used to obtain expressions for the signature and Euler characteristic in our geometric context. We then derive our main result, and present examples.

Résumé

En se basant sur des travaux de Dai et Wei, on démontre une inégalité de Hitchin–Thorpe pour variétés non-compactes de dimension 4, et munies d’une géométrie feuilletée à l’infini. Après avoir défini les notions pertinentes à cette étude, on rappelle quelques résultats concernant la formule de G-signature et l’invariant rho, qu’on utilise ici pour obtenir des expressions de la signature et de la caractéristique d’Euler dans notre cadre géométrique. On démontre ensuite nos résultats principaux avant de présenter quelques exemples.

Appendices

Appendix A: Chern–Simons corrections for $\phi$- and d-metrics

Let N and F be closed compact oriented manifolds such that $\dim N+\dim F=4k-1$ for some $k\ge 1$ and $\dim F>0$. We consider a smooth fibration $F\rightarrow W\overset{\phi }{\rightarrow }N$, where the total space W is the boundary of a compact manifold M with fixed boundary defining function $x:M\rightarrow \mathbb {R}_{+}$. In this appendix, we are interested in the behaviour of the Chern–Simons terms of the Euler characteristic and the signature of M as $x\rightarrow 0$, assuming that this space is equipped with a fibred boundary or a fibred cusp metric.

1.1 A.1. Riemannian metrics on the boundary

In addition to the fibration structure of W, we assume that we have the following geometric objects:

• A splitting $TW=T^{V}W\oplus T^{H}W$ into vertical and horizontal subbundles, where $T^{H}W$ is identified with $\phi ^{*}TN$ and $T^{V}W=\ker (d\phi )$ (i.e. a connection on W);

• A Riemannian metric $h\in \Gamma (N,S^{2}T^{*}N)$ on N;

• A family of Riemannian metrics $\{g_{F}(y)\}_{y\in N}$ on the fibre space F that is smoothly parametrized by the base N.

We thus have a metric $\phi ^{*}h$ on the subbundle $T^{H}W$, and by interpreting the family $\{g_{F}\}$ as a field of symmetric bilinear forms $\tau \in \Gamma (W,S^{2}(T^{V}W)^{*})$, we may then define $g_{W}=\phi ^{*}h+\tau$, that gives a Riemannian submersion $\phi :(W,g_{W})\rightarrow (N,h)$ for which the splitting $T^{V}W\oplus T^{H}W$ is orthogonal.

1.2 A.2. Riemannian metrics on M

In the upcoming discussion, we work on a collar neighborhood of the boundary in M that is diffeomorphic to $[0,1[_{x}\times W$. The local coordinates on N will be denoted by $\{y^{i}\}_{i=1}^{\dim N}$, and we will write $\{z^{a}\}_{a=1}^{\dim F}$ for those on the fibre F. We will use the following notational convention for tensors on M near the boundary:

• The index 0 is reserved for the boundary defining function: $\partial _{0}=\frac{\partial }{\partial x}$;

• The indices $i,j,k,l\in \{1,\ldots ,\dim N\}$ designate variables on the base of W: $\partial _{i}=\frac{\partial }{\partial y^{i}}$;

• The indices $a,b,c,d\in \{1,\ldots ,\dim F\}$ refer to coordinates on the fiber F: $\partial _{a}=\frac{\partial }{\partial z^{a}}$;

• Greek indices will be used to designate arbitrary indices in the summation convention: $\{\partial _{\alpha }\}=\{\partial _{0},\partial _{i},\partial _{a}\}$.

We now set-up the notations for the metrics that we will be dealing with, which are smooth metrics on the $\phi$- and d-tangent bundles, $^{\phi }TM$ and $^{d}TM$ ([7], section 3).

Let $\tilde{g}_{\phi }=(dx/x^{2})^{2}+(\phi ^{*}h/x^{2})+\tau$ be a product-type fibred boundary metric, and $\tilde{g}_{d}=x^{2}\tilde{g}_{\phi }$ the associated product-type fibred cusp metric ($\phi$- and d-metrics for short), where $g_{W}=\phi ^{*}h+\tau$ is the submersion metric discussed above. Near $W=\partial M$, we have the following local coordinate expressions:

$$\begin{aligned} \tilde{g}_{\phi }&=\frac{dx^{2}}{x^{4}}+\frac{h_{ij}(y)}{x^{2}}dy^{i}\otimes dy^{j}+\tau _{ab}(y,z)dz^{a}\otimes dz^{b},\\ \tilde{g}_{d}&=\frac{dx^{2}}{x^{2}}+h_{ij}(y)dy^{i}\otimes dy^{j}+x^{2}\tau _{ab}(y,z)dz^{a}\otimes dz^{b}. \end{aligned}$$

Let $A,B\in \Gamma (M,S^{2}[{}^{\phi }T^{*}M])$ be symmetric bilinear forms, with A such that $A(x^{2}\partial _{x},\cdot )\equiv 0$ and $A(x\partial _{i},x\partial _{j})=O(x)$. Asymptotic metrics are first order perturbations (in x) of product-type metrics, and will be denoted by:

$$\begin{aligned} \hat{g}_{\phi }=\tilde{g}_{\phi }+x\cdot A\text {, }\hat{g}_{d}=x^{2}\hat{g}_{\phi }. \end{aligned}$$

Exact metrics are second order perturbations of product-type metrics, and will be denoted by

$$\begin{aligned} g_{\phi }=\tilde{g}_{\phi }+x\cdot A+x^{2}\cdot B=\hat{g}_{\phi }+x^{2}\cdot B\text {, }g_{d}=x^{2}g_{\phi }. \end{aligned}$$

For asymptotic metrics, we still have an orthogonal decomposition $^{\phi }T([0,1[_{x}\times W)=\langle x^{2}\partial _{x}\rangle \oplus TW$ near the boundary. From now onward, asymptotic $\phi$-metrics will be expressed as:

$$\begin{aligned} \hat{g}_{\phi }=\frac{dx^{2}}{x^{4}}+\frac{\phi ^{*}h}{x^{2}}+\kappa \quad \text { and }\quad \hat{g}_{d}=\frac{dx^{2}}{x^{2}}+\phi ^{*}h+x^{2}\kappa , \end{aligned}$$

where $\kappa =\tau +xA\in \Gamma (W,S^{2}T^{*}W)$ is a bilinear form on the boundary, depending smoothly on the bdf x, and restricting to a metric on the fibres F. As for exact metrics, these are the most general smooth Riemannian metrics on $^{\phi }TM$ and $^{d}TM$ that we consider here (for a fixed bdf x), but all the properties we are interested in are coming from their “asymptotic part”.

For exact $\phi$- and d-metrics, the Levi–Civita covariant derivatives will be denoted by $^{\phi }\nabla$ and $^{d}\nabla ,$ the connection 1-forms by $^{\phi }\omega$ and $^{d}\omega$, and the curvature 2-forms by $^{\phi }\Omega$ and $^{d}\Omega$. To designate the correponding objects associated to the product-type and asymptotic metrics, we will use the same superscripts on the left and add a “tilde” (product) or a “hat” (asymptotic) above.

Remark

• (1) The factor $x^{2}$ in exact metrics is the smallest exponent for x that gives a well-defined covariant derivative $^{\phi }\nabla :\Gamma (\phantom {}^{\phi }TM)\rightarrow \Gamma (\phantom {}^{\phi }TM\otimes T^{*}M)$. Indeed, taking $g_{\phi }=\hat{g}_{\phi }+xB$ with $B\in \Gamma (M,S^{2}[^{\phi }T^{*}M])$ would give $\langle dz^{a},\phantom {}^{\phi }\nabla _{\partial _{x}}\partial _{z^{b}}\rangle =O(x^{-1})$, which blows-up as $x\rightarrow 0$.

• (2) In the local frame given by $\{\partial _{\alpha }\otimes dx^{\alpha }\}$ on $\text {End}TM$, an element $B\in \Gamma (M,S^{2}[^{\phi }T^{*}M])$ decomposes as

  $$\begin{aligned} B=\left( \begin{array}{ccc} \frac{1}{x^{4}}B_{00} &{}\quad \frac{1}{x^{3}}B_{0i} &{}\quad \frac{1}{x^{2}}B_{0a}\\ \frac{1}{x^{3}}B_{i0} &{}\quad \frac{1}{x^{2}}B_{ij} &{}\quad \frac{1}{x}B_{ia}\\ \frac{1}{x^{2}}B_{a0} &{}\quad \frac{1}{x}B_{ai} &{}\quad B_{ab} \end{array}\right) , \end{aligned}$$

  where the $B_{\alpha \beta }$ are all smooth on M.

Finally, we introduce the auxiliary metrics $\hat{g}_{\varepsilon }$ and $g_{\varepsilon }$ on TM with $\varepsilon \in ]0,1[$. The first one is of asymptotic type:

$$\begin{aligned} \hat{g}_{\varepsilon }:=\frac{dx^{2}}{\varepsilon ^{4}}+\frac{\phi ^{*}h}{\varepsilon ^{2}}+\kappa , \end{aligned}$$

while the second is a product metric near the boundary:

$$\begin{aligned} g_{\varepsilon }:=\frac{dx^{2}}{\varepsilon ^{4}}+\frac{\phi ^{*}h}{\varepsilon ^{2}}+\tau . \end{aligned}$$

Their restrictions to $W=\partial M$ blow-up the metrics $(\phi ^{*}h+\kappa )$ and $g_{W}=(\phi ^{*}h+\tau )$ resp. in the direction of the base, and they coincide with $\phi$-metrics on hypersurfaces $\{x=\varepsilon \}\subset M$:

$$\begin{aligned} (\hat{g}_{\varepsilon })_{|\{x=\varepsilon \}}=(\hat{g}_{\phi })_{|\{x=\varepsilon \}}\text {;\quad }(g_{\varepsilon })_{|\{x=\varepsilon \}}=(\tilde{g}_{\phi })_{|\{x=\varepsilon \}}. \end{aligned}$$

Auxiliary metrics are introduced to apply the Atiyah-Patodi-Singer index theorem. The symbols $^{\varepsilon }\nabla$, $^{\varepsilon }\omega$ and $^{\varepsilon }\Omega$ will respectively denote the Levi–Civita connection, the connection 1-form and the curvature 2-form of $g_{\varepsilon }$ and $\varepsilon ^{2}\cdot g_{\varepsilon }$, while $^{\varepsilon }\hat{\nabla }$, $^{\varepsilon }\hat{\omega }$ and $^{\varepsilon }\widehat{\Omega }$ will designate the same objects for $\hat{g}_{\varepsilon }$ and $\varepsilon ^{2}\cdot \hat{g}_{\varepsilon }$.

We now have an important technical result, that relates the connection 1-forms of the metrics defined above:

Lemma 4.4

Let $M_{\varepsilon }=\{x\ge \varepsilon \}\subset M$ with boundary $\partial M_{\varepsilon }=\{x=\varepsilon \}$ for $0<\varepsilon \ll 1$. Then:

$$\begin{aligned} (\phantom {}^{\phi }\omega -\phantom {}^{\phi }\hat{\omega })_{|\partial M_{\varepsilon }}\in \varepsilon \cdot \Omega ^{1}\left( \partial M_{\varepsilon },\text {End}\left( \phantom {}^{\phi }TM\right) |_{\partial M_{\varepsilon }}\right) , \end{aligned}$$

$$\begin{aligned} (\phantom {}^{d}\omega -\phantom {}^{d}\hat{\omega })_{|\partial M_{\varepsilon }}\in \varepsilon \cdot \Omega ^{1}\left( \partial M_{\varepsilon },\text {End}\left( \phantom {}^{d}TM\right) |_{\partial M_{\varepsilon }}\right) . \end{aligned}$$

$$\begin{aligned} (\phantom {}^{d}\hat{\omega }-\phantom {}^{\varepsilon }\hat{\omega })_{|\partial M_{\varepsilon }}\in \varepsilon \cdot \Omega ^{1}\left( \partial M_{\varepsilon },\text {End}\left( \phantom {}^{d}TM\right) |_{\partial M_{\varepsilon }}\right) . \end{aligned}$$

Remark

In all the proofs of this appendix, we will focus on the dependence on x of the objects involved rather than giving precise expressions. To obtain the entries of the connection 1-forms, one first needs to determine the Christoffel symbols of the metrics at hand. The general procedure for computing these is as follows: since $\phantom {}^{\phi }TM$ is isomorphic to TM for $x\ne 0$, the Christoffel symbols $\Gamma _{\alpha \beta }^{\mu }$ of $g_{\phi }$ with respect to the basis $\{\partial _{\alpha }\}$ are computed by means of the usual formula

$$\begin{aligned} \Gamma _{\beta \mu }^{\alpha }=\frac{(g_{\phi }^{-1})^{\alpha \nu }}{2}\left[ -\partial _{\nu }(g_{\phi })_{\beta \mu }+\partial _{\beta }(g_{\phi })_{\mu \nu }+\partial _{\mu }(g_{\phi })_{\nu \beta }\right] , \end{aligned}$$

and then re-expressed in the basis $\{x^{2}\partial _{x},x\partial _{i},\partial _{a}\}$ or in an orthonormal frame $\{x^{2}\partial _{x},xe_{i},e_{a}\}$ using the appropriate transformation rule ($\Gamma _{\beta \mu }^{\alpha }$ are not components of a tensor). To obtain the Christoffel symbols of a d-metric, one uses the fact that $g_{d}$ and $g_{\phi }$ are related by a conformal rescaling. For instance, the covariant derivatives of product-type $\phi$- and d-metrics are related by:

$$\begin{aligned} \phantom {}^{d}\widetilde{\nabla }_{X}Y=\frac{1}{x}\cdot \left[ \phantom {}^{\phi }\widetilde{\nabla }_{X}(x\cdot Y)\right] +\left( \langle \frac{dx}{x},Y\rangle X-\tilde{g}_{\phi }(X,xY)\cdot \tilde{g}_{\phi }^{-1}\left( \frac{dx}{x^{2}},\cdot \right) \right) \end{aligned}$$

for all $X\in \mathfrak {X}(M)\text { and }Y\in \Gamma (\phantom {}^{d}TM)=x^{-1}\cdot \Gamma (\phantom {}^{\phi }TM)$, and one has analogous equations for perturbed metrics.

Proof

Let $\Gamma _{\beta \mu }^{\alpha }$ and $\widehat{\Gamma }_{\beta \mu }^{\alpha }$ be the Christoffel symbols of $g_{\phi }$ and $\hat{g}_{\phi }$ in the basis $\{x^{2}\partial _{x},x\partial _{i},\partial _{a}\}$. A direct computation yields:

$$\begin{aligned} (\phantom {}^{\phi }\omega -\phantom {}^{\phi }\hat{\omega })=\left[ \left( \Gamma _{\beta \mu }^{\alpha }-\widehat{\Gamma }_{\beta \mu }^{\alpha }\right) dx^{\mu }\right] =\left( \begin{array}{ccc} 0 &{} 0 &{} f_{a}^{0}dx\\ 0 &{} 0 &{} f_{a}^{i}dx\\ f_{0}^{a}dx &{} f_{i}^{a}dx &{} f_{b}^{a}dx \end{array}\right) +x\cdot E, \end{aligned}$$

(4.1)

where $E\in \Omega ^{1}(M,\text {End}(^{\phi }TM))$ and $f_{\beta }^{\alpha }\in \mathcal {C}^{\infty }(M)$ are of order 0 in x, and where we have used the following convention for the entries of the connection 1-forms:

$$\begin{aligned} \phantom {}^{\phi }\omega _{\beta }^{\alpha }=\big <\frac{dx^{\alpha }}{x^{j_{\alpha }}},\phantom {}^{\phi }\nabla _{\partial _{\mu }}(x^{j_{\beta }}\partial _{\beta })\big >\cdot dx^{\mu }=\Gamma _{\beta \mu }^{\alpha }\cdot dx^{\mu }, \end{aligned}$$

with $x^{j_{\alpha }}\partial _{\alpha }\text { and }(dx^{\alpha }/x^{j_{\alpha }})$ being shorthands that designate the fields $x^{2}\partial _{x},x\partial _{i},\partial _{a}\in \Gamma (\phantom {}^{\phi }TM)$ and $(dx/x^{2}),(dy^{i}/x),dz^{a}\in \Gamma (\phantom {}^{\phi }T^{*}M)$. By restricting to $\partial M_{\varepsilon }=\{x=\varepsilon \}$ in Eq. (4.1), we omit the terms in dx to find that indeed

$$\begin{aligned} (\phantom {}^{\phi }\omega -\phantom {}^{\phi }\hat{\omega })_{|\partial M_{\varepsilon }}\in \varepsilon \cdot \Omega ^{1}\left( \partial M_{\varepsilon },\text {End}\left( \phantom {}^{\phi }TM\right) |_{\partial M_{\varepsilon }}\right) . \end{aligned}$$

The same computation applied to a d-metric proves the second claim, namely that

$$\begin{aligned} (\phantom {}^{d}\omega -\phantom {}^{d}\hat{\omega })_{|\partial M_{\varepsilon }}\in \varepsilon \cdot \Omega ^{1}\left( \partial M_{\varepsilon },\text {End}\left( \phantom {}^{d}TM\right) |_{\partial M_{\varepsilon }}\right) . \end{aligned}$$

Finally, in a $\hat{g}_{d}$-orthonormal frame $\{x\partial _{x},e_{i},\frac{1}{x}e_{a}\}$ with transition matrix

$$\begin{aligned} \Lambda =\left( \begin{array}{ccc} x &{}\quad 0 &{}\quad 0\\ 0 &{}\quad e_{j}^{i} &{}\quad e_{a}^{i}\\ 0 &{}\quad \frac{1}{x}e_{i}^{a} &{}\quad \frac{1}{x}e_{b}^{a} \end{array}\right) , \end{aligned}$$

we find that the only nonvanishing entries of $(\phantom {}^{d}\hat{\omega }-\phantom {}^{\varepsilon }\hat{\omega })_{|x=\varepsilon }$ are:

$$\begin{aligned} (\phantom {}^{d}\hat{\omega }-\phantom {}^{\varepsilon }\hat{\omega })_{\alpha }^{0}|_{x=\varepsilon }=-(\phantom {}^{d}\hat{\omega }-\phantom {}^{\varepsilon }\hat{\omega })_{0}^{\alpha }|_{x=\varepsilon }=-\varepsilon \cdot e_{\alpha }^{a}(\kappa _{a\beta })_{|x=\varepsilon }\cdot dx^{\beta }+O(\varepsilon ^{2})\text {, }\forall \alpha ,\beta \ne 0 \end{aligned}$$

and the third claim follows. $\square$

The next proposition is used to prove Lemma 3.4:

Proposition 4.5

Considering a product metric ${\tilde{g}}_{\phi }=(dx/x^{2})^{2}+(\phi ^{*}h/x^{2})+\tau$ and the asymptotic metric,

$$\begin{aligned} {\hat{g}}_{\phi }=\frac{dx^{2}}{x^{4}}+\frac{\phi ^{*}h}{x^{2}}+\kappa =\tilde{g}_{\phi }+xA\text {, } \end{aligned}$$

suppose that $\forall Y\in \mathfrak {X}(N)$ and $\forall Z,V\in \Gamma (M,T^{V}W)$, the tensor $\kappa =\tau +xA\in \Gamma (W,S^{2}T^{*}W)$ satisfies the identity:

$$\begin{aligned} \left( Z\cdot \kappa (V,Y^{H})-V\cdot \kappa (Z,Y^{H})-\kappa ([Z,V],Y^{H})\right) _{|x=0}=0, \end{aligned}$$

(4.2)

where $Y^{H}\in \Gamma (M,T^{H}W)$ is the lift of Y. Then the difference of connection 1-forms $(\phantom {}^{\phi }\hat{\omega }-\phantom {}^{\phi }\widetilde{\omega })$ is such that:

$$\begin{aligned} (\phantom {}^{\phi }\hat{\omega }-\phantom {}^{\phi }\widetilde{\omega })|_{\partial M_{\varepsilon }}\in \varepsilon \cdot \Omega ^{1}\left( \partial M_{\varepsilon },\text {End}\left( \phantom {}^{\phi }TM\right) |_{\partial M_{\varepsilon }}\right) . \end{aligned}$$

Proof

Define $\tilde{\theta }:=(\phantom {}^{\phi }\hat{\omega }-\phantom {}^{\phi }\widetilde{\omega })$, and write

$$\begin{aligned} \kappa _{ab}=\tau {}_{ab}+x\cdot A_{ab}. \end{aligned}$$

Now consider a local orthonormal frame $\{x^{2}\partial _{x},xe_{i},e_{a}\}\subset \Gamma (^{\phi }TM)$ for $\hat{g}_{\phi }$ with transition matrix

$$\begin{aligned} \Lambda =\left( \begin{array}{ccc} x^{2} &{}\quad 0 &{}\quad 0\\ 0 &{} \quad xe_{j}^{i} &{}\quad xe_{a}^{i}\\ 0 &{}\quad e_{i}^{a} &{}\quad e_{b}^{a} \end{array}\right) . \end{aligned}$$

A direct computation of the entries of $\tilde{\theta }$ yields that:

$$\begin{aligned} \tilde{\theta }_{|x=\varepsilon }=\left( \begin{array}{ccc} 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad \tilde{\theta }_{j}^{i}|_{x=\varepsilon } &{}\quad \tilde{\theta }_{a}^{i}|_{x=\varepsilon }\\ 0 &{}\quad -\tilde{\theta }_{a}^{i}|_{x=\varepsilon } &{}\quad 0 \end{array}\right) +\varepsilon \cdot E, \end{aligned}$$

(4.3)

with $E\in \Omega ^{1}\left( \partial M_{\varepsilon },\text {End}\left( \phantom {}^{\phi }TM\right) |_{\partial M_{\varepsilon }}\right)$, and such that for some $C\in \Omega ^{1}(\partial M_{\varepsilon })$ and any $\alpha \ne 0$:

$$\begin{aligned} \tilde{\theta }_{\alpha }^{i}|_{x=\varepsilon }=\frac{1}{2}\Lambda _{i}^{a}\Lambda _{\alpha }^{b}\left( -\partial _{a}\kappa _{bj}+\partial _{b}\kappa _{ja}+\varepsilon \cdot \partial _{j}A_{ab}\right) dy^{j}+\varepsilon \cdot C. \end{aligned}$$

(4.4)

In local coordinates, Eq. (4.2) becomes:

$$\begin{aligned}&\left( \kappa ([Z,V],Y^{H})-\left[ Z\cdot \kappa (V,Y^{H})-V\cdot \kappa (Z,Y^{H})\right] \right) _{|x=0}\nonumber \\&\quad =Z^{a}V^{b}Y^{j}(-\partial _{a}\kappa _{bj}+\partial _{b}\kappa _{ja})_{|x=0}=0, \end{aligned}$$

for any $Z^{a},V^{b}\in \mathcal {C}^{\infty }(F)\text { and }Y^{j}\in \mathcal {C}^{\infty }(N)$, which simply means that $\forall j\in 1,\ldots ,\dim N$ and $\forall a,b\in \{1,\ldots ,\dim F\}$, we have $(\partial _{a}\kappa _{bj}-\partial _{b}\kappa _{aj})=O(x)$, and we get $\tilde{\theta }_{|x=\varepsilon }\in \varepsilon \cdot \Omega ^{1}\left( \partial M_{\varepsilon },\text {End}\left( \phantom {}^{\phi }TM\right) |_{\partial M_{\varepsilon }}\right)$. $\square$

Remark

The condition $A(x\partial _{i},x\partial _{j})=O(x)$ on the tensor $A\in \Gamma (M,S^{2}[^{\phi }T^{*}M])$ is necessary for the proof above to work. If $A(x\partial _{i},x\partial _{j})=O(x^{0})$, on the one hand $\tilde{\theta }_{|x=\varepsilon }\ne O(\varepsilon )$, but more importantly, we can’t apply the Atiyah-Patodi-Singer theorem to $(M_{\varepsilon },\hat{g}_{\varepsilon })$ anymore.

A central object in the upcoming computations is the restriction to $\partial M_{\varepsilon }$ of the curvature $^{\varepsilon }\Omega$ associated to the auxiliary metric $g_{\varepsilon }$. We have the following fact:

Lemma 4.6

Let $^{h}\Omega \in \Omega ^{2}(N,\text {End }TN)$ be the connection 2-form of the metric h on the base space N, and let $^{\kappa }\Omega (y)$ be the curvature 2-form of the metric $\kappa _{|TF_{y}}$ on the fibre $F_{y}=\phi ^{-1}(\{y\})\subset W$. Then:

$$\begin{aligned} ^{\varepsilon }\Omega |_{\partial M_{\varepsilon }}=\left( \begin{array}{ccc} 0 &{} \quad 0 &{}\quad 0\\ 0 &{}\quad \phi ^{*}\left( \phantom {}{}^{h}\Omega \right) &{}\quad 0\\ 0 &{}\quad 0 &{}\quad ^{\kappa }\Omega (y)+\alpha (y) \end{array}\right) +\varepsilon \cdot \left( \begin{array}{cc} 0 &{}\quad 0\\ 0 &{}\quad E \end{array}\right) , \end{aligned}$$

where $E\in \Omega ^{2}(\partial M_{\varepsilon },\text {End}(T\partial M_{\varepsilon }))$ and $\alpha (y)=\alpha _{ia}dy^{i}\wedge dz^{a}$ locally.

Proof

First, regarding $\phantom {}^{\varepsilon }\nabla :\Gamma (TM)\rightarrow \Gamma (T^{*}M\otimes TM)$, we have

$$\begin{aligned} \langle dx^{0},\phantom {}^{\varepsilon }\nabla _{\partial _{\beta }}\partial _{\alpha }\rangle =\langle dx^{\mu },\phantom {}^{\varepsilon }\nabla _{\partial _{0}}\partial _{\alpha }\rangle =0, \end{aligned}$$

$$\begin{aligned} \langle dx^{\mu },\phantom {}^{\varepsilon }\nabla _{\partial _{\beta }}\partial _{\alpha }\rangle =\left( \mathring{\widetilde{\Gamma }}_{\alpha \beta }^{\mu }\right) _{|x=\varepsilon }\text {, }\quad \forall \mu ,\alpha ,\beta \ne 0. \end{aligned}$$

Once we express $^{\varepsilon }\omega$ in a $g_{\varepsilon }$-orthonormal frame $\{\varepsilon ^{2}\partial _{x},\varepsilon e_{i},e_{a}\}$ near the boundary W, we find:

$$\begin{aligned} \phantom {}^{\varepsilon }\omega _{j}^{i}=\Gamma _{jk}^{i}dy^{k}+\varepsilon \gamma _{ja}^{i}dz^{a}\text {, }\phantom {}^{\varepsilon }\omega _{b}^{a}=\varepsilon \gamma _{bk}^{a}dy^{k}+\Gamma _{bc}^{a}(y)dz^{c}, \end{aligned}$$

$$\begin{aligned} \phantom {}^{\varepsilon }\omega _{a}^{i}=-\phantom {}^{\varepsilon }\omega _{i}^{a}=\varepsilon \gamma _{ib}^{a}dz^{b}\text {, }\phantom {}^{\varepsilon }\omega _{\alpha }^{0}=-\phantom {}^{\varepsilon }\omega _{0}^{\alpha }=0, \end{aligned}$$

with $\gamma _{\beta \mu }^{\alpha }\in \mathcal {C}^{\infty }(M)$ of order 0 in $\varepsilon$. We observe that $\Gamma _{jk}^{i}dy^{k}$ give the entries of the connection 1-form associated to the metric h on N, while $\Gamma _{bc}^{a}(y)dz^{c}$ are those of the Levi–Civita connection form of the metric $\kappa _{|TF_{y}}$on the fibre $F_{y}=\phi ^{-1}(\{y\})$. Using the Maurer-Cartan equation

$$\begin{aligned} \phantom {}^{\varepsilon }\Omega =d\phantom {}^{\varepsilon }\omega +\phantom {}^{\varepsilon }\omega \wedge \phantom {}^{\varepsilon }\omega , \end{aligned}$$

one obtains the stated result:

$$\begin{aligned} \phantom {}^{\varepsilon }\Omega _{j}^{i}=\phantom {}^{h}\Omega _{j}^{i}+O(\varepsilon )\text {, }\phantom {}^{\varepsilon }\Omega _{a}^{i}=-\phantom {}^{\varepsilon }\Omega _{i}^{a}=O(\varepsilon ), \end{aligned}$$

$$\begin{aligned} \phantom {}^{\varepsilon }\Omega _{b}^{a}=\phantom {}^{\kappa }\Omega _{b}^{a}(y)+\partial _{i}\Gamma _{bc}^{a}(y)dy^{i}\wedge dz^{c}+O(\varepsilon ), \end{aligned}$$

where

$$\begin{aligned} \phantom {}^{h}\Omega _{j}^{i}&=(\partial _{l}\Gamma _{jk}^{i}+\Gamma _{ls}^{i}\Gamma _{jk}^{s})dy^{l}\wedge dy^{k},\\ \phantom {}^{\kappa }\Omega _{b}^{a}(y)&=(\partial _{c}\Gamma _{bd}^{a}(y)+\Gamma _{cf}^{a}(y)\Gamma _{bd}^{f}(y))dz^{c}\wedge dz^{d}. \end{aligned}$$

$\square$

1.3 A.3. Vanishing of Chern–Simons terms

The first result here is valid for arbitrary even dimensions $\dim M=2m$. With the same notations as in Lemma 4.4, we have:

Proposition 4.7

Let $\hat{g}_{\phi }$ and $\hat{g}_{d}=x^{2}\hat{g}_{\phi }$ be asymptotic metrics, and for some $B\in \Gamma (M,S^{2}[^{\phi }T^{*}M])$, consider the exact metrics

$$\begin{aligned} g_{\phi }=\hat{g}_{\phi }+x^{2}B\quad \text { and }\quad g_{d}=\hat{g}_{d}+x^{4}B, \end{aligned}$$

and an asymptotic auxiliary metric $\hat{g}_{\varepsilon }$ on TM such that $(\hat{g}_{\varepsilon })_{|\partial M_{\varepsilon }}\equiv (\hat{g}_{\phi })_{|\partial M_{\varepsilon }}$. Then for a given invariant polynomial $P\in S^{m}(\mathfrak {so}_{2m}^{*}(\mathbb {R}))$, the Chern–Simons boundary correction to P obtained from an exact or an asymptotic $\phi$-metric are the same:

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )=\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\hat{\nabla }), \end{aligned}$$

and these corrections vanish in the case of d-metrics:

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{d}\nabla )=\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{d}\hat{\nabla })=0. \end{aligned}$$

Proof

Recall that:

$$\begin{aligned} P(M,\phantom {}^{\phi }\nabla )\equiv P(\underbrace{\phantom {}^{\phi }\Omega ,\ldots ,\phantom {}^{\phi }\Omega }_{m\text { times}})\text {, }TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )\equiv m\int _{0}^{1}dtP(\phantom {}^{\phi }\omega -\phantom {}^{\varepsilon }\hat{\omega },\underbrace{\phantom {}^{\phi }\Omega _{t},\ldots ,\phantom {}^{\phi }\Omega _{t}}_{m-1\text { times}}), \end{aligned}$$

where $\phantom {}^{\phi }\Omega _{t}$ is the curvature 2-form of the interpolation connection $^{\phi }\nabla _{t}=\phantom {}^{\varepsilon }\hat{\nabla }+t(\phantom {}^{\phi }\nabla -\phantom {}^{\varepsilon }\hat{\nabla })$. As in the proof of Theorem 3.1, we have

$$\begin{aligned} \int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )=\int _{M_{\varepsilon }}P(M,\phantom {}^{\phi }\nabla )-\int _{M_{\varepsilon }}P(M,\phantom {}^{\varepsilon }\hat{\nabla }), \end{aligned}$$

which leads to

$$\begin{aligned} \int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )&=\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\hat{\nabla })+\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\phi }\hat{\nabla },\phantom {}^{\phi }\nabla )\\&=\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\hat{\nabla })\\&\phantom {=}+\int _{\partial M_{\varepsilon }}m\left[ \int _{0}^{1}P(\phantom {}^{\phi }\omega -\phantom {}^{\phi }\hat{\omega },\phantom {}^{t}\Omega ,\ldots ,\phantom {}^{t}\Omega )dt\right] _{|\partial M_{\varepsilon }} \end{aligned}$$

with $^{t}\Omega$ the curvature of the interpolation connection $t\phantom {}^{\phi }\nabla +(1-t)\phantom {}^{\phi }\hat{\nabla }$. We note that for some $E(t)\in \Omega ^{2m-2}(M_{\varepsilon })$:

$$\begin{aligned} P(\phantom {}^{\phi }\omega -\phantom {}^{\phi }\hat{\omega },\phantom {}^{t}\Omega ,\ldots ,\phantom {}^{t}\Omega )\equiv P(\phantom {}^{\phi }\omega -\phantom {}^{\phi }\hat{\omega },\phantom {}^{t}\Omega )=P(\tilde{\theta },\phantom {}^{t}\Omega )_{|x=\varepsilon }+dx\wedge E(t), \end{aligned}$$

so that

$$\begin{aligned} \left[ \int _{0}^{1}P(\phantom {}^{\phi }\omega -\phantom {}^{\phi }\hat{\omega },\phantom {}^{t}\Omega ,\ldots ,\phantom {}^{t}\Omega )dt\right] _{|\partial M_{\varepsilon }}=\int _{0}^{1}P(\tilde{\theta },\phantom {}^{t}\Omega )_{|x=\varepsilon }dt, \end{aligned}$$

and by lemma 4.4:

$$\begin{aligned} \int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )=\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })+\varepsilon \cdot \int _{\partial M_{\varepsilon }}Q. \end{aligned}$$

Since this is also valid for $\phantom {}^{d}\hat{\nabla }$ and $\phantom {}^{d}\nabla$, we get:

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )=\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\hat{\nabla }), \end{aligned}$$

and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{d}\nabla )=\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{d}\hat{\nabla }). \end{aligned}$$

On the other hand, we also have $(\phantom {}^{d}\hat{\omega }-\phantom {}^{\varepsilon }\hat{\omega })_{|x=\varepsilon }=O(\varepsilon )$ by Lemma 4.4, so that $TP(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{d}\hat{\nabla })=\varepsilon Q$ for some $Q\in \Omega ^{m-1}(\partial M)$, and the vanishing of the last two limits above follows. $\square$

The next proposition links the Chern–Simons corrections of exact and product-type $\phi$-metrics:

Proposition 4.8

Let M be a 2m-dimensional manifold with fibred boundary and $P\in S^{m}(\mathfrak {so}_{2m}^{*}(\mathbb {R}))$ an invariant polynomial. Let $\tilde{g}_{\phi }$ and $g_{\varepsilon }$ be product metrics on $^{\phi }TM$ and TM resp., and consider the metrics

$$\begin{aligned} g_{\phi }=\tilde{g}_{\phi }+xA+x^{2}B\text { and }\hat{g}_{\varepsilon }=g_{\varepsilon }+\varepsilon \cdot A_{|x=\varepsilon }, \end{aligned}$$

with $B\in \Gamma (S^{2}[^{\phi }T^{*}M])$, and $A\in \Gamma (S^{2}[^{\phi }T^{*}M])$ a symmetric bilinear form such that

$$\begin{aligned} \mathrm{{(i)}}&A(x^{2}\partial _{x},\cdot )\equiv 0\text { and }A(xY_{1},xY_{2})=O(x),\nonumber \\ \mathrm{{(ii)}}&\left( V_{1}\cdot A(V_{2},xY_{1})-V_{2}\cdot A(V_{1},xY_{1})-A([V_{1},V_{2}],xY_{1})\right) _{|x=0}=0, \end{aligned}$$

(4.5)

for all $Y_{1},Y_{2}\in \mathfrak {X}(N)$ and all $V_{1},V_{2}\in \Gamma (W,T^{V}W)$. One then has:

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )=\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla }). \end{aligned}$$

Remark

Let $\kappa =x\cdot A$. Condition (i) in this statement means that $\kappa \in \Gamma (W,S^{2}T^{*}W)$, while condition (ii) is equivalent to Eq. (4.2). Also, we identified the vectors $Y\in \mathfrak {X}(N)$ with their horizontal lifts to W.

Proof

With the usual notations for the Levi–Civita connections, we start by writing:

$$\begin{aligned} \left[ P(M,\phantom {}^{\phi }\nabla )-P(M,\phantom {}^{\varepsilon }\hat{\nabla })\right]&=\left[ P(M,\phantom {}^{\phi }\nabla )-P(M,\phantom {}^{\phi }\hat{\nabla })\right] +\left[ P(M,\phantom {}^{\phi }\hat{\nabla })-P(M,\phantom {}^{\phi }\widetilde{\nabla })\right] \\&\phantom {=}-\left[ P(M,\phantom {}^{\varepsilon }\hat{\nabla })-P(M,\phantom {}^{\varepsilon }\nabla )\right] +\left[ P(M,\phantom {}^{\phi }\widetilde{\nabla })-P(M,\phantom {}^{\varepsilon }\nabla )\right] , \end{aligned}$$

then, integrating on $M_{\varepsilon }$ and applying Stokes theorem, we have for some $E(t)\in \Omega ^{2m-2}(\partial M_{\varepsilon })$ and curvature forms $\phantom {}^{\phi }\widehat{\Omega }_{t}\text {, }\phantom {}^{\phi }\widetilde{\Omega }_{t}\text { and }\phantom {}^{\varepsilon }\Omega _{t}$ of appropriate interpolation connections that:

$$\begin{aligned} \int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )-\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })\\ =\int _{\partial M_{\varepsilon }}m\Bigg [\int _{0}^{1}dt\cdot \Big \{ dx\wedge E(t)+&P(\phantom {}^{\phi }\omega -\phantom {}^{\phi }\hat{\omega },\phantom {}^{\phi }\widehat{\Omega }_{t})_{|x=\varepsilon }\\ \phantom {=\int _{\partial M_{\varepsilon }}m\int _{0}^{1}dt\cdot }+P(\phantom {}^{\phi }\hat{\omega }-\phantom {}^{\phi }\widetilde{\omega },\phantom {}^{\phi }\widetilde{\Omega }_{t})_{|x=\varepsilon }&-P(\phantom {}^{\varepsilon }\hat{\omega }-\phantom {}^{\varepsilon }\omega ,\phantom {}^{\varepsilon }\Omega _{t})_{|x=\varepsilon }\Big \}\Bigg ]_{\big |\partial M_{\varepsilon }}, \end{aligned}$$

by Lemma 4.4, we have $(\phantom {}^{\phi }\omega -\phantom {}^{\phi }\hat{\omega })_{|x=\varepsilon }\in \varepsilon \cdot \Omega ^{1}(\partial M_{\varepsilon },\text {End}\left( \phantom {}^{\phi }TM\right) |_{\partial M_{\varepsilon }})$, and by Proposition 4.5:

$$\begin{aligned} (\phantom {}^{\phi }\hat{\omega }-\phantom {}^{\phi }\widetilde{\omega })|_{x=\varepsilon },(\phantom {}^{\varepsilon }\hat{\omega }-\phantom {}^{\varepsilon }\omega )|_{x=\varepsilon }\in \varepsilon \cdot \Omega ^{1}\left( \partial M_{\varepsilon },\text {End}\left( \phantom {}^{\phi }TM\right) |_{\partial M_{\varepsilon }}\right) , \end{aligned}$$

which means that for some $Q\in \Omega ^{2m-1}(\partial M_{\varepsilon })$, we get:

$$\begin{aligned} \int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )-\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })=\varepsilon \cdot \int _{\partial M_{\varepsilon }}Q, \end{aligned}$$

and the result follows once we take the limit as $\varepsilon \rightarrow 0$. $\square$

The last result focuses on the cases needed for our Hitchin–Thorpe inequality, namely when $P\in S^{m}(\mathfrak {so}_{2m}^{*}(\mathbb {R}))$ is the Hirzebruch L-polynomial or the Pfaffian.

Proposition 4.9

Let M be compact a 4k-manifold with fibred boundary $F\rightarrow \partial M\overset{\phi }{\rightarrow }N$ and consider a splitting $TW=T^{V}W\oplus T^{H}W$ for $W=\partial M$.

Let $g_{\phi }$ be an exact metric on M of the form

$$\begin{aligned} g_{\phi }=\tilde{g}_{\phi }+x\cdot A+x^{2}\cdot B,\quad \text { with }A,B\in \Gamma (M,S^{2}[^{\phi }T^{*}M]), \end{aligned}$$

where $\tilde{g}_{\phi }$ is a product metric, and A satisfies the following:

$$\begin{aligned} \mathrm{{(i)}}&A(x^{2}\partial _{x},\cdot )\equiv 0\text { and }A(xY_{1},xY_{2})=O(x),\\ \mathrm{{(ii)}}&\left( V_{1}\cdot A(V_{2},xY_{1})-V_{2}\cdot A(V_{1},xY_{1})-A([V_{1},V_{2}],xY_{1})\right) _{|x=0}=0, \end{aligned}$$

for all $Y_{1},Y_{2}\in \mathfrak {X}(N)$ and all $V_{1},V_{2}\in \Gamma (W,T^{V}W)$.

If the dimension of the fibre F is odd, and if $\hat{g}_{\varepsilon }$ is an asymptotic auxiliary metric such that $(\hat{g}_{\varepsilon })_{|\partial M_{\varepsilon }}\equiv (\tilde{g}_{\phi }+xA)_{|\partial M_{\varepsilon }}$, then the Chern–Simons correction associated to the Euler form vanishes:

$$\begin{aligned} e_{CS}(\partial M,\phantom {}^{\phi }\nabla )=&-\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}Te(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )=0. \end{aligned}$$

Furthermore, when F is one-dimensional, the Chern–Simons correction associated to the Hirzebruch L-polynomial also vanishes:

$$\begin{aligned} L_{CS}(\partial M,\phantom {}^{\phi }\nabla )=-\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TL(M,\phantom {}^{\varepsilon }\hat{\nabla },\phantom {}^{\phi }\nabla )=0. \end{aligned}$$

Proof

It is sufficient to consider $g_{\phi }=\tilde{g}_{\phi }$ and $\hat{g}_{\varepsilon }=\tilde{g}_{\varepsilon }$ by Proposition 4.8, and we thus carry-out the computations in a $\tilde{g}_{\phi }$-orthonormal frame $\{x^{2}\partial _{x},xe_{i},e_{a}\}$ of $^{\phi }TM$ near the boundary (using the same notations as above for indices, decompositions of matrices, etc). Defining $\tilde{\theta }:=(\phantom {}^{\phi }\widetilde{\omega }-\phantom {}^{\varepsilon }\omega )$, we employ the interpolation connection $\phantom {}^{\phi }\widetilde{\nabla }_{t}=^{\varepsilon }\nabla +t\tilde{\theta }$, and write $\phantom {}^{\phi }\widetilde{\omega }_{t}$ and $\phantom {}^{\phi }\widetilde{\Omega }_{t}$ for its connection and curvature forms. On $\partial M_{\varepsilon }=\{x=\varepsilon \}$, the only nonvanishing entries of $\tilde{\theta }|_{x=\varepsilon }$ are found to be

$$\begin{aligned} \tilde{\theta }_{i}^{0}|_{x=\varepsilon }=-\tilde{\theta }_{0}^{i}|_{x=\varepsilon }=dy^{i}\quad \text { and }\quad \tilde{\theta }_{\beta }^{\alpha }|_{x=\varepsilon }=0\text { }\forall (\alpha ,\beta )\ne (0,i) \end{aligned}$$

(4.6)

and since $\phantom {}^{\phi }\widetilde{\omega }_{t}=\phantom {}^{\varepsilon }\omega +t\tilde{\theta }$, the Maurer–Cartan equation yields:

$$\begin{aligned} \phantom {}^{\phi }\widetilde{\Omega }_{t}&=d(\phantom {}^{\varepsilon }\omega +t\tilde{\theta })+(\phantom {}^{\varepsilon }\omega +t\tilde{\theta })\wedge (\phantom {}^{\varepsilon }\omega +t\tilde{\theta })\\&=\left( d\phantom {}^{\varepsilon }\omega +\phantom {}^{\varepsilon }\omega \wedge \phantom {}^{\varepsilon }\omega \right) +t(d\tilde{\theta }+\phantom {}^{\varepsilon }\omega \wedge \tilde{\theta }+\tilde{\theta }\wedge \phantom {}^{\varepsilon }\omega )+t^{2}(\tilde{\theta }\wedge \tilde{\theta }), \end{aligned}$$

so that by Lemma 4.6:

$$\begin{aligned} \phantom {}^{\phi }\widetilde{\Omega }_{t}|_{x=\varepsilon }=\left( \begin{array}{lll} 0 &{} t[dy^{k}\wedge \phantom {}^{\varepsilon }\omega _{k}^{i}] &{} O(\varepsilon )\\ -t[dy^{k}\wedge \phantom {}^{\varepsilon }\omega _{k}^{i}] &{} \phi ^{*}\left( \phantom {}{}^{h}\Omega \right) +t^{2}[\delta _{kj}dy^{i}\wedge dy^{j}]+O(\varepsilon ) &{} O(\varepsilon )\\ O(\varepsilon ) &{} O(\varepsilon ) &{} ^{\kappa }\Omega (y)+\alpha (y)+O(\varepsilon ) \end{array}\right) . \end{aligned}$$

(4.7)

Pfaffian: For a 2m-dimensional manifold, the Euler form is given by

for a constant $c_{m}\in \mathbb {R}{\backslash }\{0\}$. If we write

$$\begin{aligned} \int _{\partial M_{\varepsilon }}Te(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })&=\int _{\partial M_{\varepsilon }}\left[ \int _{0}^{1}\left( P(\phantom {}^{\phi }\widetilde{\omega }-\phantom {}^{\varepsilon }\omega ,\phantom {}^{\phi }\widetilde{\Omega }_{t})_{|\partial M_{\varepsilon }}+dx\wedge E(t)\right) dt\right] _{|\partial M_{\varepsilon }}\\&=\int _{\partial M_{\varepsilon }}\int _{0}^{1}P(\widetilde{\theta },\phantom {}^{\phi }\widetilde{\Omega }_{t})_{|\partial M_{\varepsilon }}dt, \end{aligned}$$

where $E(t)\in \Omega ^{2m-2}(M_{\varepsilon })$, and

$$\begin{aligned} P(\tilde{\theta },\phantom {}^{\phi }\widetilde{\Omega }_{t})_{|\partial M_{\varepsilon }}&=\frac{c_{m}}{(2m)!}\sum _{\sigma \in S_{2m}}(-1)^{|\sigma |}\left[ \tilde{\theta }_{\sigma (1)}^{\sigma (2)}\wedge \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{\sigma (3)}^{\sigma (4)}\wedge \cdots \wedge \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{\sigma (2m-1)}^{\sigma (2m)}\right] _{\big |x=\varepsilon }. \end{aligned}$$

(4.8)

Now for $\dim F=2f+1$ and $\dim N=2n$, the last expression combined with Eqs. (4.8), (4.6) and (4.7) yield that the nonvanishing summands which are proportional to the volume form on $\partial M_{\varepsilon }$ necessarily come from products such as:

$$\begin{aligned} \tilde{\theta }_{0}^{i_{1}}\wedge \underbrace{\left[ \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{i_{2}}^{i_{3}}\wedge \cdots \wedge \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{i_{2n-2}}^{i_{2n-1}}\right] }_{\in \Omega ^{2n-2}(N)\oplus \varepsilon \Omega ^{2n-2}(\partial M_{\varepsilon })}\wedge \underbrace{\left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{i_{2n}}^{a_{1}}}_{\propto \varepsilon (dy^{i}\wedge dz^{a})}\wedge \underbrace{\left[ \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{a_{2}}^{a_{3}}\wedge \cdots \wedge \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{a_{2f}}^{a_{2f+1}}\right] }_{\in \Omega ^{2f}(F)\oplus \Omega ^{2f}(\partial M_{\varepsilon })}. \end{aligned}$$

(4.9)

where the $i_{j}$’s are indices coming from the base N and the $a_{j}$’s are associated to the fibre F. Again by Eq. (4.7), the presence of factors for odd dimensional fibres leads to:

for some $Q\in \Omega ^{2m-1}(\partial M)$.

L -polynomial: Let M be 4k-dimensional with one-dimensional fibres of the boundary. If we write

then the summands of that are proportional to a volume form on $\partial M_{\varepsilon }$ are obtained from products of the form:

and

which means that if $\dim F=1$ then for some $Q\in \Omega ^{4k-1}(\partial M_{\varepsilon })$, and the claim follows. $\square$

1.4 A.4. Counter-examples

We discuss some cases in which Propositions 4.7 and 4.9 do not hold anymore, as well as the mistakes in [7]. We use the same notations as above.

• (1) As shown in Propositions 4.8 and 4.5, the conditions (4.5) on the perturbation term $A\in \Gamma (S^{2}[^{\phi }T^{*}M])$ for an asymptotic metric are necessary for the vanishing of the Chern–Simons corrections, and for the equality

  $$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\hat{\nabla })=\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla }), \end{aligned}$$

  to hold. The analog of Proposition 4.8 in [7] is Lemma 3.7, and claims the same equality, but the statement doesn’t hold without the conditions 4.5, and this causes theorem 3.6 of [7] to be erroneous too. At the end of p.563, the equation

  $$\begin{aligned} S=xS_{1}+dx\otimes S' \end{aligned}$$

  is false (in our notations, $S\equiv \tilde{\theta }=\phantom {}^{\phi }\hat{\nabla }-\phantom {}^{\phi }\widetilde{\nabla }$). The correct expressions are Eqs. (4.3) and (4.4) in the proof of Proposition 4.5.

• (2) In lemma 4.2 of [7], the claim is that for an asymptotic metric $g_{\phi }=\tilde{g}_{\phi }+xA$ with $A\in \Gamma (S^{2}[^{\phi }T^{*}M])$ such that $A(x^{2}\partial _{x},\cdot )\equiv 0$, one has.

  $$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TP(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\hat{\nabla })=0. \end{aligned}$$

  Here, the conditions (4.5) of Proposition 4.8 along with the restrictions on $\dim F$ in Proposition 4.9 are necessary for this statement to be true, as well as for theorem 4.3 in [7]. The mistake in the proof of lemma 4.2 of [7] is the formula

  $$\begin{aligned} P(\theta ,\Omega _{t},\ldots ,\Omega _{t})=\pi ^{*}(\alpha )+O(\varepsilon ) \end{aligned}$$

  on page 566, which comes from computational mistakes in the decomposition of $\Omega _{t}$. The correct decomposition of $\Omega _{t}$ is given in Eq. (4.7) of the proof of 4.9.

• (3) If the fibre F is even-dimensional, then $\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}Te(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })$ does not vanish in general. Going back to the proof of Proposition 4.9, if we take $\dim F=2f$ and $\dim N=2n+1$ and employ the same notations, we find that the summands in the integrand of $Te(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })|_{\partial M_{\varepsilon }}$ that are multiples of a volume form on $\partial M_{\varepsilon }$ come from products such as

  $$\begin{aligned} \tilde{\theta }_{0}^{i_{1}}\wedge \underbrace{\left[ \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{i_{2}}^{i_{3}}\wedge \cdots \wedge \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{i_{2n}}^{i_{2n+1}}\right] }_{\in \Omega ^{2n}(N)\oplus \varepsilon \Omega ^{2n}(\partial M_{\varepsilon })}\wedge \underbrace{\left[ \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{a_{1}}^{a_{2}}\wedge \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{a_{3}}^{a_{4}}\wedge \cdots \wedge \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{a_{2f-1}}^{a_{2f}}\right] }_{\in \Omega ^{2f}(F)\oplus \Omega ^{2f}(\partial M_{\varepsilon })}, \end{aligned}$$

  which implies that in all generality, $Te(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })|_{\partial M_{\varepsilon }}$ is not of order at least one in $\varepsilon$ [c.f. Eq. (4.9)].

• (4) If $\dim F>1$, then $\lim _{\varepsilon \rightarrow 0}\int _{\partial M_{\varepsilon }}TL(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })$ does not vanish in general. Recall that for a 4k-dimensional manifold $(M,\tilde{g}_{\phi })$, the Hirzebruch L-polynomial is given by

  $$\begin{aligned} L(M,\phantom {}^{\phi }\widetilde{\nabla })\equiv L_{k}[p_{1},\ldots ,p_{k}]\left( \frac{\phantom {}^{\phi }\widetilde{\Omega }}{2\pi }\right) =\sum _{\alpha }a_{\alpha }\bigwedge _{j=1}^{k}\left( p_{j}(\phantom {}^{\phi }\widetilde{\Omega }/2\pi )\right) ^{\wedge \alpha _{j}}, \end{aligned}$$

where the multi-indices $\alpha =(\alpha _{1},\ldots ,\alpha _{k})\in \mathbb {N}^{k}$ are such that

$$\begin{aligned} \alpha _{1}+2\alpha _{2}+\cdots +j\alpha _{j}+\cdots +k\alpha _{k}=k, \end{aligned}$$

and where the coefficients $a_{\alpha }$ are rational, the $p_{j}(\phantom {}^{\phi }\widetilde{\Omega }/2\pi )$ denotes the j-th Pontrjagin form, defined as the form of degree 4j in the expansion

$$\begin{aligned} \det \left( \text {Id}_{4k}+\frac{\phantom {}^{\phi }\widetilde{\Omega }}{2\pi }\right) =1+p_{1}\left( \frac{\phantom {}^{\phi }\widetilde{\Omega }}{2\pi }\right) +\cdots +p_{j}\left( \frac{\phantom {}^{\phi }\widetilde{\Omega }}{2\pi }\right) +\cdots +p_{\dim M/2}\left( \frac{\phantom {}^{\phi }\widetilde{\Omega }}{2\pi }\right) , \end{aligned}$$

and $L_{k}[p_{1},\ldots ,p_{k}]$ is homogeneous of degree 2k in the entries of $(\phantom {}^{\phi }\widetilde{\Omega }/2\pi )$. We look at the case where $k=2$ ($\dim \partial M=7$) and $\dim F=3$, in which one has:

$$\begin{aligned} p_{1}(\Omega /2\pi )&=-\frac{1}{2(2\pi )^{2}}\text {Tr}(\Omega ^{2}),\\ p_{2}(\Omega /2\pi )&=\frac{1}{8(2\pi )^{4}}[\{\text {Tr}(\Omega ^{2})\}^{2}-\text {Tr}(\Omega ^{4})], \end{aligned}$$

and

$$\begin{aligned} L_{2}[p_{1},p_{2}](\Omega /2\pi )=\frac{1}{45}[7p_{2}-(p_{1})^{2}](\Omega /2\pi )=\frac{(2\pi )^{-4}}{360}\left[ 5\left( \text {Tr}(\Omega ^{2})\right) ^{2}-7\text {Tr}(\Omega ^{4})\right] . \end{aligned}$$

Using the notations in the proof of Proposition 4.9, analyzing the integrand of $TL(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })|_{\partial M_{\varepsilon }}=\int _{0}^{1}dt[P(\tilde{\theta },\phantom {}^{\phi }\widetilde{\Omega }_{t})|_{\partial M_{\varepsilon }}]$ yields that for some constant $c\ne 0$:

$$\begin{aligned} P(\tilde{\theta },\phantom {}^{\phi }\widetilde{\Omega }_{t})=c\cdot \underbrace{\left[ \sum _{i=1}^{\dim N=4}\tilde{\theta }_{i}^{0}\wedge \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{i}^{0}\right] }_{\propto (dy^{i}\wedge dy^{j}\wedge dy^{k})}\wedge \underbrace{\left[ \sum _{a,b=1}^{\dim F=3}\left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{a}^{b}\wedge \left( \phantom {}^{\phi }\widetilde{\Omega }_{t}\right) _{a}^{b}\right] }_{\propto (dy^{l}\wedge dz^{1}\wedge dz^{2}\wedge dz^{3})}+O(\varepsilon ), \end{aligned}$$

and by Eqs. (4.6) and (4.7) above, $TL(M,\phantom {}^{\varepsilon }\nabla ,\phantom {}^{\phi }\widetilde{\nabla })|_{\partial M_{\varepsilon }}$ is not of order 1 in $\varepsilon$, and may not necessarily vanish when $\varepsilon \rightarrow 0$.

Appendix B: Multi Taub-NUT metrics

As above, we have k points $\{p_{j}\}_{j=1}^{k}$ in $\mathbb {R}^{3}$, a principal circle bundle $M\overset{\pi }{\longrightarrow }\mathbb {R}^{3}{\backslash }\{p_{j}\}$ of degree -1 near each $p_{j}$, and we are considering the following metric on M:

$$\begin{aligned} g=\pi ^{*}[V\cdot ((dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2})]+\pi ^{*}(V^{-1})\cdot (d\theta +\pi ^{*}\omega )^{2}, \end{aligned}$$

where $\theta$ is a coordinate on the fibres, $\omega$ is a connection 1-form on $\mathbb {R}^{3}{\backslash }\{p_{j}\}$ with curvature $d\omega =*_{\mathbb {R}^{3}}dV$, and the function $V:\mathbb {R}^{3}{\backslash }\{p_{j}\}_{j=1}^{k}\rightarrow \mathbb {R}$ is defined as:

$$\begin{aligned} V(x)=1+\frac{1}{2}{\displaystyle {\sum _{j=1}^{k}}\frac{1}{|x-p_{j}|}}. \end{aligned}$$

The harmonic function V determines $g_{0}$ uniquely, since for any gauge transformation $\omega \mapsto \omega +df$ with $f\in \mathcal {C}^{\infty }(\mathbb {R}^{3}{\backslash }\{p_{j}\})$, we can make the change of variable $\theta \mapsto \theta +f$ to keep the same metric ([27], section 6.2).

Now we briefly recall the construction of the smooth completion $(M_{0},g_{0})$, following section 3 of [10] (see also [2, 19]). For $0<\delta<\min _{1\le j<i\le k}|p_{i}-p_{j}|$, we consider the punctured open balls $B_{i}=\mathbb {B}_{\delta }^{3}(p_{i}){\backslash }\{p_{i}\}\subset \mathbb {R}^{3}$ ($i=1,\ldots ,k$), and note that $\pi ^{-1}(B_{i})$ are diffeomorphic to $\mathbb {B}_{\delta }^{4}(0){\backslash }\{0\}$ in such a way that the $S^{1}$ action coincides with scalar multiplication on $\mathbb {R}^{4}\simeq \mathbb {C}^{2}$. We then define

$$\begin{aligned} M_{0}:=\left( M\sqcup \bigsqcup _{j=1}^{k}\mathbb {B}_{\delta }^{4}(q_{i})\right) /\sim , \end{aligned}$$

where $\sim$ stands for the identification of the $\pi ^{-1}(B_{i})$ with punctured 4-balls $\mathbb {B}_{\delta }^{4}(q_{i}){\backslash }\{q_{i}\}$. The map $\pi :M\rightarrow \mathbb {R}^{3}{\backslash }\{p_{j}\}$ is then smoothly extended to $\pi :M_{0}\rightarrow \mathbb {R}^{3}$ with $q_{i}=\pi ^{-1}(p_{i})$, and $\pi$ acts as the projection to the space of $S^{1}$-orbits (under scalar multiplication) when restricted to the balls $\mathbb {B}_{\delta }^{4}(q_{i})$. To see that we can also extend g to a smooth metric $g_{0}$ on $M_{0}$, we note that in the vicinity of the points $q_{i}$, we can write $g=g_{i}^{F}+\alpha _{i}$ where $g_{i}^{F}$ is isometric to the flat Euclidean metric on $\mathbb {B}_{\delta }^{4}(q_{i})$, and $\alpha _{i}$ is a symmetric bilinear form which is smooth on the same neighbourhood. Indeed, let $(r,\varphi ,\psi )$ be the spherical coordinates centred at $p_{i}\in \mathbb {R}^{3}$, and write $V=V_{i}+f_{i}$ with $V_{i}(x)=(2|x-p_{i}|)^{-1}=1/(2r)$ and $f_{i}$ smooth on $\mathbb {B}_{\delta }^{3}(p_{i})$. Since

$$\begin{aligned} *_{\mathbb {R}^{3}}dV_{i}=-\frac{1}{2}\sin \varphi d\varphi \wedge d\psi =d\left( \frac{\cos \varphi d\psi }{2}\right) , \end{aligned}$$

we may take $\omega _{i}(x)=(\cos \varphi d\psi )/2$ as a local connection 1-form for the curvature $*dV_{i}$, and the restriction $g_{|\mathbb {B}_{\delta }^{4}(q_{i}){\backslash }\{q_{i}\}}=g_{i}^{F}+\alpha _{i}$ is then given by:

$$\begin{aligned} g_{i}^{F}&=\pi ^{*}[V_{i}\cdot ((dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2})]+\pi ^{*}(V_{i}^{-1})\cdot (d\theta +\pi ^{*}\omega _{i})^{2}\\&=\frac{(dr)^{2}}{2r}+\frac{r}{2}\left[ (d\varphi )^{2}+(d\psi )^{2}+(d(2\theta ))^{2}+2\cos \varphi d(2\theta )\odot d\psi \right] ,\\ \alpha _{i}&=f_{i}\pi ^{*}((dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2})+V^{-1}\cdot \big [-(f_{i}/V_{i})(d\theta +\pi ^{*}\omega _{i})^{2}\\&\phantom {=V^{-1}\cdot \big [}+\pi ^{*}(\omega -\omega _{i})^{2}+2(d\theta +\pi ^{*}\omega _{i})\odot \pi ^{*}(\omega -\omega _{i})\big ]. \end{aligned}$$

On $\mathbb {B}_{\delta }^{4}(q_{i})$, we introduce the following coordinates [9]:

$$\begin{aligned} y^{1}+iy^{2}&=\sqrt{2r}\cos \left( \frac{\varphi }{2}\right) \exp \left[ i\left( \theta +\frac{\psi }{2}\right) \right] \\ y^{3}+iy^{4}&=\sqrt{2r}\sin \left( \frac{\varphi }{2}\right) \exp \left[ i\left( \theta -\frac{\psi }{2}\right) \right] , \end{aligned}$$

and we obtain that $g_{i}^{F}=(dy^{1})^{2}+(dy^{2})^{2}+(dy^{3})^{2}+(dy^{4})^{2}$. Letting $y\rightarrow q_{i}$ in $M_{0}$, we have $\alpha _{i}\rightarrow f_{i}\pi ^{*}((dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2})_{|p_{i}}$, and it is then obvious that $g=g_{i}^{F}+\alpha _{i}$ can be smoothly extended to $q_{i}$.

Finally, the condition $d\omega =*_{\mathbb {R}^{3}}dV$ implies that g is Ricci-flat, and we can define 3 compatible parallel complex structures $\{J_{l}\}_{l=1}^{3}$ on M by taking

$$\begin{aligned} J_{1}:\text { }\pi ^{*}dx^{1}\mapsto V^{-1}(d\theta +\pi ^{*}\omega )\text {, }\pi ^{*}dx^{2}\mapsto \pi ^{*}dx^{3}, \end{aligned}$$

and permuting the action on these forms for $J_{2}$ and $J_{3}$. Since $M_{0}$ is simply connected, we obtain that $g_{0}$ is a complete Ricci-flat hyper-Kähler metric for this space.