Relative Reshetikhin–Turaev Invariants, Hyperbolic Cone Metrics and Discrete Fourier Transforms I

Abstract

We propose the Volume Conjecture for the relative Reshetikhin–Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern–Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient 3-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston’s result that all compact oriented 3-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen–Yang’s Volume Conjecture for the Reshetikhin–Turaev invariants of closed oriented hyperbolic 3-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented 3-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin–Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.

A Proof of Proposition 5.1

The goal of this appendix is to prove Proposition 5.1. We need the following two Lemmas whose proofs are included at the end of the appendix.

Lemma A.1

For any $\epsilon >0,$ there exists a $\delta > 0$ such that

• (1)
  $$\begin{aligned} \int _{-\epsilon }^\epsilon e^{-r z^2} dz = \sqrt{\frac{\pi }{r}} + O(e^{-\delta r}), \end{aligned}$$

  and

• (2)
  $$\begin{aligned} \int _{-\epsilon }^\epsilon z^2 e^{-r z^2} dz = \frac{1}{2}\sqrt{\frac{\pi }{r^3}} + O(e^{-\delta r}). \end{aligned}$$

Lemma A.2

Let $D_{{\textbf{z}}}$ be a region in ${\mathbb {C}}^n$ containing the origin ${\textbf{0}}$ and let $g^{{\textbf{a}}}$ a family of complex valued functions on $D_{{\textbf{z}}}$ smoothly parametrized by ${\textbf{a}}$ in a region $D_{{\textbf{a}}}$ of ${\mathbb {R}}^k.$ Then there exist families of functions $h_1^{{\textbf{a}}},\dots , h_n^{{\textbf{a}}},$ and $k_1^{{\textbf{a}}},\dots ,k_n^{{\textbf{a}}}$ such that

• (1) all of $h_i^{{\textbf{a}}}$’s and $k_i^{{\textbf{a}}}$’s are smoothly parametrized by ${\textbf{a}}$ in $D_{{\textbf{a}}},$

• (2) for each ${\textbf{a}}\in D_{{\textbf{a}}},$ $h_i^{{\textbf{a}}}$ has variables $z_{i+1},\dots ,z_n$ and is holomorphic in them,

• (3) for each ${\textbf{a}}\in D_{{\textbf{a}}},$ $k_i^{{\textbf{a}}}$ has variables $z_i,\dots ,z_n$ and is holomorphic in them, and

• (4)
  $$\begin{aligned} g^{{\textbf{a}}}( z_1,\dots , z_n)=g^{{\textbf{a}}}({\textbf{0}})+\sum _{i=1}^nh_i^{{\textbf{a}}}(z_{i+1},\dots , z_n)z_i+\sum _{i=1}^nk_i^{{\textbf{a}}}(z_i,\dots ,z_n)z_i^2. \end{aligned}$$

Lemma A.3

(Complex Morse Lemma) Let $D_{{\textbf{z}}}$ be a region in ${\mathbb {C}}^n,$ let $D_{{\textbf{a}}}$ be a region in ${\mathbb {R}}^k,$ and let $f: D_{{\textbf{z}}} \times D_{{\textbf{a}}} \rightarrow {\mathbb {C}}$ be a complex valued function that is holomorphic in ${\textbf{z}}\in D_{{\textbf{z}}}$ and smooth in ${\textbf{a}}\in D_{{\textbf{a}}}.$ For ${\textbf{a}}\in D_{{\textbf{a}}},$ let $f^{{\textbf{a}}}: D_{{\textbf{z}}}\rightarrow {\mathbb {C}}$ be the function defined by $f^{{\textbf{a}}}({\textbf{z}})=f({\textbf{z}},{\textbf{a}}).$ Suppose for each ${\textbf{a}}\in D_{{\textbf{a}}},$ $f^{{\textbf{a}}}$ has a non-degenerate critical point $c_{{\textbf{a}}}$ which smoothly depends on ${\textbf{a}}.$ Then for each ${\textbf{a}}_0\in D_{{\textbf{a}}},$ there exists an open set $V \subset {\mathbb {C}}^n$ containing ${\textbf{0}},$ an open set $A \subset D_{{\textbf{a}}}$ containing ${\textbf{a}}_0,$ and a smooth function $\psi : V \times A \rightarrow D_{{\textbf{z}}}$ such that, if we denote $\psi ^{{\textbf{a}}}({\textbf{Z}}) = \psi ({\textbf{Z}}, {\textbf{a}}),$ then for each ${\textbf{a}}\in D_{{\textbf{a}}},$ ${\textbf{z}} = \psi ^{{\textbf{a}}}({\textbf{Z}})$ is a holomorphic change of variable on V such that

$$\begin{aligned} \psi ^{{\textbf{a}}}({\textbf{0}}) = {\textbf{c}}_{{\textbf{a}}}, \end{aligned}$$

$$\begin{aligned} f^{{\textbf{a}}}(\psi ^{{\textbf{a}}}({\textbf{Z}})) = f^{{\textbf{a}}}({\textbf{c}}_{{\textbf{a}}}) - Z_1^2 - \dots - Z_n^2, \end{aligned}$$

and

$$\begin{aligned} \det \Big ( D (\psi ^{{\textbf{a}}})({\textbf{0}}) \Big ) = \frac{2^{\frac{n}{2}}}{\sqrt{- \det \textrm{Hess}(f^{\textbf{a}})({\textbf{c}}_{{\textbf{a}}})}}. \end{aligned}$$

Proof of Proposition 5.1

We write ${\textbf{z}}=(z_1,\dots , z_n) \in {\mathbb {C}}^n,$ ${\textbf{Z}} = (Z_1,\dots , Z_n) \in {\mathbb {C}}^n,$ ${\textbf{W}} = (W_1,\dots , W_n) \in {\mathbb {C}}^n,$ $d{\textbf{z}}=dz_1\dots dz_n$ and ${\textbf{0}}=(0,\dots ,0) \in {\mathbb {C}}^n.$

We first consider the special case ${\textbf{c}}_r={\textbf{0}},$ $S_r=[-\epsilon ,\epsilon ]^n\subset {\mathbb {R}}^n\subset {\mathbb {C}}^n,$ and

$$\begin{aligned} f^{{\textbf{a}}_r}({\textbf{z}})=-\sum _{i=1}^nz_i^2 \end{aligned}$$

for each r. In this case, let

$$\begin{aligned} \sigma ^{{\textbf{a}}_r}_{ r}({\textbf{z}})=\upsilon _{r}({\textbf{z}}, {\textbf{a}}_r)\int _0^1e^{\frac{\upsilon _{r}({\textbf{z}}, {\textbf{a}}_r)}{r}s}ds. \end{aligned}$$

Then we can write

$$\begin{aligned} e^{\frac{\upsilon _{r}({\textbf{z}}, {\textbf{a}}_r)}{r}}=1+\frac{\sigma _{r}^{{\textbf{a}}_r}({\textbf{z}})}{r}, \end{aligned}$$

and

$$\begin{aligned} g^{{\textbf{a}}_r}({\textbf{z}})e^{rf^{{\textbf{a}}_r}_{r}(\textbf{z})}=g^{{\textbf{a}}_r}({\textbf{z}})e^{rf^{{\textbf{a}}_r}( \textbf{z})}+\frac{1}{r}g^{{\textbf{a}}_r}({\textbf{z}})\sigma ^{\textbf{a}_r}_{r}({\textbf{z}})e^{rf^{{\textbf{a}}_r}({\textbf{z}})}. \end{aligned}$$

(A.1)

Since $|\upsilon _{r}({\textbf{z}}, {\textbf{a}}_r)|<M$ for some $M>0$ independent of r, 

$$\begin{aligned} |\sigma ^{{\textbf{a}}_r}_{r}({\textbf{z}})|<M\int _0^1e^{\frac{M}{r}s}ds=M\bigg (\frac{e^{\frac{M}{r}}-1}{\frac{M}{r}}\bigg )<2M. \end{aligned}$$

Since $\{{\textbf{a}}_r\}$ is convergent and g is smooth in $\textbf{a},$ if M is big enough, then $|g^{{\textbf{a}}_r}({\textbf{z}})|<M$ for all $z \in S_r=[-\epsilon ,\epsilon ]^n$ for r large enough. By Lemma A.1 (1), we have

$$\begin{aligned} \begin{aligned} \Big |\int _{S_r}\frac{1}{r}g^{{\textbf{a}}_r} ({\textbf{z}})\sigma ^{{\textbf{a}}_r}_r({\textbf{z}})e^{rf^{{\textbf{a}}_r}({\textbf{z}})}d{\textbf{z}}\Big |&<\frac{2M^2}{r}\int _{S_r}e^{rf^{{\textbf{a}}_r}({\textbf{z}})}d{\textbf{z}}\\&=\frac{2M^2}{r}\Big (\frac{\pi }{r}\Big )^{\frac{n}{2}}+O(e^{-\delta r})=O\Big (\frac{1}{\sqrt{r^{n+2}}}\Big ). \end{aligned}\nonumber \\ \end{aligned}$$

(A.2)

By Lemma A.2, we have

$$\begin{aligned} g^{{\textbf{a}}_r}( z_1,\dots , z_n)=g^{{\textbf{a}}_r}({\textbf{0}})+\sum _{i=1}^nh_i^{{\textbf{a}}_r}(z_{i+1},\dots , z_n)z_i+\sum _{i=1}^nk_i^{{\textbf{a}}_r}(z_i,\dots ,z_n)z_i^2 \end{aligned}$$

for some holomorphic functions $\{h^{{\textbf{a}}_r}_{i}\}$ and $\{k^{{\textbf{a}}_r}_i\},$ $i\in \{1,\dots ,n\}.$ Then by Lemma A.1 (1), we have

$$\begin{aligned} \int _{S_r} g^{{\textbf{a}}_r}({\textbf{0}})e^{rf^{{\textbf{a}}_r}(\textbf{z})}d{\textbf{z}}=g^{{\textbf{a}}_r} (\textbf{0})\Big (\frac{\pi }{r}\Big )^{\frac{n}{2}}+O\Big (\frac{1}{r}\Big ). \end{aligned}$$

(A.3)

Since each $z_ie^{-rz_i^2}$ is odd, we have

$$\begin{aligned} \int _{-\epsilon }^\epsilon z_ie^{-rz_i^2}dz_i=0. \end{aligned}$$

As a consequence, for each i,  we have

$$\begin{aligned} \begin{aligned}&\int _{[-\epsilon ,\epsilon ]^n} h_i^{{\textbf{a}}_r}(z_{i+1},\dots ,z_n)z_ie^{rf^{{\textbf{a}}_r}({\textbf{z}})}d{\textbf{z}}\\&\quad =\int _{[-\epsilon ,\epsilon ]^{n-1}}h_i^{\textbf{a}_r}(z_{i+1},\dots ,z_n)e^{-r\sum _{j\ne i}z_j^2}\prod _{j\ne i}dz_j\cdot \int _{-\epsilon }^\epsilon z_ie^{-rz_i^2}dz_i =0. \end{aligned}\nonumber \\ \end{aligned}$$

(A.4)

Since $\{{\textbf{a}}_r\}$ is convergent and $k^{{\textbf{a}}}_{i}$ is smooth in ${\textbf{a}},$ if M is big enough, then for r large enough, $|k^{{\textbf{a}}_r}_i({\textbf{z}})|<M$ for all ${\textbf{z}} \in S_r,$ $i\in \{1,\dots , n\}.$ By Lemma A.1 we have for each $i\in \{1,\dots , n\}$

$$\begin{aligned} \Big |\int _{S_r} k^{{\textbf{a}}_r}_{i}({\textbf{z}})z_i^2e^{rf^{\textbf{a}_r}({\textbf{z}})}d{\textbf{z}}\Big |< & {} M\Big (\int _{-\epsilon }^\epsilon z_i^2e^{-rz_i^2}dz_i\Big )\prod _{j\ne i}\Big ( \int _{-\epsilon }^\epsilon e^{-rz_j^2}dz_j\Big ) \nonumber \\= & {} O \Big (\frac{1}{\sqrt{r^{n+2}}}\Big ). \end{aligned}$$

(A.5)

Putting (A.3), (A.4) and (A.5) together, we have the result for this special case.

For the general case, by assumption (6) of Proposition 5.1, for ${\textbf{a}}$ sufficiently close to ${\textbf{a}}_0,$ $f^{\textbf{a}}$ has a unique non-degenerate critical point ${\textbf{c}}_{\textbf{a}}$ in a sufficiently small neighborhood of ${\textbf{c}}_0.$ Then we can apply Lemma A.3 to the function f and ${\textbf{a}}_0.$ Let V, A and $\psi$ respectively be the two open sets and the change of variable function as described in Lemma A.3. For r sufficiently large, let

$$\begin{aligned} U_r=\psi ^{{\textbf{a}}_r}\Big (\prod _{i=1}^n\big \{Z_i\in {\mathbb {C}}\ \big |\ -\epsilon<\textrm{Re}(Z_i)<\epsilon , -\epsilon<\textrm{Im}(Z_i)<\epsilon \big \}\Big ) \end{aligned}$$

for some sufficiently small $\epsilon >0 .$ Let $\textrm{Vol}(S_r\setminus U_r)$ be the Euclidean volume of $S_r\setminus U_r.$ By the compactness of $S_r\setminus U_r$ and by assumptions (2), (4) and (5) of Proposition 5.1, there exist constants $M>0$ and $\delta >0$ independent of r such that $|g^{{\textbf{a}}_r}({\textbf{z}})|<M,$ $\textrm{Vol}(S_r\setminus U_r)<M$ and

$$\begin{aligned} \textrm{Re} f_r^{{\textbf{a}}_r}({\textbf{z}})<\textrm{Re} f^{\textbf{a}_r}({\textbf{c}}_r)-\delta \end{aligned}$$

(A.6)

on $S_r\setminus U$ for r large enough. Then

$$\begin{aligned} \Big |\int _{S_r\setminus U}g^{{\textbf{a}}_r}({\textbf{z}})e^{rf_r^{\textbf{a}_r}({\textbf{z}})}d{\textbf{z}}\Big | < M^2\Big (e^{r(\textrm{Re}f^{\textbf{a}_r}({\textbf{z}})({\textbf{c}}_r)-\delta )}\Big ) = O\Big (e^{r(\textrm{Re}f^{{\textbf{a}}_r}({\textbf{z}})(\textbf{c}_r)-\delta )}\Big ) . \end{aligned}$$

(A.7)

In Fig. 22 below, the shaded region is where $\textrm{Re}(-\sum _{i=1}^nZ_i^2)<0.$ For each ${\textbf{a}}_r,$ in $\overline{(\psi ^{{\textbf{a}}_r})^{-1}(U_r)}$ there is a homotopy $H_{r}$ from $\overline{(\psi ^{{\textbf{a}}_r})^{-1}(S_r\cap U_r)}$ to $[-\epsilon ,\epsilon ]^n\subset {\mathbb {R}}^n$ defined by “pushing everything down” to the real part. This is where we use condition (3). Let $S_r'=H_r(\partial (\psi ^{{\textbf{a}}_r})^{-1}(S_r\cap U_r)\times [0,1]).$ Then $\overline{(\psi ^{\textbf{a}_r})^{-1}(S_r\cap U_r)}$ is homotopic to $S_r'\cup [-\epsilon ,\epsilon ]^n.$

Fig. 22

.

Then by analyticity,

$$\begin{aligned}&\int _{S_r\cap U}g^{{\textbf{a}}_r}({\textbf{z}})e^{rf_r^{{\textbf{a}}_r}({\textbf{z}})}d{\textbf{z}} \nonumber \\&\quad =\int _{(\psi ^{{\textbf{a}}_r})^{-1}(S_r\cap U)}g^{{\textbf{a}}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))\det \mathrm D(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))e^{rf_r^{{\textbf{a}}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))}d{\textbf{Z}} \nonumber \\&\quad =\int _{S_r'}g^{{\textbf{a}}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))\det \mathrm D(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))e^{rf_r^{\textbf{a}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))}d{\textbf{Z}}\nonumber \\&\qquad +\int _{[-\epsilon ,\epsilon ]^n}g^{{\textbf{a}}_r}(\psi ({\textbf{Z}}))\det \mathrm D(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))e^{rf_r^{\textbf{a}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))}d{\textbf{Z}}. \end{aligned}$$

(A.8)

Since $\psi ^{{\textbf{a}}_r}(S')\subset S_r\setminus U_r,$ by (A.6)

$$\begin{aligned}{} & {} \int _{S_r'}g^{{\textbf{a}}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))\det \mathrm D(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))e^{rf_r^{\textbf{a}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))}d{\textbf{Z}} \nonumber \\{} & {} \quad = \int _{\psi ^{\textbf{a}_r}(S_r')} g^{{\textbf{a}}_r}({\textbf{z}})e^{rf_r^{{\textbf{a}}_r}(\textbf{z})} d{\textbf{z}} = O\Big (e^{r(\textrm{Re}f^{{\textbf{a}}_r} (\textbf{c}_r)-\delta )}\Big ); \end{aligned}$$

(A.9)

and by the special case

$$\begin{aligned} \begin{aligned}&\int _{[-\epsilon ,\epsilon ]^n}g^{{\textbf{a}}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))\det \mathrm D(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))e^{rf_r^{{\textbf{a}}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))}d{\textbf{Z}}\\&\quad =e^{rf^{{\textbf{a}}_r}({\textbf{c}}_r)}\int _{[-\epsilon ,\epsilon ]^n}g^{{\textbf{a}}_r}(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))\det \mathrm D(\psi ^{{\textbf{a}}_r}({\textbf{Z}}))e^{r\big (-\sum _{i=1}^nZ_i^2 + \frac{\upsilon _{ r}({\textbf{z}},{\textbf{a}}_r)}{r^2} \big )}d{\textbf{Z}}\\&\quad =e^{rf^{{\textbf{a}}_r}({\textbf{c}}_r)}g^{{\textbf{a}}_r}({\psi }^{{\textbf{a}}_r}({\textbf{0}}))\det \mathrm D(\psi ^{{\textbf{a}}_r}({\textbf{0}}))\Big (\frac{\pi }{r}\Big )^{\frac{n}{2}}\Big (1+O\Big (\frac{1}{r}\Big )\Big )\\&\quad =\Big (\frac{2\pi }{r}\Big )^{\frac{n}{2}}\frac{g^{\textbf{a}_r}({\textbf{c}}_r)}{\sqrt{-\det \textrm{Hess}(f^{{\textbf{a}}_r}) (\textbf{c}_r)}} e^{rf^{{\textbf{a}}_r}({\textbf{c}}_r)}\Big ( 1 + O \Big ( \frac{1}{r} \Big ) \Big ). \end{aligned} \end{aligned}$$

Together with (A.7), (A.8) and (A.9), we have the result. $\square$

Proof of Lemma A.1

For (1), we have

$$\begin{aligned} \int _{-\epsilon }^\epsilon e^{-rz^2}dz=\int _{-\infty }^\infty e^{-rz^2}dz-\int _{-\infty }^{-\epsilon } e^{-rz^2}dz-\int _{\epsilon }^\infty e^{-rz^2}dz, \end{aligned}$$

where the first term

$$\begin{aligned} \int _{-\infty }^\infty e^{-rz^2}dz=\sqrt{\frac{\pi }{r}} \end{aligned}$$

is a Gaussian integral, and the other two terms

$$\begin{aligned} \int _{-\infty }^{-\epsilon } e^{-rz^2}dz=\int _{\epsilon }^\infty e^{-rz^2}dz\leqslant \int _{\epsilon }^\infty e^{-r\epsilon z}dz=\frac{e^{-r\epsilon ^2}}{r\epsilon }=O(e^{-\delta r}). \end{aligned}$$

For (2), by integration by parts, we have

$$\begin{aligned} \int _{-\epsilon }^\epsilon e^{-rz^2}dz=ze^{-rz^2}\Big |_{-\epsilon }^\epsilon +2r\int _{-\epsilon }^\epsilon z^2e^{-rz^2}dz, \end{aligned}$$

hence by (1)

$$\begin{aligned} \int _{-\epsilon }^\epsilon z^2e^{-rz^2}dz=\frac{1}{2r}\Big (\int _{-\epsilon }^\epsilon e^{-rz^2}dz-2\epsilon e^{-r\epsilon ^2}\Big )=\frac{1}{2}\sqrt{\frac{\pi }{r^3}}+O(e^{-\delta r}). \end{aligned}$$

$\square$

Proof of Lemma A.2

We use induction on n. For $n=1,$ if $z_1\ne 0,$ then we can write

$$\begin{aligned} g^{{\textbf{a}}}(z_1)=g^{{\textbf{a}}}(0)+\frac{dg^{{\textbf{a}}}}{dz_1}(0)z_1+\bigg (\frac{g^{{\textbf{a}}}(z_1)-g^{{\textbf{a}}}(0)-\frac{dg^{{\textbf{a}}}}{dz_1}(0)z_1}{z_1^2}\bigg )z_1^2, \end{aligned}$$

and let

$$\begin{aligned} h_1^{{\textbf{a}}}=g^{{\textbf{a}}}(0) \end{aligned}$$

and

$$\begin{aligned} k_1^{{\textbf{a}}}(z_1)=\frac{g^{{\textbf{a}}}(z_1)-g^{{\textbf{a}}}(0)-\frac{dg^{{\textbf{a}}}}{dz_1}(0)z_1}{z_1^2}. \end{aligned}$$

By computing the Laurent expansion of $k_1^{{\textbf{a}}}(z_1),$ one sees $z_1=0$ is a removable singularity, and $k_1^{{\textbf{a}}}(z_1)$ extends as a holomorphic function. From the formulas, we also see that $h_1^{{\textbf{a}}}$ and $k_1^{{\textbf{a}}}$ smoothly depend on ${\textbf{a}}.$ This proves the case $n=1.$

Now assume that the result holds when $n=l.$ For $n=l+1,$ if $z_1\ne 0,$ then we have

$$\begin{aligned} \begin{aligned} g^{{\textbf{a}}}(z_1,\dots ,z_{l+1})&=g^{{\textbf{z}}}(0,z_2,\dots ,z_{l+1})+\frac{\partial g^{{\textbf{a}}}}{\partial z_1}(0,z_2,\dots ,z_{l+1})z_1\\&\quad +\bigg (\frac{g^{{\textbf{a}}}(z_1,\dots ,z_{l+1})-g^{\textbf{a}}(0,z_2,\dots ,z_{l+1})-\frac{\partial g^{{\textbf{a}}}}{\partial z_1}(0,z_2,\dots ,z_{l+1})z_1}{z_1^2}\bigg )z_1^2, \end{aligned} \end{aligned}$$

and let

$$\begin{aligned} h_1^{{\textbf{a}}}(z_2,\dots ,z_{l+1})=\frac{\partial g^{{\textbf{a}}}}{\partial z_1}(0,z_2,\dots ,z_{l+1}) \end{aligned}$$

and

$$\begin{aligned} k_1^{{\textbf{a}}}(z_1,\dots ,z_{l+1})=\frac{g^{{\textbf{a}}}(z_1,\dots ,z_{l+1})-g^{{\textbf{a}}}(0,z_2,\dots ,z_{l+1})-\frac{\partial g^{{\textbf{a}}}}{\partial z_1}(0,z_2,\dots ,z_{l+1})z_1}{z_1^2}. \end{aligned}$$

By computing the Laurent expansion again, one can see that $k_1^{{\textbf{a}}}$ holomorphically extends to $z_1=0;$ and from the formulas, $h_1^{{\textbf{a}}}$ and $k_1^{{\textbf{a}}}$ smoothly depend on ${\textbf{a}}.$ Since $g^{{\textbf{a}}}(0,z_2,\dots , z_{l+1})$ has l variables, by the induction assumption,

$$\begin{aligned} g^{{\textbf{a}}}(0,z_2,\dots ,z_{l+1})=g^{{\textbf{a}}}({\textbf{0}})+\sum _{i=2}^{l+1}h_i^{{\textbf{a}}}(z_{i+1},\dots ,z_{l+1})z_i+\sum _{i=2}^{l+1}k_i^{{\textbf{a}}}(z_i,\dots ,z_{l+1})z_i^2 \end{aligned}$$

for holomorphic functions $\{h_i^{{\textbf{a}}}\}$ and $\{k_i^{\textbf{a}}\}$ smoothly depending on ${\textbf{a}}.$ As a consequence, we have

$$\begin{aligned} g^{{\textbf{a}}}(z_1,z_2,\dots ,z_{l+1})=g^{{\textbf{a}}}({\textbf{0}})+\sum _{i=1}^{l+1}h_i^{{\textbf{a}}}(z_{i+1},\dots ,z_{l+1})z_i+\sum _{i=1}^{l+1}k_i^{{\textbf{a}}}(z_i,\dots ,z_{l+1})z_i^2. \end{aligned}$$

This completes the proof. $\square$

Proof of Lemma A.3

By doing the linear transformation $({\textbf{z}},{\textbf{a}})\mapsto ({\textbf{z}}+{\textbf{c}}_{{\textbf{a}}},{\textbf{a}}),$ we may assume that ${\textbf{c}}_{{\textbf{a}}} = {\textbf{0}}$ for all ${\textbf{a}}\in D_{{\textbf{a}}}.$ Then by the Taylor Theorem, for each ${\textbf{a}}\in D_{{\textbf{a}}}$ and ${\textbf{z}} \in D_{{\textbf{z}}},$ we can write

$$\begin{aligned} f^{{\textbf{a}}} ({\textbf{z}}) = f^{{\textbf{a}}}({\textbf{0}})+ \sum _{i=1}^n z_i b^{{\textbf{a}}}_{i}({\textbf{z}}) \end{aligned}$$

for some holomorphic functions $b^{{\textbf{a}}}_{i},$ $i =1,\dots , n.$ Since ${\textbf{0}}$ is a critical point of $f^{{\textbf{a}}},$ we have

$$\begin{aligned} b^{{\textbf{a}}}_{i}({\textbf{0}}) = \frac{\partial }{\partial z_i} f^{{\textbf{a}}} ({\textbf{0}}) = 0. \end{aligned}$$

As a result, by Taylor theorem again, we can write

$$\begin{aligned} f^{{\textbf{a}}} ({\textbf{z}}) = f^{{\textbf{a}}}({\textbf{0}})+ \sum _{i=1}^n z_i b^{\textbf{a}}_{i}({\textbf{z}}) = f^{{\textbf{a}}}({\textbf{0}})+\sum _{i, j=1}^n z_i z_j h^{{\textbf{a}}}_{ij}({\textbf{z}}) \end{aligned}$$

for some holomorphic functions $h^{{\textbf{a}}}_{ij},$ $i , j =1,\dots , n.$ Since

$$\begin{aligned} \sum _{i, j=1}^n z_i z_j h^{{\textbf{a}}}_{ij}({\textbf{z}}) = \sum _{i, j=1}^n z_i z_j \Big (\frac{h^{{\textbf{a}}}_{ij}({\textbf{z}}) + h^{{\textbf{a}}}_{ji}({\textbf{z}})}{2}\Big ), \end{aligned}$$

we may assume that $h^{{\textbf{a}}}_{ij}$ is symmetric in i and j. Since ${\textbf{0}}$ is a non-degenerate critical point of $f^{\textbf{a}},$ and

$$\begin{aligned} \frac{\partial ^2}{\partial z_i z_j} f^{{\textbf{a}}} ({\textbf{0}}) = 2h^{{\textbf{a}}}_{ij}({\textbf{0}}), \end{aligned}$$

we have $\det (h^{{\textbf{a}}}_{ij}({\textbf{0}})) \ne 0.$

Next, suppose for some m with $0 \leqslant m \leqslant n,$ there exist an open set $V_m \subset {\mathbb {C}}^n$ containing ${\textbf{0}},$ an open set $A_m \subset D_{{\textbf{a}}}$ containing ${\textbf{a}}_0,$ and a smooth function $\psi _m: V_m \times A_m \rightarrow {\mathbb {C}}^n$ such that, if we denote $\psi ^{{\textbf{a}}}_m({\textbf{Z}}) = \psi _m(\textbf{Z}, {\textbf{a}}),$ then $\psi ^{{\textbf{a}}}_m$ gives a holomorphic change of variable with

$$\begin{aligned} f^{{\textbf{a}}} (\psi ^{{\textbf{a}}}_m({\textbf{Z}}) ) = f^{{\textbf{a}}}({\textbf{0}})- Z_1^2 - \dots - Z_{m-1}^2 + \sum _{i,j = m}^n Z_i Z_j H^{{\textbf{a}}}_{m, ij} ({\textbf{Z}}) , \end{aligned}$$

where $H^{{\textbf{a}}}_{m, ij} ({\textbf{Z}})$ is holomorphic in ${\textbf{Z}}$ and symmetric in i and j. Based on this, we are going to find an open set $V_{m+1}$ of ${\mathbb {C}}^n$ containing ${\textbf{0}},$ an open set $A_{m+1} \subset A_m$ containing $\textbf{a}_0,$ and a smooth function $\psi _{m+1}: V_{m+1} \times A_{m+1} \rightarrow {\mathbb {C}}^n$ such that, if we denote $\psi ^{{\textbf{a}}}_{m+1}(\textbf{Z}) = \psi _{m+1}({\textbf{Z}}, {\textbf{a}}),$ then $\psi ^{\textbf{a}}_{m+1}$ gives a holomorphic change of variable with

$$\begin{aligned} f^{{\textbf{a}}} (\psi ^{{\textbf{a}}}_{m+1} ({\textbf{Z}}) ) = f^{{\textbf{a}}}({\textbf{0}})- Z_1^2 - \dots - Z_{m}^2 + \sum _{i,j = m+1}^n Z_i Z_j H^{{\textbf{a}}}_{m+1, ij} ({\textbf{Z}}) \end{aligned}$$

for some holomorphic functions $H^{{\textbf{a}}}_{m+1, ij} ({\textbf{Z}})$ that are symmetric in i and j.

To do so, we by the Chain Rule have

$$\begin{aligned} \frac{\partial ^2 f^{{\textbf{a}}}}{\partial Z_i Z_j} (\psi ^{{\textbf{a}}}_m({\textbf{0}})) = (D \psi ^{{\textbf{a}}}_m ({\textbf{0}}))^T \Big (\frac{\partial ^2 f^{{\textbf{a}}}}{\partial z_i z_j} ({\textbf{0}}) \Big ) (D \psi ^{\textbf{a}}_m ({\textbf{0}})) , \end{aligned}$$

where $D \psi ^{{\textbf{a}}}_m ({\textbf{0}})$ is the Jacobian matrix of $\psi ^{{\textbf{a}}}_m$ at ${\textbf{0}}.$ Thus, we have

$$\begin{aligned} 2^{m-1}\det (2H^{{\textbf{a}}}_{m, ij}({\textbf{0}}))=\det \Big (\frac{\partial ^2 f^{{\textbf{a}}} }{\partial Z_i Z_j} (\psi ^{{\textbf{a}}}_m({\textbf{0}})) \Big )\ne 0, \end{aligned}$$

implying that $(H^{{\textbf{a}}}_{m, ij}({\textbf{0}}))$ is a $(n-m+1)\times (n-m+1)$ non-singular matrix. Therefore, there exists $k \geqslant m$ such that $H^{{\textbf{a}}}_{m, km}({\textbf{0}})\ne 0.$ Reordering the variables if necessary, we may assume that $H^{{\textbf{a}}}_{m, mm}({\textbf{0}}) \ne 0.$ By continuity of $H^{{\textbf{a}}}_{m, mm}({\textbf{Z}})$ in ${\textbf{Z}}$ and ${\textbf{a}},$ there exists an open set $V'_m \subset V_m$ containing ${\textbf{0}}$ and an open set $A'_m \subset A_m$ containing ${\textbf{a}}_0$ such that $H^{{\textbf{a}}}_{m, mm}({\textbf{0}}) \ne 0$ for all $({\textbf{Z}}, {\textbf{a}}) \in V'_m \times A'_m.$ Then we can let

$$\begin{aligned} {\widetilde{H}}^{{\textbf{a}}}_{m, ij}({\textbf{Z}}) = \frac{H^{{\textbf{a}}}_{m, ij}({\textbf{Z}})}{H^{{\textbf{a}}}_{m, mm}({\textbf{Z}})} \end{aligned}$$

and have

$$\begin{aligned} f^{{\textbf{a}}}(\psi ^{{\textbf{a}}}_m ({\textbf{Z}})) =&f^{\textbf{a}}({\textbf{0}}) -Z_1^2 - \dots - Z_{m-1}^2 + \sum _{i,j = m}^n Z_i Z_j H^{{\textbf{a}}}_{m, ij} ({\textbf{Z}}) \\ =&f^{{\textbf{a}}}({\textbf{0}}) -Z_1^2 - \dots -Z_{m-1}^2 + H^{{\textbf{a}}}_{m, mm}({\textbf{Z}})\sum _{i,j = m}^n Z_i Z_j {{\widetilde{H}}}^{{\textbf{a}}}_{m, ij}({\textbf{Z}}) \\ =&f^{{\textbf{a}}}({\textbf{0}}) -Z_1^2 - \dots -Z_{m-1}^2 + H^{\textbf{a}}_{m,mm}({\textbf{Z}})\Big ( Z_m + \sum _{j=m+1}^n Z_j \widetilde{H}^{{\textbf{a}}}_{m, mj} ({\textbf{Z}}) \Big )^2 \\&- H^{{\textbf{a}}}_{m,mm}({\textbf{Z}})\Big [\Big ( \sum _{j=m+1}^n Z_j {{\widetilde{H}}}^{{\textbf{a}}}_{m, mj} ({\textbf{Z}}) \Big )^2 + \sum _{i,j = m+1}^n Z_i Z_j {{\widetilde{H}}}^{{\textbf{a}}}_{m, ij}({\textbf{Z}}) \Big ]. \end{aligned}$$

Define ${\textbf{W}} ={\textbf{W}}({\textbf{Z}})$ by

$$\begin{aligned} W_l = Z_l \end{aligned}$$

for $l \ne m,$ and

$$\begin{aligned} W_m = \sqrt{-H^{{\textbf{a}}}_{m,mm}({\textbf{Z}})} \Big ( Z_m + \sum _{j=m+1}^n Z_j {{\widetilde{H}}}^{{\textbf{a}}}_{m, mj} ({\textbf{Z}}) \Big ). \end{aligned}$$

We note that

$$\begin{aligned} \frac{\partial W_l}{\partial Z_k} ({\textbf{0}}) = \delta _{l,k} \end{aligned}$$

for $l\ne m,$ and

$$\begin{aligned} \frac{\partial W_m}{\partial Z_k} ({\textbf{0}}) = \sqrt{-H^{\textbf{a}}_{mm}({\textbf{0}})} \Big ( \delta _{m,k} + \sum _{j=m+1}^n \delta _{j,k} {{\widetilde{H}}}^{{\textbf{a}}}_{m, mj} ({\textbf{0}}) \Big ). \end{aligned}$$

Then the Jacobian matrix $DW({\textbf{0}})$ is an upper triangular matrix with the (m, m)-th entry $\sqrt{-H^{{\textbf{a}}}_{m,mm}({\textbf{0}})}\ne 0$ and all the other diagonal entries 1,  hence $\det DW(\textbf{0})\ne 0.$

Now consider the map $G: V'_m \times A'_m \rightarrow {\mathbb {C}}^n \times {\mathbb {R}}^k$ defined by

$$\begin{aligned} G({\textbf{Z}}, {\textbf{a}}) = ({\textbf{W}}({\textbf{Z}}), {\textbf{a}}). \end{aligned}$$

Then the Jacobian matrix $DG ({\textbf{0}}, {\textbf{a}}_0)$ is of the form

$$\begin{aligned} DG({\textbf{0}}, {\textbf{a}}_0) = \begin{pmatrix} DW({\textbf{0}}) &{} * \\ 0 &{} I_k \end{pmatrix}, \end{aligned}$$

where $I_k$ is the $k\times k$ identity matrix. Moreover, $\det (DG({\textbf{0}}, {\textbf{a}}_0)) = \det (DW({\textbf{0}})) \ne 0.$ Thus, by the Inverse Function Theorem, there exists an open set $V''_m\subset V'_m$ and an open subset with compact closure $A_{m+1} \subset A'_m$ containing ${\textbf{a}}_0$ such that $G: V''_{m} \times A_{m+1} \rightarrow {\mathbb {C}}^n \times A_m$ is a diffeomorphism to its image. By slightly shrinking $A_{m+1}$ if necessary, $G(V''_{m} \times A_{m+1})$ contains an open subset of the form $V_{m+1}\times A_{m+1}.$ For each ${\textbf{a}}\in A_{m+1},$ let $\psi ^{\textbf{a}}_{m+1} = \psi ^{{\textbf{a}}}_m \circ {\textbf{W}}^{-1}:V_{m+1}\rightarrow D_{{\textbf{z}}}.$ Then we have

$$\begin{aligned} f^{{\textbf{a}}}( \psi ^{{\textbf{a}}}_{m+1} ({\textbf{W}})) = f^{\textbf{a}}({\textbf{0}})-W_1^2 - \dots -W_{m}^2 + \sum _{i,j = m+1}^n W_i W_j H^{{\textbf{a}}}_{m+1, ij} ({\textbf{W}}) \end{aligned}$$

for some holomorphic functions $H^{{\textbf{a}}}_{m+1, ij} ({\textbf{Z}})$ that are symmetric in i and j.

Inductively doing the above procedure on m,  and letting $V=V_{n},$ $A=A_{n}$ and $\psi ^{{\textbf{a}}}=\psi ^{{\textbf{a}}}_n,$ we prove the result. Moreover, by the Chain Rule, we have

$$\begin{aligned} \textrm{Hess}( f^{{\textbf{a}}} \circ \psi ^{{\textbf{a}}})({\textbf{0}})) = (D \psi ^{{\textbf{a}}} ({\textbf{0}}))^T \Big (\textrm{Hess} (f^{\textbf{a}}) ({\textbf{0}}) \Big ) (D \psi ^{{\textbf{a}}} ({\textbf{0}})). \end{aligned}$$

Since $\textrm{Hess}( f^{{\textbf{a}}} \circ \psi ^{{\textbf{a}}})(\textbf{0})$ is equal to the negative of the $n\times n$ identity matrix, by taking the determinant on both sides, we get

$$\begin{aligned} \det \Big ( D (\psi ^{{\textbf{a}}})({\textbf{0}}) \Big ) = \frac{2^{\frac{n}{2}}}{\sqrt{- \det \textrm{Hess}(f^{\textbf{a}})({\textbf{0}})}}. \end{aligned}$$

$\square$