1.1 Divisor with simple normal crossing support

For $I\subseteq \{1,\ldots,n\}$, we denote

(1.1)\begin{equation} \mathbb{C}^n_I = \{ (z_1,\ldots,z_n)\in \mathbb{C}^n : z_i = 0 \text{ for } i\in I \} \subseteq \mathbb{C}^n . \end{equation}

Let $X$ be an $n$-dimensional complex manifold.

Let $D$ be a divisor on $X$. We denote

(1.2)\begin{equation} D = \sum_{j=1}^l m_j D_j , \end{equation}

where $m_j\in \mathbb {Z}\backslash \{0\}$, $D_1,\ldots,D_l \subseteq X$ are mutually distinct and irreducible.

For $J\subseteq \{1,\ldots,l\}$, let $w_d^J$ and $D_J$ be as in (0.9), let $\chi (D_J)$ be the topological Euler characteristic of $D_J$.

Definition 1.3 If $D$ is a divisor with simple normal crossing support, we define

(1.3)\begin{equation} \chi_d(X,D) = \sum_{J\subseteq\{1,\ldots,l\}} w_d^J \chi(D_J) . \end{equation}

Moreover, if there is a meromorphic section $\gamma$ of a holomorphic line bundle over $X$ such that $\mathrm {div}(\gamma ) = D$, we define

(1.4)\begin{equation} \chi_d(X,\gamma) = \chi_d(X,D) . \end{equation}

Now we assume that $D$ is a divisor with simple normal crossing support. Let $L$ be a holomorphic line bundle over $X$ together with $\gamma \in \mathscr {M}(X,L)$ such that

(1.5)\begin{equation} \mathrm{div}(\gamma) = D . \end{equation}

Let $\gamma ^{-1}\in \mathscr {M}(X,L^{-1})$ be the inverse of $\gamma$.

We denote by $(T^*X\oplus \overline {T^*X})^{\otimes k}$ the $k$th tensor power of $T^*X\oplus \overline {T^*X}$. We denote

(1.6)\begin{equation} E^\pm_k = (T^*X\oplus\overline{T^*X})^{\otimes k} \otimes L^{\pm 1} . \end{equation}

In particular, we have $E_0^{\pm } = L^{\pm 1}$. Let $\nabla ^{E^\pm _k}$ be a connection on $E^\pm _k$.

Let $L_j$ be the normal line bundle of $D_j \hookrightarrow X$.

Definition 1.4 We define $\mathrm {Res}_{D_j}(\gamma )\in \mathscr {M}(D_j,L \otimes L_j^{-m_j})$ as follows:

(1.7) \begin{equation} \mathrm{Res}_{D_j}(\gamma) = \left\{\begin{array}{@{}ll} \dfrac{1}{m_j!}\Big(\nabla^{E^+_{m_j-1}} \cdots \nabla^{E^+_0} \gamma\Big) \Big|_{D_j} & \text{if } m_j > 0 ,\\ \dfrac{1}{|m_j|!} \Big(\Big(\nabla^{E^-_{|m_j|-1}} \cdots \nabla^{E^-_0} \gamma^{-1}\Big) \Big|_{D_j} \Big)^{-1} & \text{if } m_j < 0 . \end{array} \right. \end{equation}

Here $\mathrm {Res}_{D_j}(\gamma )$ is independent of $(\nabla ^{E^\pm _k})_{k\in \mathbb {N}}$.

For $j \in \{1,\ldots,l\}$, we have

(1.8)\begin{equation} \mathrm{div}\big(\mathrm{Res}_{D_j}(\gamma)\big) = \sum_{k\in\{1,\ldots,l\}\backslash\{j\}} m_k ( D_j \cap D_k ) . \end{equation}

For distinct $j,k \in \{1,\ldots,l\}$, we have

(1.9)\begin{align} & \mathrm{Res}_{D_j \cap D_k}\big(\mathrm{Res}_{D_j}(\gamma)\big) = \mathrm{Res}_{D_j \cap D_k}\big(\mathrm{Res}_{D_k}(\gamma)\big) \nonumber\\ & \quad \in \mathscr{M}(D_j \cap D_k,L \otimes L_j^{-m_j} \otimes L_k^{-m_k}) . \end{align}

1.2 Some characteristic classes

For an $(m\times m)$-matrix $A$, we define

(1.10)\begin{equation} \mathrm{ch}(A) = \operatorname{Tr}[e^A] ,\quad \mathrm{Td}(A) = \det\bigg(\frac{A}{\operatorname{Id}-e^{-A}}\bigg) ,\quad c(A) = \det(\operatorname{Id}+A) . \end{equation}

We have

(1.11)\begin{equation} c(tA) = 1 + \sum_{k=1}^m t^kc_k(A) , \end{equation}

where $c_k(A)$ is the $k$th elementary symmetric polynomial of the eigenvalues of $A$.

Let $V$ be an $m$-dimensional complex vector space. Let $R\in \mathrm {End}(V)$. Let $V^*$ be the dual of $V$. Let $R^*\in \mathrm {End}(V^*)$ be the dual of $R$. For $r=1,\ldots,m$, we construct $R_r\in \mathrm {End}(\Lambda ^rV^*)$ by induction,

(1.12)\begin{equation} R_1 = -R^* ,\quad R_r = R_1 \wedge \operatorname{Id}_{\Lambda^{r-1}V^*} + \operatorname{Id}_{V^*} \wedge R_{r-1} . \end{equation}

We use the convention $\Lambda ^0V^* = \mathbb{C}$ and $R_0 = 0$.

Let $\lambda _1,\ldots,\lambda _m$ be the eigenvalues of $R$. For $p\in \mathbb {N}$ and $F$ a polynomial of $\lambda _1,\ldots,\lambda _m$, we denote by $\{F\}^{[p]}$ the component of $F$ of degree $p$.

Proposition 1.5 The following identities hold:

(1.13)\begin{equation} \begin{aligned} \mathrm{Td}(R) \bigg( \sum_{r=0}^m (-1)^r \mathrm{ch}(R_r) \bigg) & = c_m(R) ,\\ \bigg\{ \mathrm{Td}(R) \bigg( \sum_{r=1}^m (-1)^r r \mathrm{ch}(R_r) \bigg) \bigg\}^{[\leqslant m]} & = -c_{m-1}(R) + \frac{m}{2}c_m(R) ,\\ \bigg\{ \mathrm{Td}(R) \bigg( \sum_{r=2}^m (-1)^r r(r-1) \mathrm{ch}(R_r) \bigg) \bigg\}^{[m]} & = \frac{1}{6}(c_1c_{m-1})(R) + \frac{m(3m-5)}{12}c_m(R) . \end{aligned} \end{equation}

Proof. Note that the eigenvalues of $R_r$ are given by $((-1)^r \lambda _{j_1}\cdots \lambda _{j_r})_{1 \leqslant j_1 < \cdots < j_r \leqslant m}$, we have

(1.14)\begin{equation} \mathrm{Td}(R) = \prod_{j=1}^m \frac{\lambda_j}{1-e^{-\lambda_j}} ,\quad \sum_{r=0}^m (-1)^r t^r \mathrm{ch}(R_r) = \prod_{j=1}^m (1-te^{-\lambda_j}) . \end{equation}

Taking $t=1$ in (1.14), we obtain the first identity in (1.13).

Taking the derivative of the second identity in (1.14) at $t=1$, we obtain

(1.15)\begin{equation} \sum_{r=0}^m (-1)^r r \mathrm{ch}(R_r) = - \bigg(\sum_{j=1}^m \frac{e^{-\lambda_j}}{1-e^{-\lambda_j}}\bigg) \prod_{j=1}^m (1-e^{-\lambda_j}) . \end{equation}

From the first identity in (1.14), (1.15) and the identity

(1.16)\begin{equation} \frac{e^{-\lambda_j}}{1-e^{-\lambda_j}} = \lambda_j^{-1} - \frac{1}{2} + \frac{1}{12}\lambda_j + \cdots , \end{equation}

we obtain the second identity in (1.13).

Taking the second derivative of the second identity in (1.14) at $t=1$, we obtain

(1.17)\begin{equation} \sum_{r=0}^m (-1)^r r(r-1) \mathrm{ch}(R_r) = \bigg( \bigg(\sum_{j=1}^m \frac{e^{-\lambda_j}}{1-e^{-\lambda_j}}\bigg)^2 - \sum_{j=1}^m \bigg(\frac{e^{-\lambda_j}}{1-e^{-\lambda_j}}\bigg)^2 \bigg) \prod_{j=1}^m (1-e^{-\lambda_j}) . \end{equation}

From the first identity in (1.14), (1.16) and (1.17), we obtain the third identity in (1.13). This completes the proof.

For an $(m\times m)$-matrix $A$, we define

(1.18)\begin{equation} \mathrm{Td}'(A) = \frac{\partial}{\partial t} \mathrm{Td}(A+t\operatorname{Id}) \bigg|_{t=0} . \end{equation}

Proposition 1.6 We have

(1.19)\begin{equation} \begin{aligned} \bigg\{ \mathrm{Td}'(R) \bigg( \sum_{r=0}^m (-1)^r \mathrm{ch}(R_r) \bigg) \bigg\}^{[m]} & = \frac{m}{2}c_m(R) ,\\ \bigg\{ \mathrm{Td}'(R) \bigg( \sum_{r=0}^m (-1)^r r \mathrm{ch}(R_r) \bigg) \bigg\}^{[m]} & = \frac{1}{12}(c_1c_{m-1})(R) + \frac{m^2}{4}c_m(R). \end{aligned} \end{equation}

Proof. Let $c_k'$ be as in (1.18) with $\mathrm {Td}$ replaced by $c_k$. We have

(1.20)\begin{equation} c_1'(R) = m ,\quad c_2'(R) = (m-1)c_1(R) . \end{equation}

On the other hand, we have

(1.21)\begin{equation} \big\{\mathrm{Td}(R)\big\}^{[\leqslant 2]} = 1 + \tfrac{1}{2}c_1(R) + \tfrac{1}{12}\big(c_1^2(R)+c_2(R)\big) . \end{equation}

By (1.20) and (1.21), we have

(1.22)\begin{equation} \bigg\{\frac{\mathrm{Td}'(R)}{\mathrm{Td}(R)}\bigg\}^{[\leqslant 1]} = \frac{m}{2} - \frac{1}{12}c_1(R) . \end{equation}

From (1.13) and (1.22), we obtain (1.19). This completes the proof.

1.3 Chern form and Bott–Chern form

Let $S$ be a compact Kähler manifold. We denote

(1.23)\begin{equation} \begin{aligned} Q^S & = \bigoplus_{p=0}^{\dim S} \Omega^{p,p}(S) ,\\ Q^{S,0} & = \bigoplus_{p=1}^{\dim S} \big(\partial\Omega^{p-1,p}(S)+\bar{\partial}\Omega^{p,p-1}(S)\big) \subseteq Q^S . \end{aligned} \end{equation}

Let $E$ be a holomorphic vector bundle over $S$. Let $g^E$ be a Hermitian metric on $E$. Let $R^E \in \Omega ^{1,1}(S,\mathrm {End}(E))$ be the curvature of the Chern connection on $(E,g^E)$. Recall that $c(\cdot )$ was defined in (1.10). The total Chern form of $(E,g^E)$ is defined by

(1.24)\begin{equation} c(E,g^E) = c\bigg(-\frac{R^E}{2\pi i}\bigg) \in Q^S . \end{equation}

The total Chern class of $E$ is defined by

(1.25)\begin{equation} c(E) = \big[c(E,g^E)\big]\in H^\mathrm{even}_\mathrm{dR}(S) , \end{equation}

which is independent of $g^E$.

Let $E'\subseteq E$ be a holomorphic subbundle. Let $E''=E/E'$. We have a short exact sequence of holomorphic vector bundles over $S$,

(1.26)\begin{equation} 0 \rightarrow E' \xrightarrow{\alpha} E \xrightarrow{\beta} E'' \rightarrow 0 , \end{equation}

where $\alpha$ (respectively, $\beta$) is the canonical embedding (respectively, projection). We have

(1.27)\begin{equation} c(E) = c(E')c(E'') . \end{equation}

Let $g^{E'}$ be a Hermitian metric on $E'$. Let $g^{E''}$ be a Hermitian metric on $E''$. The Bott–Chern form [Reference Bismut, Gillet and SouléBGS88a, § 1f)]

(1.28)\begin{equation} \tilde{c}(g^{E'},g^E,g^{E''})\in Q^S/Q^{S,0} \end{equation}

is such that

(1.29)\begin{align} \frac{\bar{\partial}\partial}{2\pi i} \tilde{c}(g^{E'},g^E,g^{E''}) & = c(E,g^E) - c(E' \oplus E'',g^{E'} \oplus g^{E''}) \nonumber\\ & = c(E,g^E) - c(E',g^{E'})c(E'',g^{E''}) . \end{align}

Let $\alpha ^*g^E$ be the Hermitian metric on $E'$ induced by $g^E$ via the embedding $\alpha : E'\rightarrow E$. Let $\beta _*g^E$ be the quotient Hermitian metric on $E''$ induced by $g^E$ via the surjection $\beta : E\rightarrow E''$. We denote

(1.30)\begin{equation} \tilde{c}(E',E,g^E) = \tilde{c}(\alpha^*g^E,g^E,\beta_*g^E) . \end{equation}

Let $\beta ^*g^{E''}$ be the Hermitian pseudometric on $E$ induced by $g^{E''}$ via the surjection $\beta : E\rightarrow E''$. For $\varepsilon >0$, set

(1.31)\begin{equation} g^E_\varepsilon = g^E + \frac{1}{\varepsilon} \beta^*g^{E''} . \end{equation}

We equip $Q^S \subseteq \Omega ^{\bullet,\bullet }(S)$ with the compact-open topology. We equip $Q^S/Q^{S,0}$ with the quotient topology.

Proposition 1.7 As $\varepsilon \rightarrow 0$,

(1.32)\begin{equation} c(E,g^E_\varepsilon) \rightarrow c(E',\alpha^*g^E)c(E'',g^{E''}) ,\quad \tilde{c}(E',E,g^E_\varepsilon) \rightarrow 0 . \end{equation}

Proof. We follow the proof of [Reference Bismut, Gillet and SouléBGS88a, Theorem 1.29].

Let $\mathrm {pr}: S \times \mathbb{C} \rightarrow S$ be the canonical projection. Let

(1.33)\begin{equation} \tilde{\alpha}: \mathrm{pr}^* E' \rightarrow \mathrm{pr}^* E \end{equation}

be the pull-back of $\alpha : E' \rightarrow E$. Let $(s,z)\in S\times \mathbb{C}$ be coordinates. Let $\sigma \in H^0(S\times \mathbb{C},\mathbb{C} )$ be the holomorphic function $\sigma (s,z) = z$. Let

(1.34)\begin{equation} \tilde{\sigma}: \mathrm{pr}^* E' \rightarrow \mathrm{pr}^* E' \end{equation}

be the multiplication by $\sigma$. Set

(1.35)\begin{equation} \mathcal{E}' = \mathrm{pr}^* E' ,\quad \mathcal{E} = \mathrm{Coker} ( \tilde{\alpha} \oplus \tilde{\sigma}: \mathrm{pr}^* E' \rightarrow \mathrm{pr}^* E \oplus \mathrm{pr}^* E' ) . \end{equation}

We get a short exact sequence of holomorphic vector bundles over $S\times \mathbb{C}$,

(1.36)\begin{equation} 0 \rightarrow \mathcal{E}' \rightarrow \mathcal{E} \rightarrow \mathcal{E}'' \rightarrow 0 , \end{equation}

where $\mathcal {E}' \rightarrow \mathcal {E}$ is induced by the embedding $0 \oplus \operatorname {Id}_{\mathrm {pr}^* E'} : \mathrm {pr}^* E' \hookrightarrow \mathrm {pr}^* E \oplus \mathrm {pr}^* E'$, and $\mathcal {E} \rightarrow \mathcal {E}'' := \mathrm {Coker}(\mathcal {E}'\rightarrow \mathcal {E})$ is the canonical projection. For $z\in \mathbb{C}$, let

(1.37)\begin{equation} 0 \rightarrow \mathcal{E}_z' \rightarrow \mathcal{E}_z \rightarrow \mathcal{E}_z'' \rightarrow 0 \end{equation}

be the restriction of (1.36) to $S\times \{z\}$. For $z\neq 0$, let

(1.38)\begin{equation} \phi_z: E \rightarrow \mathcal{E}_z = \mathrm{Coker} ( \alpha \oplus z \operatorname{Id}_{E'}: E' \rightarrow E \oplus E' ) \end{equation}

be the isomorphism induced by the embedding $\operatorname {Id}_E \oplus 0: E \hookrightarrow E \oplus E'$. We obtain a commutative diagram

(1.39)

where the vertical maps are induced by $\phi _z$. Let

(1.40)\begin{equation} \phi_0: E' \oplus E'' \rightarrow \mathcal{E}_0 = \mathrm{Coker} ( \alpha \oplus 0 : E' \rightarrow E \oplus E' ) = E'' \oplus E' \end{equation}

be the obvious isomorphism. We obtain a commutative diagram

(1.41)

where the vertical maps are induced by $\phi _0$.

We can construct a Hermitian metric $g^\mathcal {E}$ on $\mathcal {E}$ such that

(1.42)\begin{equation} \phi_z^* g^\mathcal{E} = |z|^2 g^E + \beta^* g^{E''} \quad \text{for } z \neq 0 ,\quad \phi_0^* g^\mathcal{E} = \alpha^*g^E \oplus g^{E''} . \end{equation}

To show that $g^\mathcal {E}$ is a smooth metric, we consider the metric $g^{\mathrm {pr}^* E \oplus \mathrm {pr}^* E'}$ on $\mathrm {pr}^* E \oplus \mathrm {pr}^* E'$ defined by

(1.43)\begin{equation} g^{\mathrm{pr}^* E \oplus \mathrm{pr}^* E'}|_{S\times\{z\}} = (1+|z|^2)(g^E \oplus \alpha^*g^E) . \end{equation}

We can directly verify that $g^\mathcal {E}$ is the quotient metric induced by $g^{\mathrm {pr}^* E \oplus \mathrm {pr}^* E'}$ via the canonical projection $\mathrm {pr}^* E \oplus \mathrm {pr}^* E' \rightarrow \mathcal {E}$.

By (1.39) and (1.42), for $\varepsilon = |z|^2 > 0$, we have

(1.44)\begin{equation} c(\mathcal{E}_z,g^{\mathcal{E}_z}) = c(E,g^E_\varepsilon) ,\quad \tilde{c}(\mathcal{E}_z',\mathcal{E}_z,g^{\mathcal{E}_z}) = \tilde{c}(E',E,g^E_\varepsilon) . \end{equation}

By [Reference Bismut, Gillet and SouléBGS88a, Theorem 1.29 iii)], (1.41) and (1.42), we have

(1.45)\begin{equation} c(\mathcal{E}_0,g^{\mathcal{E}_0}) = c(E',\alpha^*g^E)c(E'',g^{E''}) ,\quad \tilde{c}(\mathcal{E}_0',\mathcal{E}_0,g^{\mathcal{E}_0}) = 0 . \end{equation}

On the other hand, by [Reference Bismut, Gillet and SouléBGS88a, Theorem 1.29 ii)], we have

(1.46)\begin{equation} \lim_{z\rightarrow 0} c(\mathcal{E}_z,g^{\mathcal{E}_z}) = c(\mathcal{E}_0,g^{\mathcal{E}_0}) ,\quad \lim_{z\rightarrow 0} \tilde{c}(\mathcal{E}_z',\mathcal{E}_z,g^{\mathcal{E}_z}) = \tilde{c}(\mathcal{E}_0',\mathcal{E}_0,g^{\mathcal{E}_0}) . \end{equation}

From (1.44)–(1.46), we obtain (1.32). This completes the proof.

Let $F \subseteq E$ be a holomorphic subbundle. Set $F' = \alpha ^{-1}(F) \subseteq E'$, $F'' = \beta (F) \subseteq E''$.

Proposition 1.9 If $F'=E'$, as $\varepsilon \rightarrow 0$,

(1.47)\begin{equation} \tilde{c}(F,E,g^E_\varepsilon) \rightarrow c(E',\alpha^*g^E) \tilde{c}(F'',E'',g^{E''}) . \end{equation}

If $F''=E''$, as $\varepsilon \rightarrow 0$,

(1.48)\begin{equation} \tilde{c}(F,E,g^E_\varepsilon) \rightarrow c(E'',g^{E''}) \tilde{c}(F',E',\alpha^*g^E) . \end{equation}

Proof. We use the notation from the proof of Proposition 1.7. Set

(1.49)\begin{equation} \mathcal{F} = \mathrm{Coker} ( \tilde{\alpha} \oplus \tilde{\sigma}|_{\mathrm{pr}^* F'} : \mathrm{pr}^* F' \rightarrow \mathrm{pr}^* F \oplus \mathrm{pr}^* F' ) \subseteq \mathcal{E} . \end{equation}

For $z\in \mathbb{C}$, let $\mathcal {F}_z$ be the restriction of $\mathcal {F}$ to $S\times \{z\}$.

For $z\neq 0$, we have $\phi _z(F) = \mathcal {F}_z \subseteq \mathcal {E}_z$. By (1.42), for $\varepsilon = |z|^2 > 0$, we have

(1.50)\begin{equation} \tilde{c}(\mathcal{F}_z,\mathcal{E}_z,g^{\mathcal{E}_z}\!) = \tilde{c}(F,E,g^E_\varepsilon). \end{equation}

We have $\phi _0(F) = F' \oplus F'' \subseteq E' \oplus E'' = \mathcal {E}_0$. By (1.42), we have

(1.51)\begin{equation} \tilde{c}(\mathcal{F}_0,\mathcal{E}_0,g^{\mathcal{E}_0}) = \tilde{c}(F'\oplus F'',E' \oplus E'',\alpha^*g^E \oplus g^{E''}) . \end{equation}

By [Reference Bismut, Gillet and SouléBGS88a, Theorem 1.29], we have

(1.52)\begin{equation} \begin{aligned} & \tilde{c}(F'\oplus F'',E' \oplus E'',\alpha^*g^E \oplus g^{E''}) = c(E',\alpha^*g^E) \tilde{c}(F'',E'',g^{E''}) \quad \text{if } F' = E' ,\\ & \tilde{c}(F'\oplus F'',E' \oplus E'',\alpha^*g^E \oplus g^{E''}) = c(E'',g^{E''})\tilde{c}(F',E',\alpha^*g^E) \quad \text{if } F'' = E'' . \end{aligned} \end{equation}

On the other hand, by [Reference Bismut, Gillet and SouléBGS88a, Theorem 1.29 ii)], we have

(1.53)\begin{equation} \lim_{z\rightarrow 0} \tilde{c}(\mathcal{F}_z,\mathcal{E}_z,g^{\mathcal{E}_z}) = \tilde{c}(\mathcal{F}_0,\mathcal{E}_0,g^{\mathcal{E}_0}) . \end{equation}

From (1.50)–(1.53), we obtain (1.47) and (1.48). This completes the proof.

Recall that $\mathrm {Td}(\cdot )$ was defined in (1.10). The Bott–Chern form [Reference Bismut, Gillet and SouléBGS88a, § 1f)]

(1.54)\begin{equation} \widetilde{\mathrm{Td}}(g^{E'},g^E,g^{E''}) \in Q^S/Q^{S,0} \end{equation}

is such that

(1.55)\begin{equation} \frac{\bar{\partial}\partial}{2\pi i} \widetilde{\mathrm{Td}}(g^{E'},g^E,g^{E''}) = \mathrm{Td}(E,g^E) - \mathrm{Td}(E',g^{E'})\mathrm{Td}(E'',g^{E''}) . \end{equation}

Recall that $\mathrm {ch}(\cdot )$ was defined in (1.10). The Bott–Chern form [Reference Bismut, Gillet and SouléBGS88a, § 1f)]

(1.56)\begin{equation} \widetilde{\mathrm{ch}}(g^{E'},g^E,g^{E''}) \in Q^S/Q^{S,0} \end{equation}

is such that

(1.57)\begin{equation} \frac{\bar{\partial}\partial}{2\pi i} \widetilde{\mathrm{ch}}(g^{E'},g^E,g^{E''}) = \mathrm{ch}(E',g^{E'}) - \mathrm{ch}(E,g^E) + \mathrm{ch}(E'',g^{E''}) . \end{equation}

For another Hermitian metric $\hat {g}^E$ on $E$, let

(1.58)\begin{equation} \widetilde{\mathrm{ch}}(\hat{g}^E,g^E) \in Q^S/Q^{S,0} \end{equation}

be the Bott–Chern form [Reference Bismut, Gillet and SouléBGS88a, § 1f)] such that

(1.59)\begin{equation} \frac{\bar{\partial}\partial}{2\pi i} \widetilde{\mathrm{ch}}(\hat{g}^E,g^E) = \mathrm{ch}(E,\hat{g}^E) - \mathrm{ch}(E,g^E) . \end{equation}

The following proposition is a direct consequence of the construction of the Bott–Chern form [Reference Bismut, Gillet and SouléBGS88a, § 1f)].

Proposition 1.11 For another Hermitian metric $\hat {g}^E$ (respectively, $\hat {g}^{E'}$, $\hat {g}^{E''}$) on $E$ (respectively, $E'$, $E''$), we have

(1.60)\begin{equation} \widetilde{\mathrm{ch}}(\hat{g}^{E'},\hat{g}^E,\hat{g}^{E''}) = \widetilde{\mathrm{ch}}(g^{E'},g^E,g^{E''}) + \widetilde{\mathrm{ch}}(\hat{g}^{E'},g^{E'}) - \widetilde{\mathrm{ch}}(\hat{g}^E,g^E) + \widetilde{\mathrm{ch}}(\hat{g}^{E''},g^{E''}) . \end{equation}

For $a,b>0$, we have

(1.61)\begin{equation} \widetilde{\mathrm{ch}}(a g^E,bg^E) = \mathrm{ch}(E,g^E) (\log b - \log a) . \end{equation}

For $(g^E_t)_{t\in \mathbb {R}}$ a smooth family of Hermitian metrics on $E$, the map $t \mapsto \widetilde {\mathrm {ch}}(g^E_t,g^E_0)$ is continuous. In particular, we have

(1.62)\begin{equation} \widetilde{\mathrm{ch}}(g^E_t,g^E_0) \rightarrow 0 \quad \text{as } t \rightarrow 0 . \end{equation}

Let $E^*$ be the dual of $E$. Following [Reference Berthomieu and BismutBB94, § 1a)], for $p=0,\ldots,\dim E$ and $s=0,\ldots,p-1$, set

(1.63)\begin{align} I^p_s = \big\{ u \in \Lambda^pE^* : u(v_1,\ldots,v_p) = 0 \text{ for any } v_1,\ldots,v_{s+1} \in E', v_{s+2},\ldots,v_p \in E \big\} . \end{align}

For convenience, we denote $I^p_p = \Lambda ^pE^*$ and $I^p_{-1} = 0$. We obtain a filtration

(1.64)\begin{equation} \Lambda^pE^* = I^p_p \hookleftarrow I^p_{p-1} \hookleftarrow \cdots \hookleftarrow I^p_{-1} = 0 . \end{equation}

For $r=0,\ldots,\dim E''$ and $s=0,\ldots,\dim E'$, we denote $E_{r,s} = \Lambda ^s{E'}^* \otimes \Lambda ^r{E''}^*$. We have a short exact sequence of holomorphic vector bundles over $S$,

(1.65)\begin{equation} 0 \rightarrow I^{r+s}_{s-1} \rightarrow I^{r+s}_s \rightarrow E_{r,s} \rightarrow 0 . \end{equation}

Recall that $g^E_\varepsilon$ was defined in (1.31). Let $g^{\Lambda ^pE^*}_\varepsilon$ be the Hermitian metric on $\Lambda ^pE^*$ induced by $g^E_\varepsilon$. Let $g^{I^{r+s}_s}_\varepsilon$ be the restriction of $g^{\Lambda ^pE^*}_\varepsilon$ to $I^{r+s}_s$. Let $g^{E_{r,s}}_\varepsilon$ be the quotient metric on $E_{r,s}$ induced by $g^{I^{r+s}_s}_\varepsilon$ via the surjection $I^{r+s}_s \rightarrow E_{r,s}$.

Similarly to Proposition 1.7, we have the following proposition.

Proposition 1.12 As $\varepsilon \rightarrow 0$,

(1.66)\begin{equation} \widetilde{\mathrm{ch}}\Big(g^{I^{r+s}_{s-1}}_\varepsilon, g^{I^{r+s}_s}_\varepsilon,g^{E_{r,s}}_\varepsilon\Big) \rightarrow 0 . \end{equation}

Proof. Let $0 \rightarrow \mathcal {E}' \rightarrow \mathcal {E} \rightarrow \mathcal {E}'' \rightarrow 0$ be as in (1.36). Let $\mathcal {I}^p_s \subseteq \Lambda ^p\mathcal {E}^*$ be as in (1.63) with $E$ replaced by $\mathcal {E}$ and $E'$ replaced by $\mathcal {E}'$. We denote $\mathcal {E}_{r,s} = \Lambda ^s{\mathcal {E}'}^* \otimes \Lambda ^r{\mathcal {E}''}^*$. We have a short exact sequence of holomorphic vector bundles over $S\times \mathbb{C}$,

(1.67)\begin{equation} 0 \rightarrow \mathcal{I}^{r+s}_{s-1} \rightarrow \mathcal{I}^{r+s}_s \rightarrow \mathcal{E}_{r,s} \rightarrow 0 . \end{equation}

Proceeding in the same way as in the proof of Proposition 1.7 with (1.36) replaced by (1.67), we obtain (1.66). This completes the proof.

1.4 Quillen metric

Let $X$ be an $n$-dimensional compact Kähler manifold. Let $E$ be a holomorphic vector bundle over $X$. Let $\bar {\partial }^E$ be the Dolbeault operator on

(1.68)\begin{equation} \Omega^{0,\bullet}(X,E) = {\mathscr{C}^\infty}\big(X,\Lambda^\bullet(\overline{T^*X})\otimes E\big) . \end{equation}

For $q=0,\ldots,n$, we have $H^q(X,E) = H^q(\Omega ^{0,\bullet }(X,E),\bar {\partial }^E)$. Set

(1.69)\begin{equation} \lambda(E) = \det H^\bullet(X,E) := \bigotimes_{q=0}^n \big(\!\det H^q(X,E)\big)^{(-1)^q} . \end{equation}

Let $g^{TX}$ be a Kähler metric on $TX$. Let $g^E$ be a Hermitian metric on $E$. Let $\langle \cdot,\cdot \rangle _{\Lambda ^\bullet (\overline {T^*X})\otimes E}$ be the Hermitian product on $\Lambda ^\bullet (\overline {T^*X})\otimes E$ induced by $g^{TX}$ and $g^E$. Let $dv_X$ be the Riemannian volume form on $X$ induced by $g^{TX}$. For $s_1,s_2\in \Omega ^{0,\bullet }(X,E)$, set

(1.70)\begin{equation} \langle s_1,s_2 \rangle = (2\pi)^{-n} \int_X \langle s_1,s_2 \rangle_{\Lambda^\bullet(\overline{T^*X})\otimes E}\, dv_X , \end{equation}

which we call the $L^2$-product.

Let $\bar {\partial }^{E,*}$ be the formal adjoint of $\bar {\partial }^E$ with respect to the Hermitian product (1.70). The Kodaira Laplacian on $\Omega ^{0,\bullet }(X,E)$ is defined by

(1.71)\begin{equation} \square^E = \bar{\partial}^E\bar{\partial}^{E,*} + \bar{\partial}^{E,*}\bar{\partial}^E . \end{equation}

Let $\square ^E_q$ be the restriction of $\square ^E$ to $\Omega ^{0,q}(X,E)$.

By the Hodge theorem, we have

(1.72)\begin{equation} \mathrm{Ker}(\square^E_q) = \{s\in\Omega^{0,q}(X,E): \bar{\partial}^Es=0,\bar{\partial}^{E,*}s=0\} . \end{equation}

Still by the Hodge theorem, the following map is bijective:

(1.73)\begin{equation} \begin{aligned} \mathrm{Ker}(\square^E_q) & \rightarrow H^q(X,E) \\ s & \mapsto [s] . \end{aligned} \end{equation}

Let $|\cdot |_{\lambda (E)}$ be the $L^2$-metric on $\lambda (E)$ induced by the metric (1.70) via (1.69) and (1.73).

Let $\mathrm {Sp}(\square ^E_q)$ be the spectrum of $\square ^E_q$, which is a multiset.Footnote ¹ For $z\in \mathbb{C}$ with $\mathrm {Re}(z)>n$, set

(1.74)\begin{equation} \theta(z) = \sum_{q=1}^n (-1)^{q+1}q \sum_{\lambda\in\mathrm{Sp}(\square^E_q),\lambda\neq 0} \lambda^{-z} . \end{equation}

By [Reference SeeleySee67], the function $\theta (z)$ extends to a meromorphic function of $z\in \mathbb{C}$, which is holomorphic at $z=0$.

The following definition is due to Quillen [Reference QuillenQui85] and Bismut, Gillet and Soulé [Reference Bismut, Gillet and SouléBGS88b, § 1d)].

Definition 1.13 The Quillen metric on $\lambda (E)$ is defined by

(1.75)\begin{equation} \lVert\cdot\rVert_{\lambda(E)} = \exp\big(\tfrac{1}{2}\theta'(0)\big) |\cdot|_{\lambda(E)} . \end{equation}

1.5 Analytic torsion form

Let $\pi : X \rightarrow Y$ be a holomorphic submersion between Kähler manifolds with compact fiber $Z$.

Let $E$ be a holomorphic vector bundle over $X$. Let $R^\bullet \pi _*E$ be the derived direct image of $E$, which is a graded analytic coherent sheaf on $Y$. We assume that $R^\bullet \pi _*E$ is a graded holomorphic vector bundle. Let $H^\bullet (Z,E)$ be the fiberwise cohomology. More precisely, its fiber at $y\in Y$ is given by $H^\bullet (Z_y,E|_{Z_y})$. We have a canonical identification $R^\bullet \pi _*E = H^\bullet (Z,E)$. We have the Grothendieck–Riemann–Roch formula,

(1.76)\begin{equation} \mathrm{ch}(H^\bullet(Z,E)) := \sum_j (-1)^j \mathrm{ch}(H^j(Z,E)) = \int_Z \mathrm{Td}(TZ) \mathrm{ch}(E) \in H^\mathrm{even}_\mathrm{dR}(Y) .\end{equation}

Let $\omega \in \Omega ^{1,1}(X)$ be a Kähler form. Let $g^{TZ}$ be the Hermitian metric on $TZ$ associated with $\omega$. Let $g^E$ be a Hermitian metric on $E$. Let $g^{H^\bullet (Z,E)}$ be the $L^2$-metric on $H^\bullet (Z,E)$ associated with $g^{TZ}$ and $g^E$ via (1.73).

We use the notation in (1.23). Let $\mathrm {ch}(H^\bullet (Z,E),g^{H^\bullet (Z,E)}) \in Q^Y$ be the Chern character form of $(H^\bullet (Z,E),g^{H^\bullet (Z,E)})$. We introduce $\mathrm {Td}(TZ,g^{TZ}) \in Q^X$ and $\mathrm {ch}(E,g^E) \in Q^X$ in the same way.

Bismut and Köhler [Reference Bismut and KöhlerBK92, Definition 3.8] defined the analytic torsion forms. The analytic torsion form associated with $(\pi : X \rightarrow Y, \omega, E, g^E)$ is a differential form on $Y$, which we denote by $T(\omega,g^E)$. Moreover, we have

(1.77)\begin{equation} T(\omega,g^E) \in Q^Y . \end{equation}

We sometimes view $T(\omega,g^E)$ as an element in $Q^Y/Q^{Y,0}$. By [Reference Bismut and KöhlerBK92, Theorem 3.9], we have

(1.78)\begin{equation} \frac{\bar{\partial}\partial}{2\pi i} T(\omega,g^E) = \mathrm{ch}\big(H^\bullet(Z,E),g^{H^\bullet(Z,E)}\big) - \int_Z \mathrm{Td}(TZ,g^{TZ}) \mathrm{ch}(E,g^E) . \end{equation}

The identity (1.78) is a refinement of the Grothendieck–Riemann–Roch formula (1.76).

For $y\in Y$, let $\theta _y(z)$ be as in (1.74) with $(X,g^{TX},E,g^E)$ replaced by $(Z_y,g^{TZ_y},E|_{Z_y},g^E|_{Z_y})$. Let $\theta '(0)$ be the function $y \mapsto \theta _y'(0)$ on $Y$. By the construction of the analytic torsion forms, we have

(1.79)\begin{equation} \big\{T(\omega,g^E)\big\}^{(0,0)} = \theta'(0) \in {\mathscr{C}^\infty}(Y), \end{equation}

where $\{\cdot \}^{(0,0)}$ means the component of degree $(0,0)$.

Let $F$ be a holomorphic vector bundle over $Y$. Let $\pi ^*F$ be its pull-back via $\pi$, which is a holomorphic vector bundle over $X$. Let $g^F$ be a Hermitian metric on $F$. Let $g^{E\otimes \pi ^*F}$ be the Hermitian metric on $E\otimes \pi ^*F$ induced by $g^E$ and $g^F$. Let

(1.80)\begin{equation} T(\omega,g^{E\otimes\pi^*F}) \in Q^Y \end{equation}

be the analytic torsion form associated with $(\pi : X \rightarrow Y, \omega, E\otimes \pi ^*F, g^{E\otimes \pi ^*F})$.

The following proposition is a direct consequence of the construction of the analytic torsion forms.

Proposition 1.15 The following identity holds in $Q^Y/Q^{Y,0}$:

(1.81)\begin{equation} T(\omega,g^{E\otimes\pi^*F}) = \mathrm{ch}(F,g^F) T(\omega,g^E) . \end{equation}

For $p=0,\ldots,\dim Z$, let $g^{\Lambda ^p(T^*Z)}$ be the metric on $\Lambda ^p(T^*Z)$ induced by $g^{TZ}$. Let

(1.82)\begin{equation} T(\omega,g^{\Lambda^p(T^*Z)}) \in Q^Y \end{equation}

be the analytic torsion form associated with $(\pi : X \rightarrow Y, \omega, \Lambda ^p(T^*Z), g^{\Lambda ^p(T^*Z)})$.

Theorem 1.16 (Bismut [Reference BismutBis04, Theorem 4.15])

The following identity holds in $Q^Y/Q^{Y,0}$,

(1.83)\begin{equation} \sum_{p=0}^{\dim Z} (-1)^p T(\omega,g^{\Lambda^p(T^*Z)}) = 0 . \end{equation}

1.6 Properties of the Quillen metric

In this subsection, we state several results describing the behavior of the Quillen metric under submersion, resolution, immersion and blow-up.

Submersion. Let $\pi : X \rightarrow Y$, $Z$, $E$ and $H^\bullet (Z,E)$ be as in § 1.5. We assume that $X$ and $Y$ are compact. We further assume that the Leray spectral sequence for $E$ and $\pi$ degenerates at $E_2$, i.e.

(1.84)\begin{equation} H^q(X,E) \simeq \bigoplus_{j+k = q} H^j(Y,H^k(Z,E)) \quad \text{for } q = 0,\ldots,\dim X . \end{equation}

We denote

(1.85) \begin{align} \det H^\bullet(Y,H^\bullet(Z,E)) & = \bigotimes_{k=0}^{\dim Z} \big(\!\det H^\bullet\big(Y,H^k(Z,E)\big)\big)^{(-1)^k} \nonumber\\ & = \bigotimes_{j=0}^{\dim Y} \bigotimes_{k=0}^{\dim Z} \big(\!\det H^j\big(Y,H^k(Z,E)\big)\big)^{(-1)^{j+k}} . \end{align}

Let

(1.86)\begin{equation} \sigma \in \det H^\bullet(X,E) \otimes \big(\!\det H^\bullet\big(Y,H^\bullet(Z,E)\big)\big)^{-1} \end{equation}

be the canonical section induced by (1.84).

Let $\omega _X\in \Omega ^{1,1}(X)$ and $\omega _Y\in \Omega ^{1,1}(Y)$ be Kähler forms. For $\varepsilon >0$, set

(1.87)\begin{equation} \omega_\varepsilon = \omega_X + \frac{1}{\varepsilon}\pi^*\omega_Y . \end{equation}

Let $g^E$ be a Hermitian metric on $E$.

Let $g^{TX}_\varepsilon$ be the metric on $TX$ associated with $\omega _\varepsilon$. Let

(1.88)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X,E),\varepsilon} \end{equation}

be the Quillen metric on $\det H^\bullet (X,E)$ associated with $g^{TX}_\varepsilon$ and $g^E$. Let $g^{TY}$ be the metric on $TY$ associated with $\omega _Y$. Let $g^{TZ}$ be the metric on $TZ$ associated with $\omega _X|_Z$. Let $g^{H^\bullet (Z,E)}$ be the $L^2$-metric on $H^\bullet (Z,E)$ associated with $g^{TZ}$ and $g^E$. For $k=0,\ldots,\dim Z$, let

(1.89)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(Y,H^k(Z,E))} \end{equation}

be the Quillen metric on $\det H^\bullet (Y,H^k(Z,E))$ associated with $g^{TY}$ and $g^{H^k(Z,E)}$. Let

(1.90)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(Y,H^\bullet(Z,E))} \end{equation}

be the metric on $\det H^\bullet (Y,H^\bullet (Z,E))$ induced by the Quillen metrics (1.89) via (1.85). Let $\lVert \sigma \rVert _\varepsilon$ be the norm of $\sigma$ with respect to the metrics (1.88) and (1.90).

We use the notation in (1.23). Let $\mathrm {Td}(TY,g^{TY}) \in Q^Y$ be the Todd form of $(TY,g^{TY})$. Let

(1.91)\begin{equation} T(\omega,g^E) \in Q^Y \end{equation}

be the analytic torsion form (see § 1.5) associated with $(\pi : X \rightarrow Y, \omega _X, E, g^E)$.

Recall that $\mathrm {Td}'(\cdot )$ was defined by (1.18).

Theorem 1.17 (Berthomieu and Bismut [Reference Berthomieu and BismutBB94, Theorem 3.2])

As $\varepsilon \rightarrow 0$,

(1.92)\begin{equation} \log \lVert\sigma\rVert^2_\varepsilon + \int_Y \mathrm{Td}'(TY) \int_Z \mathrm{Td}(TZ)\mathrm{ch}(E) \log \varepsilon \rightarrow \int_Y \mathrm{Td}(TY,g^{TY}) T(\omega,g^E) . \end{equation}

Resolution. Let $X$ be a compact Kähler manifold. Let

(1.93)\begin{equation} 0 \rightarrow E^0 \rightarrow E^1 \rightarrow E^2 \rightarrow 0 \end{equation}

be a short exact sequence of holomorphic vector bundles over $X$. Let

(1.94)\begin{equation} \sigma \in \bigotimes_{k=0}^2 \big(\!\det H^\bullet(X,E^k)\big)^{(-1)^{k+1}} \end{equation}

be the canonical section induced by the long exact sequence induced by (1.93).

Let $g^{TX}$ be a Kähler metric on $TX$. For $k=0,1,2$, let $g^{E^k}$ be a Hermitian metric on $E^k$. Let

(1.95)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X,E^k)} \end{equation}

be the Quillen metric on $\det H^\bullet (X,E^k)$ associated with $g^{TX}$ and $g^{E^k}$. Let $\lVert \sigma \rVert$ be the norm of $\sigma$ with respect to the metrics (1.95).

We use the notation in (1.23). Let $\mathrm {Td}(TX,g^{TX}) \in Q^X$ be the Todd form of $(TX,g^{TX})$. Let $\mathrm {ch}(E^k,g^{E^k}) \in Q^X$ be the Chern character form of $(E^k,g^{E^k})$. Let

(1.96)\begin{equation} \widetilde{\mathrm{ch}}(g^{E^\bullet}) \in Q^X/Q^{X,0} \end{equation}

be the Bott–Chern form [Reference Bismut, Gillet and SouléBGS88a, § 1f)] such that

(1.97)\begin{equation} \frac{\bar{\partial}\partial}{2\pi i} \widetilde{\mathrm{ch}}(g^{E^\bullet}) = \sum_{k=0}^2 (-1)^k \mathrm{ch}(E^k,g^{E^k}) . \end{equation}

Theorem 1.18 (Bismut, Gillet and Soulé [Reference Bismut, Gillet and SouléBGS88b, Theorem 1.23])

The following identity holds:

(1.98)\begin{equation} \log \lVert\sigma\rVert^2 = \int_X \mathrm{Td}(TX,g^{TX}) \widetilde{\mathrm{ch}}(g^{E^\bullet}). \end{equation}

Immersion. Let $X$ be a compact Kähler manifold. Let $Y \subseteq X$ be a complex submanifold of codimension one. Let $i: Y \hookrightarrow X$ be the canonical embedding. Let $F$ be a holomorphic vector bundle over $Y$. Let $v: E_1 \rightarrow E_0$ be a map between holomorphic vector bundles over $X$ which, together with a restriction map $r: E_0|_Y \rightarrow F$, provides a resolution of $i_*\mathscr {O}_Y(F)$. More precisely, we have an exact sequence of analytic coherent sheaves on $X$,

(1.99)\begin{equation} 0 \rightarrow \mathscr{O}_X(E_1) \xrightarrow{v} \mathscr{O}_X(E_0) \xrightarrow{r} i_*\mathscr{O}_Y(F) \rightarrow 0 . \end{equation}

Let

(1.100)\begin{equation} \sigma \in \big(\!\det H^\bullet(X,E_1)\big)^{-1} \otimes \det H^\bullet(X,E_0) \otimes \big(\!\det H^\bullet(Y,F)\big)^{-1} \end{equation}

be the canonical section induced by the long exact sequence induced by (1.99).

Let $\omega \in \Omega ^{1,1}(X)$ be a Kähler form. For $k=0,1$, let $g^{E_k}$ be a Hermitian metric on $E_k$. Let $g^F$ be a Hermitian metric on $F$. Assume that there is an open neighborhood $Y \subseteq U \subseteq X$ such that $v|_{X\backslash U}$ is isometric, i.e.

(1.101)\begin{equation} g^{E_1}|_{X\backslash U} = v^*g^{E_0}|_{X\backslash U} . \end{equation}

Let $g^{TX}$ be the metric on $TX$ associated with $\omega$. For $k=0,1$, let

(1.102)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X,E_k)} \end{equation}

be the Quillen metric on $\det H^\bullet (X,E_k)$ associated with $g^{TX}$ and $g^{E_k}$. Let $g^{TY}$ be the metric on $TY$ associated with $\omega |_Y$. Let

(1.103)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(Y,F)} \end{equation}

be the Quillen metric on $\det H^\bullet (Y,F)$ associated with $g^{TY}$ and $g^F$. Let $\lVert \sigma \rVert$ be the norm of $\sigma$ with respect to the metrics (1.102) and (1.103).

The following theorem is a direct consequence of the immersion formula due to Bismut and Lebeau [Reference Bismut and LebeauBL91, Theorem 0.1] and the anomaly formula due to Bismut, Gillet and Soulé [Reference Bismut, Gillet and SouléBGS88b, Theorem 1.23].

Theorem 1.19 We have

(1.104)\begin{equation} \log \lVert\sigma\rVert^2 = \alpha(U,\omega|_U,v|_U,g^{E_\bullet}|_U,r,g^F), \end{equation}

where $\alpha (U,\omega |_U,v|_U,r|_U,g^{E_\bullet },g^F)$ is a real number determined by

(1.105)\begin{equation} U ,\quad \omega|_U ,\quad v|_U: E_1|_U \rightarrow E_0|_U ,\quad g^{E_\bullet}|_U ,\quad r: E_0|_Y \rightarrow F ,\quad g^F . \end{equation}

More precisely, given

(1.106)\begin{equation} \tilde{Y} \subseteq \tilde{U} \subseteq \tilde{X} , \quad \tilde{\omega} ,\quad \tilde{v}: \tilde{E}_1 \rightarrow \tilde{E}_0 ,\quad \tilde{r}: \tilde{E}_0 |_{\tilde{Y}} \rightarrow \tilde{F} ,\quad g^{\tilde{E}_\bullet} ,\quad g^{\tilde{F}} \end{equation}

satisfying the same properties that

(1.107)\begin{equation} Y \subseteq U \subseteq X ,\quad \omega ,\quad v: E_1 \rightarrow E_0 ,\quad r: E_0 |_Y \rightarrow F ,\quad g^{E_\bullet} ,\quad g^F \end{equation}

satisfy, if there is a biholomorphic map $U \rightarrow \tilde {U}$ inducing an isomorphism between the restrictions of the data above to $U$ and $\tilde {U}$, then

(1.108)\begin{equation} \log \lVert\sigma\rVert^2 = \log \lVert\tilde{\sigma}\rVert^2 , \end{equation}

where

(1.109)\begin{equation} \tilde{\sigma} \in \big(\!\det H^\bullet(\tilde{X},\tilde{E}_1)\big)^{-1} \otimes \det H^\bullet(\tilde{X},\tilde{E}_0) \otimes \big(\!\det H^\bullet(\tilde{Y},\tilde{F})\big)^{-1} \end{equation}

is the canonical section, and $\lVert \tilde {\sigma }\rVert$ is its norm with respect to the Quillen metrics.

Blow-up. Let $X$ be a compact Kähler manifold. Let $Y \subseteq X$ be a complex submanifold of codimension $r \geqslant 2$. Let $f: X' \rightarrow X$ be the blow-up along $Y$. Let $E$ be a holomorphic vector bundle over $X$. Let $f^*E$ be the pull-back of $E$ via $f$, which is a holomorphic vector bundle over $X'$. Applying spectral sequence, we obtain a canonical identification

(1.110)\begin{equation} H^\bullet(X',f^*E) = H^\bullet(X,E) . \end{equation}

Let

(1.111)\begin{equation} \sigma \in \big(\!\det H^\bullet(X,E)\big)^{-1} \otimes \det H^\bullet(X',f^*E) \end{equation}

be the canonical section induced by (1.110).

Let $\omega \in \Omega ^{1,1}(X)$ and $\omega '\in \Omega ^{1,1}(X')$ be Kähler forms. Assume that there are open neighborhoods $Y \subseteq U \subseteq X$ and $f^{-1}(Y) \subseteq U' \subseteq X'$ such that

(1.112)\begin{equation} f^{-1}(U) = U' ,\quad f^*( \omega|_{X\backslash U} ) = \omega'|_{X'\backslash U'} . \end{equation}

For the existence of such $\omega$ and $\omega '$, see the proof of [Reference VoisinVoi02, Proposition 3.24]. Let $g^E$ be a Hermitian metric on $E$.

Let $g^{TX}$ be the metric on $TX$ associated with $\omega$. Let

(1.113)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X,E)} \end{equation}

be the Quillen metric on $\det H^\bullet (X,E)$ associated with $g^{TX}$ and $g^E$. Let $g^{TX'}$ be the metric on $TX'$ associated with $\omega '$. Let

(1.114)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X',f^*E)} \end{equation}

be the Quillen metric on $\det H^\bullet (X',f^*E)$ associated with $g^{TX'}$ and $f^*g^E$. Let $\lVert \sigma \rVert$ be the norm of $\sigma$ with respect to the metrics (1.113) and (1.114).

The following theorem is a direct consequence of the blow-up formula due to Bismut [Reference BismutBis97, Theorem 8.10].

Theorem 1.21 We have

(1.115)\begin{equation} \log \lVert\sigma\rVert^2 = \alpha(U,\omega|_U,U',\omega'|_{U'},E|_U,g^E|_U) , \end{equation}

where $\alpha (U,\omega |_U,U',\omega '|_{U'},E|_U,g^E|_U)$ is a real number determined by

(1.116)\begin{equation} U ,\quad \omega|_U ,\quad U' ,\quad \omega'|_{U'} ,\quad E|_U ,\quad g^E|_U . \end{equation}

1.7 Topological torsion and BCOV torsion

Let $X$ be an $n$-dimensional compact Kähler manifold. For $p=0,\ldots,n$, set

(1.117)\begin{equation} \lambda_p(X) = \det H^{p,\bullet}(X) := \bigotimes_{q=0}^n\, (\det H^{p,q}(X))^{(-1)^q} . \end{equation}

Set

(1.118)\begin{align} \eta(X) = \det H^\bullet_\mathrm{dR}(X) & := \bigotimes_{k=0}^{2n} \big(\!\det H^k_\mathrm{dR}(X)\big)^{(-1)^k} \nonumber\\ & = \bigotimes_{p=0}^n \big(\lambda_p(X)\big)^{(-1)^p} . \end{align}

Set

(1.119)\begin{equation} \begin{aligned} \lambda(X) & = \bigotimes_{0\leqslant p,q\leqslant n} \big(\!\det H^{p,q}(X) \big)^{(-1)^{p+q}p} = \bigotimes_{p=1}^n \big(\lambda_p(X)\big)^{(-1)^pp} ,\\ \lambda_\mathrm{tot}(X) & = \bigotimes_{k=1}^{2n} \big(\!\det H^k_\mathrm{dR}(X)\big)^{(-1)^kk} = \lambda(X) \otimes \overline{\lambda(X)} . \end{aligned} \end{equation}

The identities in (1.119) appeared in [Reference KatoKat14]. They were applied to the theory of BCOV invariant by Eriksson, Freixas i Montplet and Mourougane [Reference Eriksson, Freixas i Montplet and MourouganeEFM21].

For $\mathbb {A} = \mathbb {Z},\mathbb {R},\mathbb{C}$, we denote by $H^\bullet _\mathrm {Sing}(X,\mathbb {A})$ the singular cohomology of $X$ with coefficients in $\mathbb {A}$. For $k=0,\ldots,2n$, let

(1.120)\begin{equation} \sigma_{k,1},\ldots,\sigma_{k,b_k} \in \mathrm{Im}\big(H^k_\mathrm{Sing}(X,\mathbb{Z}) \rightarrow H^k_\mathrm{Sing}(X,\mathbb{R})\big) \end{equation}

be a basis of the lattice. We fix a square root of $i$. In what follows, the choice of square root is irrelevant. We identify $H^k_\mathrm {dR}(X)$ with $H^k_\mathrm {Sing}(X,\mathbb{C} )$ as follows:

(1.121)\begin{equation} \begin{aligned} H^k_\mathrm{dR}(X) & \rightarrow H^k_\mathrm{Sing}(X,\mathbb{C}) \\ [\alpha] & \mapsto \bigg[\mathfrak{a} \mapsto (2\pi i)^{-k/2} \int_\mathfrak{a}\alpha\bigg], \end{aligned} \end{equation}

where $\alpha$ is a closed $k$-form on $X$ and $\mathfrak {a}$ is a $k$-chain in $X$. Then $\sigma _{k,1},\ldots,\sigma _{k,b_k}$ form a basis of $H^k_\mathrm {dR}(X)$. Set

(1.122)\begin{equation} \begin{aligned} & \sigma_k = \sigma_{k,1}\wedge\cdots\wedge\sigma_{k,b_k} \in \det H^k_\mathrm{dR}(X) ,\\ & \epsilon_X = \bigotimes_{k=0}^{2n} \sigma_k^{(-1)^k} \in \eta(X) ,\quad \sigma_X = \bigotimes_{k=1}^{2n} \sigma_k^{(-1)^kk} \in \lambda_\mathrm{tot}(X) , \end{aligned} \end{equation}

which are well-defined up to $\pm 1$.

Let $\omega$ be a Kähler form on $X$. Let $\lVert \cdot \rVert _{\lambda _p(X),\omega }$ be the Quillen metric on $\lambda _p(X)$ associated with $\omega$. Let $\lVert \cdot \rVert _{\eta (X)}$ be the metric on $\eta (X)$ induced by $\lVert \cdot \rVert _{\lambda _p(X),\omega }$ via (1.118). The same calculation as in [Reference ZhangZha22, Theorem 2.1] together with the first identity in Proposition 1.5 shows that $\lVert \cdot \rVert _{\eta (X)}$ is independent of $\omega$.

Definition 1.23 We define

(1.123)\begin{equation} \tau_\mathrm{top}(X) = \log \lVert\epsilon_X\rVert_{\eta(X)} . \end{equation}

Indeed $\lVert \cdot \rVert _{\eta (X)}$ is the classical Ray–Singer metric up to a normalization. Later, we use this fact to show that $\tau _\mathrm {top}(X) = 0$.

Let $\lVert \cdot \rVert _{\lambda (X),\omega }$ be the metric on $\lambda (X)$ induced by $\lVert \cdot \rVert _{\lambda _p(X),\omega }$ via the first identity in (1.119). Let $\lVert \cdot \rVert _{\lambda _\mathrm {tot}(X),\omega }$ be the metric on $\lambda _\mathrm {tot}(X)$ induced by $\lVert \cdot \rVert _{\lambda (X),\omega }$ via the second identity in (1.119).

Definition 1.24 We define

(1.124)\begin{equation} \tau_\mathrm{BCOV}(X,\omega) = \log \lVert\sigma_X\rVert_{\lambda_\mathrm{tot}(X),\omega} . \end{equation}

For $p=0,\ldots,n$, let $g^{\Lambda ^p(T^*X)}_\omega$ be the metric on $\Lambda ^p(T^*X)$ induced by $\omega$. Let $g^{\Omega ^{p,q}(X)}_\omega$ be the $L^2$-metric on $\Omega ^{p,q}(X)$. More precisely, $g^{\Omega ^{p,q}(X)}_\omega$ is defined by (1.70) with $(E,g^E)$ replaced by $(\Lambda ^p(T^*X),g^{\Lambda ^p(T^*X)}_\omega )$. Let $g^{H^{p,q}(X)}_\omega$ be the $L^2$-metric on $H^{p,q}(X)$. More precisely, $g^{H^{p,q}(X)}_\omega$ is induced by $g^{\Omega ^{p,q}(X)}_\omega$ via the Hodge theorem. Let $|\cdot |_{\eta (X),\omega }$ be the metric on $\eta (X)$ induced by $(g^{H^{p,q}(X)}_\omega )_{0\leqslant p,q\leqslant n}$ via (1.117) and (1.118).

Proposition 1.25 The following identity holds,

(1.125)\begin{equation} \tau_\mathrm{top}(X) = \log |\epsilon_X|_{\eta(X),\omega} = 0 . \end{equation}

Proof. Let $\square _p$ be as in (1.71) with $( \Omega ^{0,\bullet }(X,E), \bar {\partial }^E, g^E )$ replaced by $( \Omega ^{p,\bullet }(X), \bar {\partial }, g^{\Lambda ^p(T^*X)}_\omega )$. Let $\square _{p,q}$ be the restriction of $\square _p$ to $\Omega ^{p,q}(X)$. Let $\theta _p(z)$ be as in (1.74) with $\square ^E_q$ replaced by $\square _{p,q}$. By Definition 1.13, 1.23, the first equality in (1.125) is equivalent to

(1.126)\begin{equation} \sum_{p=0}^n (-1)^p \theta_p'(0) = 0 , \end{equation}

which was indicated in [Reference BismutBis04, p. 1304].

Denote by $\mathrm {covol}(H^k_\mathrm {Sing}(X,\mathbb {Z}),\omega )$ the covolume of $\mathrm {Im}(H^k_\mathrm {Sing}(X,\mathbb {Z}) \rightarrow H^k_\mathrm {Sing}(X,\mathbb {R}))$ with respect to the metric induced by $\bigoplus _{p+q=k} g^{H^{p,q}(X)}_\omega$ via (1.121). We have

(1.127)\begin{equation} |\epsilon_X|_{\eta(X),\omega} = \prod_{k=0}^{2n} \big( \mathrm{covol}(H^k_\mathrm{Sing}(X,\mathbb{Z}),\omega) \big)^{(-1)^k} . \end{equation}

On the other hand, by [Reference Eriksson, Freixas i Montplet and MourouganeEFM21, Remark 5.5(ii)], we have

(1.128)\begin{equation} \mathrm{covol}\big(H^k_\mathrm{Sing}(X,\mathbb{Z}),\omega\big) \mathrm{covol}\big(H^{2n-k}_\mathrm{Sing}(X,\mathbb{Z}),\omega\big) = 1 . \end{equation}

Here we remark that, due to the normalization in (1.70) and (1.121), the covolume in the sense of [Reference Eriksson, Freixas i Montplet and MourouganeEFM21, Remark 5.5(ii)] equals $(2\pi )^{(n-k)b_k/2}\mathrm {covol}(H^k_\mathrm {Sing}(X,\mathbb {Z}),\omega )$, where $b_k$ is the $k$th Betti number of $X$. From (1.127) and (1.128), we obtain $|\epsilon _X|_{\eta (X),\omega } = 1$, which is equivalent to the second equality in (1.125). This completes the proof.

2.1 Kähler metric on projective bundle

For a complex vector space $V$, we denote by $\mathbb {P}(V)$ the set of complex lines in $V$. Then $\mathbb {P}(V)$ is complex manifold.

Let $Y$ be an $m$-dimensional compact Kähler manifold. Let $N$ be a holomorphic vector bundle over $Y$ of rank $n$. Let $\mathbb {1}$ be the trivial line bundle over $Y$. Set

(2.1)\begin{equation} X = \mathbb{P}(N\oplus\mathbb{1}) . \end{equation}

Let $\pi : X \rightarrow Y$ be the canonical projection. For $y\in Y$, we denote $Z_y = \pi ^{-1}(y)$, which is isomorphic to $\mathbb{C} \mathrm {P}^n$. Let $\omega _{\mathbb{C} \mathrm {P}^n}$ be the Kähler form on $\mathbb{C} \mathrm {P}^n$ associated with the Fubini–Study metric. More precisely, $-i\omega _{\mathbb{C} \mathrm {P}^n}$ is equal to the curvature of the tautological line bundle over $\mathbb{C} \mathrm {P}^n$ equipped with the standard metric.

Here $(\phi _y)_{y\in Y}$ is merely a set of maps parameterized by $y\in Y$. It is not even required to depend continuously on $y$.

Let $s\in \{1,\ldots,n\}$. We assume that there are holomorphic line bundles $L_1,\ldots,L_s$ over $Y$ together with a surjection between holomorphic vector bundles,

(2.2)\begin{equation} N \rightarrow L_1 \oplus \cdots \oplus L_s . \end{equation}

For $k=1,\ldots,s$, let $N \rightarrow L_k$ be the composition of (2.2) and the canonical projection $L_1 \oplus \cdots \oplus L_s \rightarrow L_k$. Set

(2.3)\begin{equation} N_k = \mathrm{Ker}(N\rightarrow L_k) \subseteq N ,\quad X_k = \mathbb{P}(N_k\oplus\mathbb{1}) \subseteq X ,\quad X_0 = \mathbb{P}(N) \subseteq X . \end{equation}

Let $[\xi _0:\cdots :\xi _n]$ be homogenous coordinates on $\mathbb{C} \mathrm {P}^n$. For $k=0,\ldots,n$, we denote ${H_k = \{\xi _k = 0 \} \subseteq \mathbb{C} \mathrm {P}^n}$.

2.2 Behavior under adiabatic limit

We use the notation in § 2.1. By Lemma 2.1, there exists a Kähler form $\omega _X$ on $X$ such that for any $y\in Y$, there exists an isomorphism $\phi _y: \mathbb{C} \mathrm {P}^n\rightarrow Z_y$ such that

(2.4)\begin{equation} \phi_y^*(\omega_X|_{Z_y}) = \omega_{\mathbb{C}\mathrm{P}^n} . \end{equation}

Let $\omega _{Z_y} = \omega _X|_{Z_y}$. Note that $(Z_y,\omega _{Z_y})_{y\in Y}$ are mutually isometric, we omit the index $y$ as long as there is no confusion. Let $\omega _Y$ be a Kähler form on $Y$. For $\varepsilon >0$, set

(2.5)\begin{equation} \omega_\varepsilon = \omega_X + \frac{1}{\varepsilon} \pi^*\omega_Y . \end{equation}

We denote

(2.6)\begin{equation} (c_1c_{m-1})(Y) = \int_Y c_1(TY)c_{m-1}(TY) . \end{equation}

Let $\chi (\cdot )$ be the topological Euler characteristic. Recall that $\tau _\mathrm {BCOV}(\cdot,\cdot )$ was defined in Definition 1.24.

Theorem 2.3 As $\varepsilon \rightarrow 0$,

(2.7)\begin{align} & \tau_\mathrm{BCOV}(X,\omega_\varepsilon) - \tfrac{1}{12} \chi(Z) \big( m \chi(Y) + (c_1c_{m-1})(Y) \big) \log \varepsilon \nonumber\\ & \quad \rightarrow \chi(Z) \tau_\mathrm{BCOV}(Y,\omega_Y) + \chi(Y) \tau_\mathrm{BCOV}(Z,\omega_Z) . \end{align}

Proof. The proof consists of several steps.

Recall that $\eta (\cdot )$ was constructed in (1.118) and $\lambda _\mathrm {tot}(\cdot )$ was constructed in (1.119).

Step 1. We construct two canonical sections of

(2.8)\begin{equation} \lambda_\mathrm{tot}(X) \otimes \big(\lambda_\mathrm{tot}(Y)\big)^{-\chi(Z)} \otimes \big(\eta(Y)\big)^{-n\chi(Z)} . \end{equation}

For $p=0,\ldots,m+n$ and $s=0,\ldots,p-1$, set

(2.9)\begin{equation} I^p_s = \big\{ u \in \Lambda^p(T^*X) : u(v_1,\ldots,v_p) = 0 \text{ for any } v_1,\ldots,v_{s+1} \in TZ, v_{s+2},\ldots,v_p \in TX \big\} .\end{equation}

For convenience, we denote $I^p_p = \Lambda ^p(T^*X)$ and $I^p_{-1} = 0$. We obtain a filtration

(2.10)\begin{equation} \Lambda^p(T^*X) = I^p_p \hookleftarrow I^p_{p-1} \hookleftarrow \cdots \hookleftarrow I^p_{-1} = 0 . \end{equation}

For $r=0,\ldots,m$ and $s=0,\ldots,n$, we denote

(2.11)\begin{equation} E_{r,s} = \Lambda^s(T^*Z) \otimes \pi^*\Lambda^r(T^*Y) . \end{equation}

We have a short exact sequence of holomorphic vector bundles over $X$,

(2.12)\begin{equation} 0 \rightarrow I^{r+s}_{s-1} \rightarrow I^{r+s}_s \rightarrow E_{r,s} \rightarrow 0 . \end{equation}

Let

(2.13)\begin{equation} \alpha_{r,s} \in \big(\!\det H^\bullet(X,I^{r+s}_{s-1})\big)^{-1} \otimes \det H^\bullet(X,I^{r+s}_s) \otimes \big(\!\det H^\bullet(X,E_{r,s})\big)^{-1} . \end{equation}

be the canonical section induced by the long exact sequence induced by (2.12).

Let $H^{\bullet,\bullet }(Z)$ be the fiberwise cohomology. As $Z \simeq \mathbb{C} \mathrm {P}^n$, we have

(2.14)\begin{equation} H^{p,p}(Z) = \mathbb{C} \quad\text{for } p = 0,\ldots, n ,\quad H^{p,q}(Z) = 0 \quad\text{for } p \neq q . \end{equation}

Applying spectral sequence while using (2.11) and (2.14), we obtain

(2.15)\begin{equation} H^q(X,E_{r,s}) \simeq H^{r,q-s}\big(Y,H^{s,s}(Z)\big) := H^{q-s}\big(Y,\Lambda^r(T^*Y) \otimes H^{s,s}(Z)\big) . \end{equation}

Let

(2.16)\begin{equation} \beta_{r,s} \in \det H^\bullet(X,E_{r,s}) \otimes \big(\!\det H^{r,\bullet}\big(Y,H^{s,s}(Z)\big) \big)^{-(-1)^s} \end{equation}

be the canonical section induced by (2.15).

We have a generator of lattice,

(2.17)\begin{equation} \delta_s \in H^{2s}_\mathrm{Sing}(\mathbb{C}\mathrm{P}^n,\mathbb{Z}) \subseteq H^{2s}_\mathrm{Sing}(\mathbb{C}\mathrm{P}^n,\mathbb{R}) \subseteq H^{2s}_\mathrm{Sing}(\mathbb{C}\mathrm{P}^n,\mathbb{C}) . \end{equation}

We identify $H^{2s}_\mathrm {Sing}(\mathbb{C} \mathrm {P}^n,\mathbb{C} )$ with $H^{2s}_\mathrm {dR}(\mathbb{C} \mathrm {P}^n) = H^{s,s}(\mathbb{C} \mathrm {P}^n)$ (see (1.121)). Since $H^{s,s}(Z) = H^{s,s}(\mathbb{C} \mathrm {P}^n) = H^{2s}_\mathrm {Sing}(\mathbb{C} \mathrm {P}^n,\mathbb{C} )$ is a trivial line bundle over $Y$, we have an isomorphism (cf. [Reference Griffiths and HarrisGH94, p. 607])

(2.18)\begin{equation} \begin{aligned} H^{r,\bullet}(Y) & \rightarrow H^{r,\bullet}\big(Y,H^{s,s}(Z)\big) = H^{r,\bullet}(Y) \otimes H^{s,s}(\mathbb{C}\mathrm{P}^n)\\ u & \mapsto u \otimes \delta_s . \end{aligned} \end{equation}

Let

(2.19)\begin{equation} \gamma_{r,s} \in \big(\!\det H^{r,\bullet}\big(Y,H^{s,s}(Z)\big) \big)^{(-1)^s} \otimes \big(\!\det H^{r,\bullet}(Y) \big)^{-(-1)^s} \end{equation}

be the canonical section induced by (2.18). By (2.13), (2.16) and (2.19), we have

(2.20)\begin{equation} \alpha_{r,s} \otimes \beta_{r,s} \otimes \gamma_{r,s} \in \big(\!\det H^\bullet(X,I^{r+s}_{s-1})\big)^{-1} \otimes \det H^\bullet(X,I^{r+s}_s) \otimes \big(\!\det H^{r,\bullet}(Y) \big)^{-(-1)^s} .\end{equation}

Recall that $\lambda (\cdot )$ was defined in (1.119). By (1.119) and (2.10), we have

(2.21)\begin{align} \lambda(X) & = \bigotimes_{p=1}^{m+n} \big(\!\det H^\bullet\big(X,\Lambda^p(T^*X)\big)\big)^{(-1)^pp} \nonumber\\ & = \bigotimes_{p=1}^{m+n} \big(\!\det H^\bullet(X,I^p_p)\big)^{(-1)^pp} \nonumber\\ & = \bigotimes_{r=0}^m \bigotimes_{s=0}^n \big( \big(\!\det H^\bullet(X,I^{r+s}_{s-1})\big)^{-1} \otimes \det H^\bullet(X,I^{r+s}_s) \big)^{(-1)^{r+s}(r+s)}. \end{align}

On the other hand, by (1.118), (1.119) and the identities

(2.22)\begin{equation} n+1 = \chi(Z) ,\quad \sum_{s=0}^n s = \frac{n(n+1)}{2} = \frac{n}{2}\chi(Z) , \end{equation}

we have

(2.23)\begin{equation} \bigotimes_{r=0}^m \bigotimes_{s=0}^n \big(\!\det H^{r,\bullet}(Y) \big)^{(-1)^r(r+s)} = \big(\lambda(Y)\big)^{\chi(Z)} \otimes \big(\eta(Y)\big)^{n\chi(Z)/2} . \end{equation}

By (2.20), (2.21) and (2.23), we have

(2.24)\begin{equation} \prod_{r=0}^m \prod_{s=0}^n ( \alpha_{r,s}\otimes\beta_{r,s}\otimes\gamma_{r,s} )^{(-1)^{r+s}(r+s)} \in \lambda(X) \otimes \big(\lambda(Y)\big)^{-\chi(Z)} \otimes \big(\eta(Y)\big)^{-n\chi(Z)/2} .\end{equation}

By (1.119) and (2.24), we have

(2.25)\begin{align} &\prod_{r=0}^m \prod_{s=0}^n ( \alpha_{r,s}\otimes\beta_{r,s}\otimes\gamma_{r,s} )^{(-1)^{r+s}(r+s)} \otimes \overline{ \prod_{r=0}^m \prod_{s=0}^n ( \alpha_{r,s}\otimes\beta_{r,s}\otimes\gamma_{r,s} )^{(-1)^{r+s}(r+s)} } \nonumber\\ &\quad \in \lambda_\mathrm{tot}(X) \otimes \big(\lambda_\mathrm{tot}(Y)\big)^{-\chi(Z)} \otimes \big(\eta(Y)\big)^{-n\chi(Z)} , \end{align}

where $\bar {\cdot }$ is the conjugation.

Let $\sigma _X \in \lambda _\mathrm {tot}(X)$, $\sigma _Y \in \lambda _\mathrm {tot}(Y)$ and $\epsilon _Y \in \eta (Y)$ be as in (1.122). Obviously, we have

(2.26)\begin{equation} \sigma_X \otimes \sigma_Y^{-\chi(Z)} \otimes \epsilon_Y^{-n\chi(Z)} \in \lambda_\mathrm{tot}(X) \otimes \big(\lambda_\mathrm{tot}(Y)\big)^{-\chi(Z)} \otimes \big(\eta(Y)\big)^{-n\chi(Z)} . \end{equation}

Step 2. We show that

(2.27)\begin{align} &\prod_{r=0}^m \prod_{s=0}^n ( \alpha_{r,s}\otimes\beta_{r,s}\otimes\gamma_{r,s} )^{(-1)^{r+s}(r+s)} \otimes \overline{ \prod_{r=0}^m \prod_{s=0}^n ( \alpha_{r,s}\otimes\beta_{r,s}\otimes\gamma_{r,s} )^{(-1)^{r+s}(r+s)} } \nonumber\\ &\quad = \pm \sigma_X \otimes \sigma_Y^{-\chi(Z)} \otimes \epsilon_Y^{-n\chi(Z)}. \end{align}

Let $\mathbb {Z}(-1)$ be the inverse of the Tate twist, which is a Hodge structure of pure weight two. For $j\in \mathbb {N}$, we denote by $\mathbb {Z}(-j)$ its $j$th tensor power. We have canonical identifications of Hodge structures,

(2.28)\begin{equation} \begin{aligned} H^{2j}_\mathrm{Sing}(\mathbb{C}\mathrm{P}^n,\mathbb{Z}) & = \mathbb{Z}(-j) \quad \text{for } j = 0,\ldots,n ,\\ H^k_\mathrm{Sing}(X,\mathbb{Z}) & = \bigoplus_{j=0}^n H^{k-2j}_\mathrm{Sing}(Y,\mathbb{Z}) \otimes H^{2j}_\mathrm{Sing}(\mathbb{C}\mathrm{P}^n,\mathbb{Z}) \\ & = \bigoplus_{j=0}^n H^{k-2j}_\mathrm{Sing}(Y,\mathbb{Z}) \otimes \mathbb{Z}(-j) . \end{aligned} \end{equation}

Complexifying (2.28) and applying Hodge decomposition, we obtain

(2.29)\begin{equation} \begin{aligned} & H^{j,j}(\mathbb{C}\mathrm{P}^n) = \mathbb{C} \quad \text{for } j = 0,\ldots,n ,\\ & H^{p,q}(X) = \bigoplus_{j=0}^n H^{p-j,q-j}(Y) \otimes H^{j,j}(\mathbb{C}\mathrm{P}^n) = \bigoplus_{j=0}^n H^{p-j,q-j}(Y) . \end{aligned} \end{equation}

We use the identifications in (2.28) and (2.29) until the end of Step 2.

Claim. For complex vector spaces $A$ and $B$, the canonical identification $\det A \otimes \det B \otimes (\det (A \oplus B))^{-1} = \mathbb{C}$ is such that the canonical section of $\det A \otimes \det B \otimes (\det (A \oplus B))^{-1}$ is identified with $1\in \mathbb{C}$.

Recall that $I^{r+s}_s$ was defined in (2.9) and $E_{r,s}$ was defined in (2.11). We have

(2.30)\begin{equation} H^q(X,I^{r+s}_s) = \bigoplus_{j=0}^s H^{r+s-j,q-j}(Y) ,\quad H^q(X,E_{r,s}) = H^{r,q-s}(Y) . \end{equation}

By (2.30), we have

(2.31)\begin{equation} H^\bullet(X,I^{r+s}_s) = H^\bullet(X,I^{r+s}_{s-1}) \oplus H^\bullet(X,E_{r,s}). \end{equation}

Applying the claim in the last paragraph to (2.31), we obtain

(2.32)\begin{equation} \big(\!\det H^\bullet(X,I^{r+s}_{s-1})\big)^{-1} \otimes \det H^\bullet(X,I^{r+s}_s) \otimes \big(\!\det H^\bullet(X,E_{r,s})\big)^{-1} = \mathbb{C} , \quad \alpha_{r,s} = 1 . \end{equation}

A similar argument shows that

(2.33)\begin{equation} \begin{aligned} \det H^\bullet(X,E_{r,s}) \otimes \big(\!\det H^{r,\bullet}\big(Y,H^{s,s}(Z)\big) \big)^{-(-1)^s} = \mathbb{C} , \quad \beta_{r,s} = 1 , \\ \big(\!\det H^{r,\bullet}\big(Y,H^{s,s}(Z)\big) \big)^{(-1)^s} \otimes \big(\!\det H^{r,\bullet}(Y) \big)^{-(-1)^s} = \mathbb{C} , \quad \gamma_{r,s} = 1 . \end{aligned} \end{equation}

Using (1.119), (1.121) and (2.28), we can show that

(2.34)\begin{equation} \begin{aligned} \lambda_\mathrm{tot}(X) \otimes \big(\lambda_\mathrm{tot}(Y)\big)^{-\chi(Z)} \otimes \big(\eta(Y)\big)^{-n\chi(Z)} & = \mathbb{C} ,\\ \sigma_X \otimes \sigma_Y^{-\chi(Z)} \otimes \epsilon_Y^{-n\chi(Z)} & = \pm 1 . \end{aligned} \end{equation}

From (2.32)–(2.34), we obtain (2.27).

Step 3. We introduce several Quillen metrics.

• • Let $g^{TX}_\varepsilon$ be the metric on $TX$ induced by $\omega _\varepsilon$.

• • Let $g^{\Lambda ^p(T^*X)}_\varepsilon$ be the metric on $\Lambda ^p(T^*X)$ induced by $g^{TX}_\varepsilon$.

• • Let $g^{I^p_s}_\varepsilon$ be the metric on $I^p_s$ induced by $g^{\Lambda ^p(T^*X)}_\varepsilon$ via (2.10).

• • Let $g^{TY}$ be the metric on $TY$ induced by $\omega _Y$.

• • Let $g^{\Lambda ^r(T^*Y)}$ be the metric on $\Lambda ^r(T^*Y)$ induced by $g^{TY}$.

• • Let $g^{TZ}$ be the metric on $TZ$ induced by $\omega _Z = \omega _\varepsilon |_Z$.

• • Let $g^{\Lambda ^s(T^*Z)}$ be the metric on $\Lambda ^s(T^*Z)$ induced by $g^{TZ}$.

• • Let $g^{E_{r,s}}$ be the metric on $E_{r,s}$ induced by $g^{\Lambda ^r(T^*Y)}$ and $g^{\Lambda ^s(T^*Z)}$ via (2.11).

Let

(2.35)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X,I^p_s),\varepsilon} \end{equation}

be the Quillen metric on $\det H^\bullet (X,I^p_s)$ associated with $g^{TX}_\varepsilon$ and $g^{I^p_s}_\varepsilon$. Let

(2.36)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X,E_{r,s}),\varepsilon} \end{equation}

be the Quillen metric on $\det H^\bullet (X,E_{r,s})$ associated with $g^{TX}_\varepsilon$ and $g^{E_{r,s}}$. Recall that $\alpha _{r,s}$ was defined by (2.13). Let $\lVert \alpha _{r,s}\rVert _\varepsilon$ be the norm of $\alpha _{r,s}$ with respect to the metrics (2.35) and (2.36).

• • Let $g^{\Omega ^{s,s}(Z)}$ be the $L^2$-metric on $\Omega ^{s,s}(Z)$ induced by $g^{TZ}$ (see (1.70)).

• • Let $g^{H^{s,s}(Z)}$ be the metric on $H^{s,s}(Z)$ induced by $g^{\Omega ^{s,s}(Z)}$ via the Hodge theorem.

Let

(2.37)\begin{equation} \lVert\cdot\rVert_{\det H^{r,\bullet}(Y,H^{s,s}(Z))} \end{equation}

be the Quillen metric on $\det H^{r,\bullet }(Y,H^{s,s}(Z)) = \det H^\bullet (Y,\Lambda ^r(T^*Y)\otimes H^{s,s}(Z))$ associated with $g^{TY}$ and $g^{\Lambda ^r(T^*Y)} \otimes g^{H^{s,s}(Z)}$. Recall that $\beta _{r,s}$ was defined by (2.16). Let $\lVert \beta _{r,s}\rVert _\varepsilon$ be the norm of $\beta _{r,s}$ with respect to the metrics (2.36) and (2.37). Let

(2.38)\begin{equation} \lVert\cdot\rVert_{\det H^{r,\bullet}(Y)} \end{equation}

be the Quillen metric on $\det H^{r,\bullet }(Y) = \det H^\bullet (Y,\Lambda ^r(T^*Y))$ associated with $g^{TY}$ and $g^{\Lambda ^r(T^*Y)}$. Recall that $\gamma _{r,s}$ was defined by (2.19). Let $\lVert \gamma _{r,s}\rVert$ be the norm of $\gamma _{r,s}$ with respect to the metrics (2.37) and (2.38).

By (1.119) and (2.10), we have

(2.39)\begin{equation} \sigma_X \in \lambda_\mathrm{tot}(X) = \bigotimes_{p=1}^{m+n} \big(\!\det H^\bullet(X,I^p_p)\big)^{(-1)^pp} \otimes \overline{\bigotimes_{p=1}^{m+n} \big(\!\det H^\bullet(X,I^p_p)\big)^{(-1)^pp}} . \end{equation}

Let $\lVert \sigma _X\rVert _\varepsilon$ be the norm of $\sigma _X$ with respect to the metrics (2.35) with $s=p$. By (1.118) and (1.119), we have

(2.40)\begin{equation} \begin{aligned} & \epsilon_Y \in \eta(Y) = \bigotimes_{r=0}^m \big(\!\det H^{r,\bullet}(Y)\big)^{(-1)^r} ,\\ & \sigma_Y \in \lambda_\mathrm{tot}(Y) = \bigotimes_{r=1}^m \big(\!\det H^{r,\bullet}(Y)\big)^{(-1)^rr} \otimes \overline{\bigotimes_{r=1}^m \big(\!\det H^{r,\bullet}(Y)\big)^{(-1)^rr}} . \end{aligned} \end{equation}

Let $\lVert \epsilon _Y\rVert$ be the norm of $\epsilon _Y$ with respect to the metrics (2.38). Let $\lVert \sigma _Y\rVert$ be the norm of $\sigma _Y$ with respect to the metrics (2.38). By (2.27), we have

(2.41)\begin{align} & \sum_{r=0}^m \sum_{s=0}^n (-1)^{r+s}(r+s) \big( \log\lVert\alpha_{r,s}\rVert^2_\varepsilon + \log\lVert\beta_{r,s}\rVert^2_\varepsilon + \log\lVert\gamma_{r,s}\rVert^2 \big) \nonumber\\ &\quad = \log\lVert\sigma_X\rVert_\varepsilon - \chi(Z) \log\lVert\sigma_Y\rVert - n \chi(Z) \log\lVert\epsilon_Y\rVert . \end{align}

On the other hand, by Definition 1.23 and Proposition 1.25, we have

(2.42)\begin{equation} \log\lVert\epsilon_Y\rVert = 0 . \end{equation}

By Definition 1.24, (2.41) and (2.42), we have

(2.43) \begin{align} \tau_\mathrm{BCOV}(X,\omega_\varepsilon) &= \chi(Z) \tau_\mathrm{BCOV}(Y,\omega_Y) \nonumber\\ &\quad + \sum_{r=0}^m \sum_{s=0}^n (-1)^{r+s}(r+s) \big( \log\lVert\alpha_{r,s}\rVert^2_\varepsilon + \log\lVert\beta_{r,s}\rVert^2_\varepsilon + \log\lVert\gamma_{r,s}\rVert^2 \big) . \end{align}

Step 4. We estimate $\log \lVert \alpha _{r,s}\rVert ^2_\varepsilon$.

Recall that $I^{r+s}_s$ was defined in (2.9), $E_{r,s}$ was defined in (2.11), $g^{I^{r+s}_s}_\varepsilon$ and $g^{E_{r,s}}$ were defined at the beginning of Step 3. Let $g^{E_{r,s}}_\varepsilon$ be quotient metric on $E_{r,s}$ induced by $g^{I^{r+s}_s}_\varepsilon$ via the surjection $I^{r+s}_s \rightarrow E_{r,s}$ in (2.12). Note that $g^{I^{r+s}_s}_\varepsilon$ is induced by $\omega _\varepsilon$. By (2.5), as $\varepsilon \rightarrow 0$,

(2.44)\begin{equation} \varepsilon^{-r} g^{E_{r,s}}_\varepsilon \rightarrow g^{E_{r,s}} . \end{equation}

We use the notation from (1.23). Let

(2.45)\begin{equation} \tilde{T}_{r,s,\varepsilon} = \widetilde{\mathrm{ch}}\Big( g^{I^{r+s}_{s-1}}_\varepsilon, g^{I^{r+s}_s}_\varepsilon, g^{E_{r,s}}_\varepsilon \Big) \in Q^X/Q^{X,0} \end{equation}

be the Bott–Chern form (1.56) with $0\rightarrow E' \rightarrow E \rightarrow E'' \rightarrow 0$ replaced by (2.12) and $(g^{E'},g^E,g^{E''})$ replaced by $( g^{I^{r+s}_{s-1}}_\varepsilon, g^{I^{r+s}_s}_\varepsilon, g^{E_{r,s}}_\varepsilon )$. Let

(2.46)\begin{equation} T_{r,s,\varepsilon} = \widetilde{\mathrm{ch}}\Big( g^{I^{r+s}_{s-1}}_\varepsilon, g^{I^{r+s}_s}_\varepsilon, g^{E_{r,s}} \Big) \in Q^X/Q^{X,0} \end{equation}

be the Bott–Chern form (1.56) with $0\rightarrow E' \rightarrow E \rightarrow E'' \rightarrow 0$ replaced by (2.12) and $(g^{E'},g^E,g^{E''})$ replaced by $( g^{I^{r+s}_{s-1}}_\varepsilon, g^{I^{r+s}_s}_\varepsilon, g^{E_{r,s}} )$. By Proposition 1.11 and (2.44), as $\varepsilon \rightarrow 0$,

(2.47) \begin{align} T_{r,s,\varepsilon} - \tilde{T}_{r,s,\varepsilon} - \mathrm{ch}(E_{r,s},g^{E_{r,s}}) r \log \varepsilon = \widetilde{\mathrm{ch}}(g^{E_{r,s}},g^{E_{r,s}}_\varepsilon) - \mathrm{ch}(E_{r,s},g^{E_{r,s}}) r \log \varepsilon \rightarrow 0 . \end{align}

On the other hand, by Proposition 1.12, as $\varepsilon \rightarrow 0$,

(2.48)\begin{equation} \tilde{T}_{r,s,\varepsilon} \rightarrow 0 . \end{equation}

By (2.47) and (2.48), as $\varepsilon \rightarrow 0$,

(2.49)\begin{equation} T_{r,s,\varepsilon} - \mathrm{ch}(E_{r,s},g^{E_{r,s}}) r \log \varepsilon \rightarrow 0 . \end{equation}

Applying Theorem 1.18 to the short exact sequence (2.12), we obtain

(2.50)\begin{equation} \log \lVert\alpha_{r,s}\rVert_\varepsilon^2 = \int_X \mathrm{Td}(TX,g^{TX}_\varepsilon) T_{r,s,\varepsilon} . \end{equation}

By Proposition 1.10, as $\varepsilon \rightarrow 0$,

(2.51)\begin{equation} \mathrm{Td}(TX,g^{TX}_\varepsilon) \rightarrow \pi^*\mathrm{Td}(TY,g^{TY}) \mathrm{Td}(TZ,g^{TZ}) . \end{equation}

On the other hand, by the Grothendieck–Riemann–Roch formula (1.76), (2.11) and (2.14), we have

(2.52)\begin{align} & \int_X \pi^*\mathrm{Td}(TY,g^{TY}) \mathrm{Td}(TZ,g^{TZ}) \mathrm{ch}(E_{r,s},g^{E_{r,s}})\nonumber\\ &\quad = \int_Y \mathrm{Td}(TY) \mathrm{ch}\big(H^\bullet(Z,E_{r,s})\big) \nonumber\\ &\quad = \int_Y \mathrm{Td}(TY) \mathrm{ch}\big(\Lambda^r(T^*Y)\big) \mathrm{ch}\big(H^{s,\bullet}(Z)\big) \nonumber\\ &\quad = (-1)^s \int_Y \mathrm{Td}(TY) \mathrm{ch}\big(\Lambda^r(T^*Y)\big) . \end{align}

By (2.49)–(2.52), as $\varepsilon \rightarrow 0$,

(2.53)\begin{equation} \log \lVert\alpha_{r,s}\rVert_\varepsilon^2 - (-1)^s r \int_Y \mathrm{Td}(TY) \mathrm{ch}\big(\Lambda^r(T^*Y)\big) \log \varepsilon \rightarrow 0. \end{equation}

By Proposition 1.5, (2.22) and (2.53), as $\varepsilon \rightarrow 0$,

(2.54)\begin{align} & \sum_{r=0}^m \sum_{s=0}^n (-1)^{r+s}(r+s) \log\lVert\alpha_{r,s}\rVert^2_\varepsilon \nonumber\\ & \quad - \bigg( \frac{m(3m+3n+1)}{12}\chi(Y) + \frac{1}{6}(c_1c_{m-1})(Y) \bigg) \chi(Z) \log \varepsilon \rightarrow 0 . \end{align}

Step 5. We estimate $\log \lVert \beta _{r,s}\rVert ^2_\varepsilon$.

Let

(2.55)\begin{equation} T_{r,s}\in Q^Y \end{equation}

be the Bismut–Köhler analytic torsion form (see § 1.5) associated with $( \pi : X \rightarrow Y, \omega _X, E_{r,s}, g^{E_{r,s}} )$. Applying Theorem 1.17 with $E=E_{r,s}$, as $\varepsilon \rightarrow 0$,

(2.56) \begin{align} \log \lVert\beta_{r,s}\rVert_\varepsilon^2 + \int_Y \mathrm{Td}'(TY) \int_Z \mathrm{Td}(TZ) \mathrm{ch}(E_{r,s}) \log\varepsilon\rightarrow \int_Y \mathrm{Td}(TY,g^{TY}) T_{r,s}. \end{align}

Similarly to (2.52), we have

(2.57) \begin{align} \int_Y \mathrm{Td}'(TY) \int_Z \mathrm{Td}(TZ) \mathrm{ch}(E_{r,s}) = (-1)^s \int_Y \mathrm{Td}'(TY)\mathrm{ch}\big(\Lambda^r(T^*Y)\big). \end{align}

Applying Proposition 1.15 with $E = E_{0,s}$ and $F = \Lambda ^r(T^*Y)$, we obtain

(2.58)\begin{equation} T_{r,s} = \mathrm{ch}\big(\Lambda^r(T^*Y),g^{\Lambda^r(T^*Y)}\big)T_{0,s} \text{ modulo } Q^{Y,0} . \end{equation}

By (2.56)–(2.58), as $\varepsilon \rightarrow 0$,

(2.59)\begin{align} & \log \lVert\beta_{r,s}\rVert_\varepsilon^2 + (-1)^s \int_Y \mathrm{Td}'(TY)\mathrm{ch}\big(\Lambda^r(T^*Y)\big) \log\varepsilon \nonumber\\ & \quad \rightarrow \int_Y \mathrm{Td}(TY,g^{TY}) \mathrm{ch}\big(\Lambda^r(T^*Y),g^{\Lambda^r(T^*Y)}\big)T_{0,s} . \end{align}

On the other hand, by Theorem 1.16, we have

(2.60)\begin{equation} \sum_{s=0}^n (-1)^s T_{0,s} = 0 \text{ modulo } Q^{Y,0} . \end{equation}

By Propositions 1.5, 1.6, (2.22), (2.59) and (2.60), as $\varepsilon \rightarrow 0$,

(2.61) \begin{align} &\sum_{r=0}^m \sum_{s=0}^n (-1)^{r+s}(r+s) \log\lVert\beta_{r,s}\rVert^2_\varepsilon+ \bigg( \frac{m(m+n)}{4}\chi(Y) + \frac{1}{12}(c_1c_{m-1})(Y) \bigg) \chi(Z) \log \varepsilon\nonumber\\ &\quad \rightarrow \int_Y c_m(TY,g^{TY}) \sum_{s=0}^n (-1)^s s T_{0,s} \nonumber\\ &\quad = \int_Y c_m(TY,g^{TY}) \sum_{s=0}^n (-1)^s s \{ T_{0,s} \}^{(0,0)}, \end{align}

where $\{\cdot \}^{(0,0)}$ means the component of degree $(0,0)$.

Step 6. We calculate $\log \lVert \gamma _{r,s}\rVert ^2$.

Recall that $H^{s,s}(Z)$ is a trivial line bundle over $Y$. Recall that $g^{H^{s,s}(Z)}$ was constructed in the paragraph above (2.37). By our assumption (2.4), $g^{H^{s,s}(Z)}$ is a constant metric. Recall that $\delta _s \in H^{s,s}(Z)$ was constructed in (2.17). Let $|\delta _s|$ be the norm of $\delta _s$ with respect to $g^{H^{s,s}(Z)}$, which is a constant function on $Y$. In the following, we do not distinguish between a constant function and its value. We denote $\chi _r(Y) = \sum _{q=0}^m (-1)^q \dim H^{r,q}(Y)$. By Remark 1.14, we have

(2.62)\begin{equation} \log\lVert\gamma_{r,s}\rVert^2 = (-1)^s \chi_r(Y) \log |\delta_s|^2 . \end{equation}

Let $\epsilon _Z \in \eta (Z)$ be as in (1.122). We have

(2.63)\begin{equation} \epsilon_Z = \pm \bigotimes_{s=0}^n \delta_s . \end{equation}

Let $|\epsilon _Z|$ be the norm of $\epsilon _Z$ with respect to the metrics $g^{H^{s,s}(Z)}$. By Proposition 1.25 and (2.63), we have

(2.64)\begin{equation} \sum_{s=0}^n \log |\delta_s|^2 = \log |\epsilon_Z|^2 = 0 . \end{equation}

Let $\sigma _Z \in \lambda _\mathrm {tot}(Z)$ be as in (1.122). We have

(2.65)\begin{equation} \sigma_Z = \pm \bigotimes_{s=1}^n \delta_s^{2s} . \end{equation}

Let $|\sigma _Z|$ be the norm of $\sigma _Z$ with respect to the metrics $g^{H^{s,s}(Z)}$. By (2.65), we have

(2.66)\begin{equation} \sum_{s=0}^n s \log |\delta_s|^2 = \log | \sigma_Z | . \end{equation}

By (2.62), (2.64), (2.66) and the identity $\sum _{r=0}^m (-1)^r \chi _r(Y) = \chi (Y)$, we have

(2.67)\begin{equation} \sum_{r=0}^m \sum_{s=0}^n (-1)^{r+s}(r+s) \log\lVert\gamma_{r,s}\rVert^2 = \chi(Y) \log | \sigma_Z |. \end{equation}

Step 7. We conclude.

By (2.43), (2.54), (2.61) and (2.67), as $\varepsilon \rightarrow 0$,

(2.68) \begin{align} & \tau_\mathrm{BCOV}(X,\omega_\varepsilon) - \frac{1}{12} \chi(Z) \big( m \chi(Y) + (c_1c_{m-1})(Y) \big) \log \varepsilon \nonumber\\ & \quad \rightarrow \chi(Z) \tau_\mathrm{BCOV}(Y,\omega_Y) + \chi(Y) \log |\sigma_Z | \nonumber\\ & \qquad + \int_Y c_m(TY,g^{TY}) \sum_{s=0}^n (-1)^s s \{T_{0,s}\}^{(0,0)} . \end{align}

Let $\theta _s(z)$ be as in (1.74) with $(X,\omega )$ replaced by $(Z,\omega _Z)$ and $(E,g^E)$ replaced by $(\Lambda ^s(T^*Z),g^{\Lambda ^s(T^*Z)})$. By Definition 1.13, 1.24, we have

(2.69)\begin{equation} \tau_\mathrm{BCOV}(Z,\omega_Z) = \log | \sigma_Z | + \sum_{s=0}^n (-1)^s s \theta_s'(0) . \end{equation}

By (2.4), all the terms in (2.69) are constant functions on $Y$. By (1.79), we have

(2.70)\begin{equation} \{ T_{0,s} \}^{(0,0)} = \theta_s'(0) . \end{equation}

From (2.68)–(2.70), we obtain (2.7). This completes the proof.

2.3 Behavior under blow-ups

The following lemma is direct consequence of Bott formula [Reference BottBot57] (see also [Reference Okonek, Schneider and SpindlerOSS11, p. 5]).

Lemma 2.5 Let $L$ be the holomorphic line bundle of degree one over $\mathbb{C} \mathrm {P}^n$. For $k=1,\ldots,n$ and $s =1,\ldots,k$, we have

(2.71)\begin{equation} H^\bullet\big(\mathbb{C}\mathrm{P}^n,\Lambda^k(T^*\mathbb{C}\mathrm{P}^n) \otimes L^s \big) = 0 . \end{equation}

Let $X$ be an $n$-dimensional compact Kähler manifold. Let $Y\subseteq X$ be a closed complex submanifold. Let $f: X' \rightarrow X$ be the blow-up along $Y$. Let $Y \subseteq U \subseteq X$ be an open neighborhood of $Y$. Set $U' = f^{-1}(U)$. Let $\omega$ be a Kähler form on $X$. Let $\omega '$ be a Kähler form on $X'$ such that

(2.72)\begin{equation} \omega'|_{X'\backslash U'} = f^*(\omega|_{X\backslash U}) . \end{equation}

For the existence of such $\omega '$, see the proof of [Reference VoisinVoi02, Proposition 3.24].

Theorem 2.6 We have

(2.73)\begin{equation} \tau_\mathrm{BCOV}(X',\omega') - \tau_\mathrm{BCOV}(X,\omega) = \alpha(U,U',\omega|_U,\omega'|_{U'}) , \end{equation}

where $\alpha (U,U',\omega |_U,\omega '|_{U'})$ is a real number determined by $U$, $U'$, $\omega |_U$ and $\omega '|_{U'}$.

Proof. The proof consists of several steps.

Step 0. We introduce several pieces of notation.

We denote $D = f^{-1}(Y)$. Let $i: D \hookrightarrow X'$ be the canonical embedding. Let $\mathscr {I} \subseteq \mathscr {O}_{X'}$ be the ideal sheaf associated with $D$. More precisely, for open subset $U \subseteq X'$, we have

(2.74)\begin{equation} \mathscr{I}(U) = \{ \theta \in \mathscr{O}_{X'}(U) : \theta|_{U \cap D} = 0 \} . \end{equation}

For $p=0,\ldots,n$, there exist holomorphic vector bundles over $X'$ linked by holomorphic maps

(2.75)\begin{equation} f^*\Lambda^p(T^*X) = F^p_p \rightarrow F^p_{p-1} \rightarrow \cdots \rightarrow F^p_0 = \Lambda^p(T^*X') \end{equation}

such that for $s=0,\ldots,p-1$,

• • the induced map $\mathscr {O}_{X'}(F^p_{s+1}) \rightarrow \mathscr {O}_{X'}(F^p_s)$ is injective;

• • we have $\mathscr {I} \otimes \mathscr {O}_{X'}(F^p_s) \hookrightarrow \mathscr {O}_{X'}(F^p_{s+1}) \hookrightarrow \mathscr {O}_{X'}(F^p_s)$.

Set

(2.76)\begin{equation} \mathscr{G}^p_s = \mathscr{O}_{X'}(F^p_s)/\mathscr{O}_{X'}(F^p_{s+1}) . \end{equation}

Then we have a commutative diagram of analytic coherent sheaves on $X'$,

(2.77)

where the first row is exact. Now we briefly explain the existence of these $F^p_s$. We have

(2.78)\begin{equation} \mathscr{I}^{\otimes p} \otimes \mathscr{O}_{X'}\big(\Lambda^p(T^*X')\big) \hookrightarrow \mathscr{O}_{X'}\big(f^*\Lambda^p(T^*X)\big) \hookrightarrow \mathscr{O}_{X'}\big(\Lambda^p(T^*X')\big) . \end{equation}

For $s=0,\ldots,p$, let $\mathscr {F}^p_s$ be the sub-sheaf of $\mathscr {O}_{X'}(\Lambda ^p(T^*X'))$ generated by $\mathscr {I}^{\otimes s} \otimes \mathscr {O}_{X'}(\Lambda ^p(T^*X'))$ and $\mathscr {O}_{X'}(f^*\Lambda ^p(T^*X))$. Then the desired properties hold with $\mathscr {O}_{X'}(F^p_s)$ replaced by $\mathscr {F}^p_s$. It remains to show that each $\mathscr {F}^p_s$ is given by a holomorphic vector bundle. Let $r$ be the codimension of $Y\hookrightarrow X$. Let $N_Y$ be the normal bundle of $Y\hookrightarrow X$. Let $\pi : D = \mathbb {P}(N_Y) \rightarrow Y$ be the canonical projection. Let $(y_0,y_1,\ldots,y_{n-r},z_1,\ldots,z_{r-1}) \in \mathbb{C} ^n$ be local coordinates on a neighborhood of $x \in D$ such that:

• • $(y_1,\ldots,y_{n-r})$ are the coordinates on $Y$;

• • $(z_1,\ldots,z_{r-1})$ are the coordinates on the fiber of $\pi : D \rightarrow Y$;

• • $D \subseteq X'$ is given by the equation $y_0 = 0$.

Then the image of $\mathscr {O}_{X'}(f^*T^*X) \hookrightarrow \mathscr {O}_{X'}(T^*X')$ is generated by

(2.79)\begin{equation} dy_0 , dy_1 , \ldots , dy_{n-r} , y_0\,dz_1 , \ldots , y_0\,dz_{r-1} . \end{equation}

As a consequence, the image of $\mathscr {F}^p_s \hookrightarrow \mathscr {O}_{X'}(\Lambda ^p(T^*X'))$ is generated by

(2.80)\begin{equation} y_0^{\min\{s,|J|\}} \bigotimes_{i\in I}\, dy_i \otimes \bigotimes_{j\in J } \,dz_j \end{equation}

with $I \subseteq \{0,1,\ldots,n-r\}$ and $J \subseteq \{1,\ldots,r-1\}$ satisfying $|I|+|J|=p$. Each term in (2.80) yields a holomorphic line bundle. Hence, $\mathscr {F}^p_s$ is given by a holomorphic vector bundle, which we denote by $F^p_s$.

Let $TD \rightarrow \pi ^* TY$ be the derivative of $\pi$. Set

(2.81)\begin{equation} T^VD = \mathrm{Ker}(TD \rightarrow \pi^* TY) \subseteq TD \subseteq TX'|_D . \end{equation}

Set

(2.82)\begin{align} I^p_s = \big\{\alpha \in \Lambda^p(T^*X')|_D : \alpha (v_1,\ldots,v_p) = 0 \text{ for any } v_1,\ldots,v_{s+1}\in T^VD, v_{s+2},\ldots,v_p\in TX'|_D \} .\end{align}

We obtain a filtration of holomorphic vector bundles over $D$,

(2.83)\begin{equation} \Lambda^p(T^*X')|_D = I^p_p \supseteq I^p_{p-1} \supseteq \cdots \supseteq I^p_0 . \end{equation}

Let $N_D$ be the normal line bundle of $D\hookrightarrow X'$. From the calculation in local coordinates, we see that

(2.84)\begin{equation} \mathscr{G}^p_s = i_*\mathscr{O}_D\big(N_D^{-s} \otimes (I^p_p/I^{p}_s)\big) \quad \text{for } s=0,\ldots,p-1 . \end{equation}

For convenience, we denote

(2.85)\begin{equation} G^p_s = N_D^{-s} \otimes (I^p_p/I^{p}_s) . \end{equation}

Then we obtain a short exact sequence

(2.86)\begin{equation} 0 \rightarrow \mathscr{O}_{X'}(F^p_{s+1}) \rightarrow \mathscr{O}_{X'}(F^p_s) \rightarrow i_*\mathscr{O}_D(G^p_s) \rightarrow 0 . \end{equation}

Step 1. We show that

(2.87)\begin{equation} \begin{aligned} H^q(D,G^p_0) & = \bigoplus_{k=1}^{r-1} H^{k,k}(\mathbb{C}\mathrm{P}^{r-1}) \otimes H^{p-k,q-k}(Y) ,\\ H^q(D,G^p_s) & = 0 \quad \text{for } s = 1,\ldots,p-1 . \end{aligned} \end{equation}

Set

(2.88)\begin{align} J^p_s = \big\{\alpha \in \Lambda^p(T^*D) : \alpha (v_1,\ldots,v_p) = 0 \text{ for any } v_1,\ldots,v_{s+1}\in T^VD, v_{s+2},\ldots,v_p\in TD \big\} .\end{align}

Let $\phi : \Lambda ^p(T^*X')|_D \rightarrow \Lambda ^p(T^*D)$ be the canonical projection. By (2.82) and (2.88), we have

(2.89)\begin{equation} J^p_s = \phi(I^p_s) \subseteq \Lambda^p(T^*D) . \end{equation}

By (2.83) and (2.89), we have a filtration of holomorphic vector bundles over $D$,

(2.90)\begin{equation} \Lambda^p(T^*D) = J^p_p \supseteq J^p_{p-1} \supseteq \cdots \supseteq J^p_0 . \end{equation}

We also have

(2.91)\begin{equation} J^p_k/J^p_{k-1} = \pi^*\big(\Lambda^{p-k}(T^*Y)\big) \otimes \Lambda^k(T^{V,*}D) , \end{equation}

and a short exact sequence of holomorphic vector bundles over $D$,

(2.92)\begin{equation} 0 \rightarrow N_D^{-1} \otimes J^{p-1}_k \rightarrow I^p_k \rightarrow J^p_k \rightarrow 0 . \end{equation}

Combining (2.91) and (2.92), we obtain a short exact sequence,

(2.93)\begin{align} 0 & \rightarrow N_D^{-1}\otimes\pi^*\big(\Lambda^{p-k-1}(T^*Y)\big) \otimes \Lambda^k(T^{V,*}D) \rightarrow I^p_k/I^p_{k-1} \nonumber\\ & \rightarrow \pi^*\big(\Lambda^{p-k}(T^*Y)\big) \otimes \Lambda^k(T^{V,*}D) \rightarrow 0 . \end{align}

By (2.85) and (2.93), $G^p_s$ admits a filtration with factors

(2.94)\begin{equation} \big( N_D^{-s-\epsilon}\otimes\pi^*\big(\Lambda^{p-k-\epsilon}(T^*Y)\big) \otimes \Lambda^k(T^{V,*}D) \big)_{\epsilon = 0,1, k = s+1,\ldots,p} . \end{equation}

We remark that $\pi : D \rightarrow Y$ is a $\mathbb{C} \mathrm {P}^{r-1}$-bundle and the restriction of $N_D^{-1}$ to the fiber of $\pi : D \rightarrow Y$ is a holomorphic line bundle of degree one. Applying spectral sequence while using Lemma 2.5, we see that the cohomology of the holomorphic vector bundles in (2.94) vanishes unless $\epsilon = s = 0$. Hence, we obtain the second identity in (2.87). This argument also shows that

(2.95)\begin{equation} H^q(D,G^p_0) = H^q(D,I^p_p/I^p_0) = H^q(D,J^p_p/J^p_0) . \end{equation}

Using spectral sequence and (2.91), we obtain

(2.96)\begin{equation} H^q(D,J^p_k/J^p_{k-1}) = H^{k,k}(\mathbb{C}\mathrm{P}^{r-1}) \otimes H^{p-k,q-k}(Y). \end{equation}

On the other hand, it is classical that

(2.97)\begin{equation} H^q(D,J^p_p) = H^q\big(D,\Lambda^p(T^*D)\big) = \bigoplus_{k=0}^{r-1} H^{k,k}(\mathbb{C}\mathrm{P}^{r-1}) \otimes H^{p-k,q-k}(Y) . \end{equation}

From (2.95)–(2.97), we obtain the first identity in (2.87).

Set

(2.98)\begin{equation} \lambda(G^\bullet_0) = \bigotimes_{p=1}^n\big(\!\det H^\bullet(D,G^p_0)\big)^{(-1)^pp} ,\quad \lambda_\mathrm{tot}(G^\bullet_0) = \lambda(G^\bullet_0) \otimes \overline{\lambda(G^\bullet_0)} . \end{equation}

Recall that $\lambda _\mathrm {tot}(X)$ was defined in (1.119).

Step 2. We construct two canonical sections of

(2.99)\begin{equation} \big(\lambda_\mathrm{tot}(X)\big)^{-1} \otimes \lambda_\mathrm{tot}(X') \otimes \big(\lambda_\mathrm{tot}(G^\bullet_0)\big)^{-1} \end{equation}

and show that they coincide up to $\pm 1$.

Let

(2.100)\begin{equation} \mu_{p,s} \in \big(\!\det H^\bullet(X',F^p_{s+1}) \big)^{-1} \otimes \det H^\bullet(X',F^p_s) \otimes \big(\!\det H^\bullet(D,G^p_s)\big)^{-1} \end{equation}

be the canonical section induced by the long exact sequence induced by (2.86). Indeed, by (2.87), we have

(2.101)\begin{equation} \mu_{p,s} \in \big(\!\det H^\bullet(X',F^p_{s+1}) \big)^{-1} \otimes \det H^\bullet(X',F^p_s) \quad \text{for } s \neq 0 . \end{equation}

Set

(2.102)\begin{align} \mu_p &= \bigotimes_{s=0}^{p-1} \mu_{p,s} \in \big(\!\det H^\bullet(X',F^p_p) \big)^{-1} \otimes \det H^\bullet(X',F^p_0) \otimes \big(\!\det H^\bullet(D,G^p_0)\big)^{-1} \nonumber\\ & = \big(\!\det H^\bullet(X',f^*\Lambda^p(T^*X)) \big)^{-1} \otimes \det H^{p,\bullet}(X') \otimes \big(\!\det H^\bullet(D,G^p_0)\big)^{-1} . \end{align}

We remark that $f_*\mathscr {O}_{X'} = \mathscr {O}_X$ and $R^{>0}f_*\mathscr {O}_{X'} = 0$. Using spectral sequence, we obtain a canonical identification

(2.103)\begin{equation} H^{p,\bullet}(X) = H^\bullet\big(X',f^*\Lambda^p(T^*X)\big) . \end{equation}

Let

(2.104)\begin{equation} \nu_p \in \big(\!\det H^{p,\bullet}(X) \big)^{-1} \otimes \det H^\bullet\big(X',f^*\Lambda^p(T^*X)\big) \end{equation}

be the canonical section induced by (2.103).

By (2.102) and (2.104), we have

(2.105)\begin{equation} \mu_p \otimes \nu_p \in \big(\!\det H^{p,\bullet}(X) \big)^{-1} \otimes \det H^{p,\bullet}(X') \otimes \big(\!\det H^\bullet(D,G^p_0)\big)^{-1} . \end{equation}

By (1.119), (2.98) and (2.105), we have

(2.106)\begin{equation} \bigotimes_{p=1}^n (\mu_p \otimes \nu_p)^{(-1)^pp} \in \big(\lambda(X)\big)^{-1} \otimes \lambda(X') \otimes \big(\lambda(G^\bullet_0)\big)^{-1} , \end{equation}

and

(2.107) \begin{align} \bigotimes_{p=1}^n (\mu_p \otimes \nu_p)^{(-1)^pp} \otimes \overline{\bigotimes_{p=1}^n (\mu_p \otimes \nu_p)^{(-1)^pp}} \in \big(\lambda_\mathrm{tot}(X)\big)^{-1} \otimes \lambda_\mathrm{tot}(X') \otimes \big(\lambda_\mathrm{tot}(G^\bullet_0)\big)^{-1} . \end{align}

We have the Hodge decomposition

(2.108)\begin{equation} H^j_\mathrm{dR}(Y) = \bigoplus_{p+q=j} H^{p,q}(Y) . \end{equation}

Let $b_k$ be the $k$th Betti number of $Y$. By (2.87), (2.98) and (2.108), we have

(2.109)\begin{equation} \lambda_\mathrm{tot}(G^\bullet_0) = \bigotimes_{k=1}^{r-1} \bigotimes_{j=2k}^{2k+2n-2r} \Big( \big(\!\det H^{2k}_\mathrm{dR}(\mathbb{C}\mathrm{P}^{r-1})\big)^{b_{j-2k}} \otimes \det H^{j-2k}_\mathrm{dR}(Y) \Big)^{(-1)^jj} . \end{equation}

Let

(2.110)\begin{equation} \delta_j\in H^j_\mathrm{Sing}(\mathbb{C}\mathrm{P}^{r-1},\mathbb{Z}) \subseteq H^j_\mathrm{Sing}(\mathbb{C}\mathrm{P}^{r-1},\mathbb{C}) = H^j_\mathrm{dR}(\mathbb{C}\mathrm{P}^{r-1}) \end{equation}

be a generator of $H^j_\mathrm {Sing}(\mathbb{C} \mathrm {P}^{r-1},\mathbb {Z})$. Let

(2.111)\begin{equation} \tau_{j,1},\ldots,\tau_{j,b_j}\in \mathrm{Im}\big( H^j_\mathrm{Sing}(Y,\mathbb{Z}) \rightarrow H^j_\mathrm{Sing}(Y,\mathbb{R}) \big) \subseteq H^j_\mathrm{dR}(Y) \end{equation}

be a basis of the lattice. We denote $\tau _j = \tau _{j,1}\wedge \cdots \wedge \tau _{j,b_j} \in \det H^j_\mathrm {dR}(Y)$. Set

(2.112)\begin{equation} \sigma_{G^\bullet_0} = \bigotimes_{k=1}^{r-1} \bigotimes_{j=2k}^{2k+2n-2r} ( \delta_{2k}^{b_{j-2k}} \otimes \tau_{j-2k} )^{(-1)^jj} \in \lambda_\mathrm{tot}(G^\bullet_0) . \end{equation}

Let $\sigma _X \in \lambda _\mathrm {tot}(X)$ and $\sigma _{X'} \in \lambda _\mathrm {tot}(X')$ be as in (1.122). Obviously, we have

(2.113)\begin{equation} \sigma_X^{-1} \otimes \sigma_{X'} \otimes \sigma_{G^\bullet_0}^{-1} \in \big(\lambda_\mathrm{tot}(X)\big)^{-1} \otimes \lambda_\mathrm{tot}(X') \otimes \big(\lambda_\mathrm{tot}(G^\bullet_0)\big)^{-1} . \end{equation}

We have a canonical identification (cf. [Reference VoisinVoi02, Théorème 7.31])

(2.114)\begin{equation} H^j_\mathrm{Sing}(X',\mathbb{Z}) = H^j_\mathrm{Sing}(X,\mathbb{Z}) \oplus \bigoplus_{k=1}^{r-1} H^{2k}_\mathrm{Sing}(\mathbb{C}\mathrm{P}^{r-1},\mathbb{Z}) \otimes H^{j-2k}_\mathrm{Sing}(Y,\mathbb{Z}) , \end{equation}

which induces an isomorphism of Hodge structures. Similarly to Step 2 in the proof of Theorem 2.3, using (2.114), we can show that

(2.115)\begin{equation} \bigotimes_{p=1}^n (\mu_p \otimes \nu_p)^{(-1)^pp} \otimes \overline{\bigotimes_{p=1}^n (\mu_p \otimes \nu_p)^{(-1)^pp}} = \pm \sigma_X^{-1} \otimes \sigma_{X'} \otimes \sigma_{G^\bullet_0}^{-1} . \end{equation}

Step 3. We introduce Quillen metrics.

Let $g^{TX}$ be the metric on $TX$ induced by $\omega$. Let $g^{\Lambda ^p(T^*X)}$ be the metric on $\Lambda ^p(T^*X)$ induced by $g^{TX}$. Let

(2.116)\begin{equation} \lVert\cdot\rVert_{\det H^{p,\bullet}(X)} \end{equation}

be the Quillen metric on $\det H^{p,\bullet }(X) = \det H^\bullet (X,\Lambda ^p(T^*X))$ associated with $g^{TX}$ and $g^{\Lambda ^p(T^*X)}$.

Let $g^{TX'}$ be the metric on $TX'$ induced by $\omega '$. Let $g^{\Lambda ^p(T^*X')}$ be the metric on $\Lambda ^p(T^*X')$ induced by $g^{TX'}$. Let

(2.117)\begin{equation} \lVert\cdot\rVert_{\det H^{p,\bullet}(X')} \end{equation}

be the Quillen metric on $\det H^{p,\bullet }(X') = \det H^\bullet (X',\Lambda ^p(T^*X'))$ associated with $g^{TX'}$ and $g^{\Lambda ^p(T^*X')}$.

Let

(2.118)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X',f^*\Lambda^p(T^*X))} \end{equation}

be the Quillen metric on $\det H^\bullet (X',f^*\Lambda ^p(T^*X))$ associated with $g^{TX'}$ and $f^*g^{\Lambda ^p(T^*X)}$.

Let $g^{TD}$ and $g^{N_D}$ be the metrics on $TD$ and $N_D$ induced by $g^{TX'}$. Let $g^{I^p_s}$ be the metric on $I^p_s$ induced by $g^{\Lambda ^p(T^*X')}$ via (2.83). Let $g^{G^p_s}$ be the metric on $G^p_s$ induced by $g^{N_D}$ and $g^{I^p_s}$ via (2.85). Let

(2.119)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(D,G^p_s)} \end{equation}

be the Quillen metric on $\det H^\bullet (D,G^p_s)$ associated with $g^{TD}$ and $g^{G^p_s}$. By the second identity in (2.87), we have a canonical identification $\det H^\bullet (D,G^p_s) = \mathbb{C}$ for $s\neq 0$. However, the metric (2.119) with $s\neq 0$ is not necessarily the standard metric on $\mathbb{C}$.

We remark that

(2.120)\begin{align} \Lambda^p(T^*X')|_{X'\backslash U'} & = F^p_s |_{X'\backslash U'} \nonumber\\ & = f^*\Lambda^p(T^*X)|_{X'\backslash U'} \quad \text{for } s=0,\ldots,p . \end{align}

We equip $F^p_s$ with Hermitian metric $g^{F^p_s}$ such that

(2.121) \begin{equation} \begin{gathered} g^{F^p_0} = g^{\Lambda^p(T^*X')} ,\quad g^{F^p_p} = f^*g^{\Lambda^p(T^*X)} ,\\ g^{F^p_{s+1}}|_{X'\backslash U'} = g^{F^p_s}|_{X'\backslash U'} \quad \text{for } s = 0,\ldots,p-1 . \end{gathered} \end{equation}

Our assumption (2.72) implies $g^{\Lambda ^p(T^*X')}|_{X'\backslash U'} = f^*(g^{\Lambda ^p(T^*X)}|_{X\backslash U})$, which guarantees the existence of $g^{F^p_s}$ satisfying (2.121). Let

(2.122)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X',F^p_s)} \end{equation}

be the Quillen metric on $\det H^\bullet (X',F^p_s)$ associated with $g^{TX'}$ and $g^{F^p_s}$. We remark that $H^\bullet (X',F^p_0) = H^{p,\bullet }(X')$ and

(2.123)\begin{equation} \lVert\cdot\rVert_{\det H^\bullet(X',F^p_0)} = \lVert\cdot\rVert_{\det H^{p,\bullet}(X')} . \end{equation}

Recall that $\mu _{p,s}$ was defined in (2.100). Let $\lVert \mu _{p,s}\rVert$ be the norm of $\mu _{p,s}$ with respect to the metrics (2.119) and (2.122).

Recall that $\nu _p$ was defined in (2.104). Let $\lVert \nu _p\rVert$ be the norm of $\nu _p$ with respect to the Quillen metrics (2.116) and (2.118).

Recall that $\sigma _{G^\bullet _0}$ was defined in (2.112). By (2.98) and the second identity in (2.87), we can and do view $\sigma _{G^\bullet _0}$ as the section of

(2.124)\begin{equation} \lambda_\mathrm{tot}(G^\bullet_\bullet) := \bigotimes_{p=1}^n \bigotimes_{s=0}^{p-1} \big(\!\det H^\bullet(D,G^p_s)\big)^{(-1)^pp} \otimes \overline{\bigotimes_{p=1}^n \bigotimes_{s=0}^{p-1} \big(\!\det H^\bullet(D,G^p_s)\big)^{(-1)^pp}} . \end{equation}

Let $\lVert \sigma _{G^\bullet _0}\rVert _{\lambda _\mathrm {tot}(G^\bullet _\bullet )}$ be the norm of $\sigma _{G^\bullet _0}\in \lambda _\mathrm {tot}(G^\bullet _\bullet )$ with respect to the metrics (2.119).

Let $\lVert \sigma _X\rVert _{\lambda _\mathrm {tot}(X)}$ be the norm of $\sigma _X$ with respect to the metrics (2.116). Let $\lVert \sigma _{X'}\rVert _{\lambda _\mathrm {tot}(X')}$ be the norm of $\sigma _{X'}$ with respect to the metrics (2.117). By (2.102) and (2.115), we have

(2.125)\begin{align} & \log \lVert\sigma_{X'}\rVert_{\lambda_\mathrm{tot}(X')} - \log \lVert\sigma_X\rVert_{\lambda_\mathrm{tot}(X)} - \log \lVert\sigma_{G^\bullet_0}\rVert_{\lambda_\mathrm{tot} (G^\bullet_\bullet)} \nonumber\\ &\quad = \sum_{p=1}^n (-1)^pp \bigg( \log \lVert\nu_p\rVert^2 + \sum_{s=0}^{p-1} \log \lVert\mu_{p,s}\rVert^2 \bigg) . \end{align}

By Definition 1.24 and (2.125), we have

(2.126) \begin{align} &\tau_\mathrm{BCOV}(X',\omega') - \tau_\mathrm{BCOV}(X,\omega)\nonumber\\ &\quad= \log \lVert\sigma_{G^\bullet_0}\rVert_{\lambda_\mathrm{tot}(G^\bullet_\bullet)} +\sum_{p=1}^n (-1)^pp \bigg( \log \lVert\nu_p\rVert^2 + \sum_{s=0}^{p-1} \log \lVert\mu_{p,s}\rVert^2 \bigg) . \end{align}

Step 4. We conclude.

For ease of notation, we denote

(2.127)\begin{equation} \alpha_{p,s} = \log \lVert\mu_{p,s}\rVert^2 . \end{equation}

Applying Theorem 1.19 to the short exact sequence (2.86) while using the second line in (2.121), we see that $\alpha _{p,s}$ is determined by $( U', \omega '|_{U'}, g^{F^p_s}|_{U'}, g^{F^p_{s+1}}|_{U'} )$. We denote

(2.128)\begin{equation} \alpha_p = \sum_{s=0}^{p-1} \alpha_{p,s} . \end{equation}

We remark that for $s = 1,\ldots,p-1$, the contributions of the metric $\lVert \cdot \rVert _{\det H^\bullet (X',F^p_s)}$ (see (2.122)) to $\alpha _{p,s-1}$ and $\alpha _{p,s}$ cancel with each other. Thus, $\alpha _p$ is independent of $(g^{F^p_s})_{s=1,\ldots,p-1}$. Hence, $\alpha _p$ is determined by $( U', \omega '|_{U'}, g^{F^p_0}|_{U'}, g^{F^p_p}|_{U'} )$. Now, applying the first line in (2.121), we see that $\alpha _p$ is determined by $(U, U', \omega |_U, \omega '|_{U'})$.

For ease of notation, we denote

(2.129)\begin{equation} \beta_p = \log \lVert\nu_p\rVert^2 , \end{equation}

Applying Theorem 1.21 with $E = \Lambda ^p(T^*X)$ while using (2.72), we see that $\beta _p$ is determined by $(U, U', \omega |_U, \omega '|_{U'})$.

By (2.126)–(2.129), we have

(2.130) \begin{align} \tau_\mathrm{BCOV}(X',\omega') - \tau_\mathrm{BCOV}(X,\omega) = \log \lVert\sigma_{G^\bullet_0}\rVert_{\lambda_\mathrm{tot}(G^\bullet_\bullet)} + \sum_{p=1}^n (-1)^pp ( \alpha_p + \beta_p).\end{align}

Here:

• • the section $\sigma _{G^\bullet _0}\in \lambda _\mathrm {tot}(G^\bullet _\bullet )$ is determined by $D\subseteq U'$ and its normal bundle;

• • the Quillen metric $\lVert \cdot \rVert _{\lambda _\mathrm {tot}(G^\bullet _\bullet )}$ is determined by $\omega '|_{U'}$;

• • the real number $\alpha _p$ is determined by $(U, U', \omega |_U, \omega '|_{U'})$;

• • the real number $\beta _p$ is determined by $(U, U', \omega |_U, \omega '|_{U'})$.

In conclusion, the right-hand side of (2.130) is determined by $(U, U', \omega |_U, \omega '|_{U'})$. This completes the proof.