In this subsection we assume $Q\equiv D\times\mathbb{R}$ is a Riemannian product of a Kähler manifold $(D,\omega_{D},J_{D})$ of complex dimension $n-1$, and the real line $\mathbb{R}$ with coordinate $z$. Given a smooth divisor $H\subset D$, let $P\equiv H\times\{0\}\subset D\times\{0\}$. In our discussion sometimes we also naturally identify $H$ with $P$. The results of this subsection will be purely local so $D$ and $H$ are not necessarily compact.

The splitting of a line $\mathbb{R}$ allows us to study the normal exponential map in $Q$ in terms of the normal exponential map in $D$. Notice the normal bundle of $P$ in $Q$ is naturally a Riemannian direct sum

where $N_{0}$ is the normal bundle of $H$ in $D$ given as the orthogonal complement $(TH)^{\perp}$ of $TH$ in $TD|_{H}$ (with respect to $\omega_{D}$). So $N_{0}$ is naturally a hermitian line bundle. We also naturally identify $N_{0}$ with the holomorphic normal bundle $TD|_{H}/TH$, as complex line bundles. Therefore, $N_{0}$ can be viewed as a holomorphic hermitian line bundle. The normal exponential map of $H$ in $D$ is defined by

which gives a local diffeomorphism from a neighborhood of the zero section in $N_{0}$ to a tubular neighborhood of $H$ in $D$. Immediately,

is the identity map under the natural isomorphisms $TN_{0}|_{H}\cong N_{0}\oplus TH$ and $TD|_{H}\cong N_{0}\oplus TH$.

Given any point $p\in H$, we may choose local holomorphic coordinates $\{w_{i}\}_{i=1}^{n-1}$ on $D$, centered at $p$, such that $H$ is locally defined by $w_{1}=0$. Then $dw_{1},w_{2},\ldots,w_{n-1}$ induces local holomorphic coordinates on $N_{0}$, which we denote by $\{\zeta,w_{2}^{\prime},\ldots,w_{n-1}^{\prime}\}$. Given any $(p,v)\in N_{0}$, its coordinates are by definition given as

Under the normal exponential map $\Exp_{H}$, these coordinates can also be viewed as local (non-holomorphic) coordinates on $D$, and when restricted to $H$ we have $w_{j}^{\prime}=w_{j}(j\geq 2)$ and $d\zeta=dw_{1}$. In particular, $\{w_{2}^{\prime},\cdots,w_{n-1}^{\prime}\}$ still gives holomorphic coordinates on $H$.

Similarly using $\Exp_{H}$, the coordinate vector field $\partial_{\zeta}$, originally defined on the normal bundle $N_{0}$, can also be viewed as a local (non-holomorphic) vector field on $D$. When restricted to $H$, the vector field $\partial_{\zeta}$ can hence be identified with the local section $\sigma$ of $(TH)^{\perp}\subset TD|_{H}$ given by the orthogonal projection of the holomorphic vector field $\partial_{w_{1}}$. Then we obtain a local unitary frame $e$ of $(TH)^{\perp}$ given by

These generate fiber coordinates $y,\bar{y}$ on $N_{0}$ such that

In this way we obtain local coordinates $\{y,\bar{y},y_{3}=z,w_{2}^{\prime},\bar{w}_{2}^{\prime},\ldots,w_{n-1}^{\prime},\bar{w}_{n-1}^{\prime}\}$ in a neighborhood of $p$ in $Q$. To match with the notation in the previous subsection, with respect to the local orthonormal basis $\{\sigma+\bar{\sigma},\sqrt{-1}(\sigma-\bar{\sigma}),\partial_{z}\}$, the normal geodesic coordinates are given by $\{y_{1}=Re(y),y_{2}=Im(y),y_{3}=z\}$, and

Also, the convention for orientation is given such that

defines a positive volume form. In the following, we will also use $\langle\cdot,\cdot\rangle$ to denote the hermitian inner product on $(1,0)$-type vectors. The relation with the Riemannian inner product is seen as

What the notation means will be clear in the context.

By making $P$ and $Q$ smaller we get local existence of Green’s current $G_{P}$ for $P$ in $Q$, by Theorem 3.10, with the expansion given there. In our case the formula can be written in terms of the above complex coordinates

Proposition 3.24 has a quick corollary which will be used in our later calculations.

Notice that the above local coordinates $\{y,\bar{y}\}$ are not canonical, and depend on the initial choice of the local coordinates $\{w_{i}\}_{i=1}^{n-1}$ on $D$. However, a different choice of local holomorphic coordinates on $D$ will induce the coordinates $\tilde{y},\bar{\tilde{y}}$ on fibers of $N_{0}$ such that

for some real function $\phi$ on $H$. In particular, we have the transformation

and

This suggests viewing $\Gamma$ as a connection 1-form on the normal bundle. Indeed this is exactly the case.

For later applications we will need a few more local expansion results. We will also use the notation $O^{\prime}$ and $\widetilde{O}$ in Definition 3.3. The meaning is similar, but here we work on a neighborhood of $H$ in $D$, and the distance function is locally given by $|y|$. Notice the following expansions are given in the local (non-holomorphic) coordinates $\{y,\bar{y},w_{2}^{\prime},\bar{w}_{2}^{\prime},\cdots,w_{n-1},\bar{w}_{n-1}^{\prime}\}$, and by definition we have $w_{j}^{\prime}|_{H}=w_{j}|_{H}$ for $j\geq 2$.

The proof relies on the following expansions of the holomorphic coordinate functions $w_{j}$.

Now we prove an expansion result for the trace of $\psi$.

Using (3.264) it is easy to see $\Tr_{\omega_{D}}\psi$ admits an expansion of the form

for local functions $A_{0},A_{1}$ defined on $H$. It suffices to show $A_{0}\equiv 1$ and $A_{1}\equiv 0$. Since the left hand side is independent of the choice of local holomorphic coordinates, it suffices to we only need to work on the slice $z=0$ with special local holomorphic coordinates in a neighborhood of $p\in H$, and it suffices to understand the Taylor expansion along the fiber $N_{0}(p)$ of $N_{0}$ over the fixed point $p$.

Now we prove Proposition 3.28.