2.1. Poincaré–Lelong formula

Let M be an m-dimensional complex manifold. A divisor D on M is said to be of normal crossings if D is locally defined by an equation $z_{1}\cdots z_{k}=0$ for a local holomorphic coordinate system $z_{1},\cdots ,z_{m}.$ Additionally, if every irreducible component of D is smooth, one says that D is of simple normal crossings. A holomorphic line bundle $L\rightarrow M$ is said to be Hermitian if L is equipped with a Hermitian metric $h=(\{h_{\alpha }\},\{U_{\alpha }\}),$ where

$$ \begin{align*}h_{\alpha}: U_{\alpha}\rightarrow \mathbb R^{+}\end{align*} $$

are positive smooth functions such that $h_{\beta }=|g_{\alpha \beta }|^{2}h_{\alpha }$ on $U_{\alpha }\cap U_{\beta },$ and $\{g_{\alpha \beta }\}$ is a transition function system of $L.$ Let $\{e_{\alpha }\}$ be a local holomorphic frame of L; then we have $\|e_{\alpha }\|^{2}_{h}=h_{\alpha }.$ A Hermitian metric h of L defines a global, closed and smooth (1,1)-form $-dd^{c}\log h$ on $M,$ where

$$ \begin{align*}d=\partial+\bar{\partial}, \ \ d^{c}=\frac{\sqrt{-1}}{4\pi}(\bar{\partial}-\partial), \ \ dd^{c}=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}.\end{align*} $$

We call $-dd^{c}\log h$ the Chern form denoted by $c_{1}(L,h)$ associated with metric $h,$ which determines a Chern class $c_{1}(L)\in H^{2}_{{\text {DR}}}(M,\mathbb R)$ ; $c_{1}(L,h)$ is also called the curvature form of $L.$ If $c_{1}(L)>0,$ namely, there exists a Hermitian metric h such that $-dd^{c}\log h>0,$ then we say that L is positive, written as $L>0.$

Let $T M$ denote the holomorphic tangent bundle of $M.$ The canonical line bundle of M is defined by

$$ \begin{align*}K_{M}=\bigwedge^{m}T^{*}M\end{align*} $$

with transition functions $g_{\alpha \beta }=\det (\partial z^{\beta }_{j}/\partial z^{\alpha }_{i})$ on $U_{\alpha }\cap U_{\beta }.$ Given a Hermitian metric h on $K_{M}$ , it well defines a global, positive and smooth $(m,m)$ -form

$$ \begin{align*}\Omega=\frac{1}{h}\bigwedge_{j=1}^{m}\frac{\sqrt{-1}}{2\pi}dz_{j}\wedge d\bar{z}_{j}\end{align*} $$

on $M,$ which is therefore a volume form of $M.$ The Ricci form of $\Omega $ is defined by ${\text {Ric}}\Omega =dd^{c}\log h.$ Clearly, $c_{1}(K_{M},h)=-{\text {Ric}}\Omega .$ Conversely, if we let $\Omega $ be a volume form on M which is compact, there exists a unique Hermitian metric h on $K_{M}$ such that $dd^{c}\log h={\text {Ric}} \Omega .$

Let $H^{0}(M,L)$ denote the vector space of holomorphic global sections of L over M. For any $s\in H^{0}(M,L)$ , the divisor $D_{s}$ is well defined by $D_{s}\cap U_{\alpha }=(s)|_{U_{\alpha }}$ . Denote by $|L|$ the complete linear system of all effective divisors $D_{s}$ with $s\in H^{0}(M,L).$ Let D be a divisor on M; then D defines a holomorphic line bundle $L_{D}$ over M in such manner: let $(\{g_{\alpha }\},\{U_{\alpha }\})$ be the local defining function system of D; then the transition system is given by $\{g_{\alpha \beta }=g_{\alpha }/g_{\beta }\}.$ Note that $\{g_{\alpha }\}$ defines a global meromorphic section on M written as $s_{D}$ of $L_{D}$ over $M,$ called the canonical section associated with $D.$

Lemma 2.1 Poincaré–Lelong formula, [Reference Carlson and Griffiths7]

Let $L\rightarrow M$ be a holomorphic line bundle equipped with a Hermitian metric $h,$ and let s be a holomorphic section of L over M with zero divisor $D_{s}.$ Then $\log \|s\|_{h}$ is locally integrable on M and defines a current satisfying

$$ \begin{align*}dd^{c}\big[\log\|s\|_{h}^{2}\big]=D_{s}-c_{1}(L,h).\end{align*} $$

2.2. Brownian motions

Let $(M,g)$ be a Riemannian manifold with the Laplace–Beltrami operator $\Delta _{M}$ associated with metric $g.$ A Brownian motion $X_{t}$ in M is a heat diffusion process generated by $\Delta _{M}/2$ with transition density function $p(t,x,y)$ being the minimal positive fundamental solution of the heat equation

$$ \begin{align*}\frac{\partial}{\partial t}u(t,x)-\frac{1}{2}\Delta_{M}u(t,x)=0.\end{align*} $$

In particular, when $M=\mathbb R^{m}$ ,

$$ \begin{align*}p(t,x,y)=\frac{1}{(2\pi t)^{\frac{m}{2}}}e^{-\|x-y\|^{2}/2t}.\end{align*} $$

Let $X_{t}$ be the Brownian motion in M with generator $\Delta _{M}/2.$ We denote by $\mathbb P_{x}$ the law of $X_{t}$ starting from $x\in M$ and denote by $\mathbb E_{x}$ the expectation with respect to $\mathbb P_{x}$ .

A. Co-area formula

Let D be a bounded domain with the smooth boundary $\partial D$ in $M.$ Denote by $d\pi ^{\partial D}_{x}(y)$ the harmonic measure on $\partial D$ with respect to x and by $g_{D}(x,y)$ the Green function of $\Delta _{M}/2$ for D with Dirichlet boundary condition and a pole at x; that is,

$$ \begin{align*}-\frac{1}{2}\Delta_{M}g_{D}(x,y)=\delta_{x}(y), \ \ y\in D; \ \ \ g_{D}(x,y)=0, \ \ y\in \partial D.\end{align*} $$

For each $\phi \in \mathscr {C}_{\flat }(D)$ (space of bounded and continuous functions on D), the co-area formula [Reference Bass5] says that

(2) $$ \begin{align} \mathbb{E}_{x}\left[\int_{0}^{\tau_{D}}\phi(X_{t})dt\right]=\int_{D}g_{D}(x,y)\phi(y)dV(y), \end{align} $$

where $dV$ is the Riemannian volume element on $M.$ From Proposition 2.8 in [Reference Bass5], we have the relation of harmonic measures and hitting times as follows:

(3) $$ \begin{align} \mathbb{E}_{x}\left[\psi(X_{\tau_{D}})\right]=\int_{\partial D}\psi(y)d\pi_{x}^{\partial D}(y) \end{align} $$

for $\psi \in \mathscr {C}(\overline {D})$ . The co-area formulas (3) and (2) still work when $\phi , \psi $ are of a pluripolar set of singularities.

B. Itô formula

The following identity is called the Itô formula (see [Reference Atsuji1, Reference Ikeda and Watanabe17, Reference Itô and McKean18]):

$$ \begin{align*}u(X_{t})-u(x)=B\left(\int_{0}^{t}\|\nabla_{M}u\|^{2}(X_{s})ds\right)+\frac{1}{2}\int_{0}^{t}\Delta_{M}u(X_{s})dt, \ \ \mathbb P_{x}-a.s.\end{align*} $$

for $u\in \mathscr {C}_{\flat }^{2}(M)$ (space of bounded $\mathscr {C}^{2}$ -class functions on M), where $B_{t}$ is the standard Brownian motion in $\mathbb R$ and $\nabla _{M}$ is the gradient operator on M. It follows the Dynkin formula

$$ \begin{align*} \mathbb E_{x}[u(X_{T})]-u(x)=\frac{1}{2}\mathbb E_{x}\left[\int_{0}^{T}\Delta_{M}u(X_{t})dt\right] \end{align*} $$

for a stopping time T such that each term makes sense. The Dynkin formula still works if u is of a pluripolar set of singularities.

2.3. Ricci curvatures

Let $(M,g)$ be a Kähler manifold of complex dimension m. Write the Ricci curvature of M in the form ${\text {Ric}}_{M}=\sum _{i,j}R_{i\bar {j}}dz_{i}\otimes d\bar {z}_{j},$ where

(4) $$ \begin{align} R_{i\bar{j}}=-\frac{\partial^{2}}{\partial z_{i}\partial \bar{z}_{j}}\log\det(g_{s\bar{t}}). \end{align} $$

A well-known theorem by S. S. Chern asserts that the Ricci form of M

$$ \begin{align*} \mathscr{R}_{M}:=-dd^{c}\log\det(g_{s\bar{t}})=\frac{\sqrt{-1}}{2\pi}\sum_{i,j=1}^{m}R_{i\bar{j}}dz_{i}\wedge d\bar{z}_{j} \end{align*} $$

is a real and closed (1,1)-form which represents a cohomology class of the de Rham cohomology group $H^{2}_{{\text {DR}}}(M,\mathbb R).$ Let $s_{M}$ be the scalar curvature of M defined by

$$ \begin{align*}s_{M}=\sum_{i,j=1}^{m}g^{i\bar j}R_{i\bar j},\end{align*} $$

where $(g^{i\bar j})$ is the inverse of $(g_{i\bar j}).$ Since M is Kählerian, then by

$$ \begin{align*}\Delta_{M}=2\sum_{i,j=1}^{m}g^{i\bar j}\frac{\partial^{2}}{\partial z_{i}\partial\bar{z}_{j}}\end{align*} $$

acting on a function, which yields from (4) that

$$ \begin{align*} s_{M}=-\frac{1}{2}\Delta_{M}\log\det(g_{s\bar t}). \end{align*} $$

Lemma 2.2. Let $R_{M}$ be the pointwise lower bound of Ricci curvature of M. Then

$$ \begin{align*}s_{M}\geq m R_{M}.\end{align*} $$

Proof. Fix a point $x\in M$ ; we take local holomorphic coordinates $z_{1}, \cdots , z_{m}$ near x such that $g_{i\bar {j}}(x)=\delta ^{i}_{j}.$ Then we obtain

$$ \begin{align*} s_{M}(x)&=\sum_{j=1}^{m}R_{j\bar{j}}(x) =\sum_{j=1}^{m}{\text{Ric}}_{M}(\frac{\partial}{\partial z_{j}},\frac{\partial}{\partial \bar{z}_{j}})_{x}\geq mR_{M}(x), \end{align*} $$

which proves the lemma.

3.1. Nevanlinna’s functions

Let

$$ \begin{align*}f: M\rightarrow N\end{align*} $$

be a meromorphic mapping to a compact complex manifold $N,$ which means that f is defined by such a holomorphic mapping $f_{0}:M\setminus I\rightarrow N,$ where I is some analytic subset of M with $\dim _{\mathbb C}I\leq m-2,$ called the indeterminacy set of f such that the closure $\overline {G(f_{0})}$ of the graph of $f_{0}$ is an analytic subset of $M\times N$ and the natural projection $\overline {G(f_{0})}\rightarrow M$ is proper. Let $\eta $ be a (1,1)-form on M, we use the following convenient notation:

$$ \begin{align*}e_{\eta}(x)=2m\frac{\eta \wedge\alpha^{m-1}}{\alpha^{m}}.\end{align*} $$

Given a smooth (1,1)-form $\omega $ on $N,$ since I is an indeterminacy set of $f,$ one could confirm the local integrability of $g_{r}(o,x)e_{f^{*}\omega }(x)$ on M with respect to measure $\alpha ^{m}$ by using the arguments in Noguchi–Ochiai [[Reference Noguchi and Ochiai20], Subsection 5.2]. We define the characteristic function of f with respect to $\omega $ by

$$ \begin{align*} T_{f}(r,\omega)&=\frac{1}{2}\int_{B_{o}(r)}g_{r}(o,x)e_{f^{*}\omega}(x)dV(x) \\ &= \frac{\pi^{m}}{(m-1)!}\int_{B_{o}(r)}g_{r}(o,x)f^{*}\omega\wedge \alpha^{m-1}, \end{align*} $$

where $dV=\pi ^{m}\alpha ^{m}/m!$ is the Riemannian volume element on $M.$ Let $(L, h)$ be a Hermitian line bundle over $N.$ By the compactness of N, we well define

$$ \begin{align*}T_{f}(r, L): =T_{f}\big{(}r, c_{1}(L, h)\big{)}\end{align*} $$

up to a bounded term. We further remark that the indeterminacy set I does not affect the local integrability of integrands in those quantities treated and hence the definitions of the following introduced proximity function $m_{f}(r, D)$ and counting function $N_{f}(r, D)$ (including Nevanlinna’s functions in Section 5) make sense. We refer the reader to Noguchi–Ochiai [[Reference Noguchi and Ochiai20], Subsection 5.2].

In what follows, we define the proximity function and counting function.

Lemma 3.1. $\Delta _{M}\log (h\circ f)$ is well defined on $M\setminus I$ satisfying

$$ \begin{align*}\Delta_{M}\log(h\circ f)=-4m\frac{f^{*}c_{1}(L,h)\wedge\alpha^{m-1}}{\alpha^{m}}.\end{align*} $$

Hence, we have

$$ \begin{align*}e_{f^{*}c_{1}(L,h)}=-\frac{1}{2}\Delta_{M}\log(h\circ f).\end{align*} $$

Proof. Let $(\{U_{\alpha }\},\{e_{\alpha }\})$ be a local trivialisation covering of $(L,h)$ with transition function system $\{g_{\alpha \beta }\}$ of local holomorphic frames $\{e_{\alpha }\}.$ On $U_{\alpha }\cap U_{\beta },$

$$ \begin{align*} e_{\beta}=g_{\alpha\beta}e_{\alpha}, \ \ h_{\alpha}=h|_{U_{\alpha}}=\|e_{\alpha}\|^{2}, \ \ h_{\beta}=h|_{U_{\beta}}=\|e_{\beta}\|^{2}.\end{align*} $$

We get

$$ \begin{align*}\Delta_{M}\log (h_{\beta}\circ f)=\Delta_{M}\log(h_{\alpha}\circ f)+\Delta_{M}\log |g_{\alpha\beta}\circ f|^{2}\end{align*} $$

on $f^{-1}(U_{\alpha }\cap U_{\beta })\setminus I.$ Notice that $g_{\alpha \beta }$ is holomorphic and nowhere vanishing on $U_{\alpha }\cap U_{\beta }$ ; we see that $\log |g_{\alpha \beta }\circ f|^{2}$ is harmonic on $f^{-1}(U_{\alpha }\cap U_{\beta })\setminus I.$ So, $\Delta _{M}\log (h_{\beta }\circ f)=\Delta _{M}\log (h_{\alpha }\circ f)$ on $f^{-1}(U_{\alpha }\cap U_{\beta })\setminus I.$ Thus, $\Delta _{M}\log (h\circ f)$ is well defined on $M\setminus I.$ Fix $x\in M$ ; then we choose a normal holomorphic coordinate system z near x in the sense that $g_{i\bar {j}}(x)=\delta _{j}^{i}$ and all of the first-order derivatives of $g_{i\overline {j}}$ vanish at $x.$ Then at $x,$ we have

(5) $$ \begin{align} \Delta_{M}=2\sum_{j=1}^{m}\frac{\partial^{2}}{\partial z_{j}\partial\bar{z}_{j}}, \ \ \ \alpha^{m}=m!\bigwedge_{j=1}^{m}\frac{\sqrt{-1}}{\pi}dz_{j}\wedge d\bar{z}_{j} \end{align} $$

as well as

$$ \begin{align*}f^{*}c_{1}(L,h)\wedge\alpha^{m-1}=-\frac{(m-1)!}{2}{\text{tr}}\left(\frac{\partial^{2}\log(h\circ f)}{\partial z_{i}\partial\bar{z}_{j}}\right)\bigwedge_{j=1}^{m}\frac{\sqrt{-1}}{\pi}dz_{j}\wedge d\bar{z}_{j},\end{align*} $$

where ‘tr’ means the trace of a square matrix. Indeed, by (5),

$$ \begin{align*}\Delta_{M}\log(h\circ f)=2{\text{tr}}\left(\frac{\partial^{2}\log(h\circ f)}{\partial z_{i}\partial\bar{z}_{j}}\right)\end{align*} $$

at $x.$ This proves the lemma.

Take $0\not =s\in H^{0}(N, L).$ Locally, we can write $s=\tilde s e,$ where e is a local holomorphic frame of $L.$ Then

$$ \begin{align*} \Delta_{M}\log \|s\circ f\|^{2}=\Delta_{M}\log(h\circ f)+\Delta_{M}\log|\tilde{s}\circ f|^{2}. \end{align*} $$

Using similar arguments as in the proof of Lemma 3.1, we get

$$ \begin{align*} \Delta_{M}\log|\tilde{s}\circ f|^{2}=4m\frac{dd^{c}\log|\tilde{s}\circ f|^{2}\wedge\alpha^{m-1}}{\alpha^{m}}. \end{align*} $$

Proof. Locally, we can write $s=\tilde se,$ where e is a local holomorphic frame of L with $h=\|e\|^{2}.$ Then

$$ \begin{align*}\log\|s\circ f\|^{2}=\log|\tilde{s}\circ f|^{2}+\log (h\circ f).\end{align*} $$

Since $c_{1}(L,h)\geq 0,$ one obtains $-dd^{c}\log (h\circ f)\geq 0.$ Indeed, $\tilde {s}$ is holomorphic; hence, $dd^{c}\log |\tilde {s}\circ f|^{2}\geq 0.$ This follows $(i)$ . The Poincaré–Lelong formula implies that $dd^{c}[\log |\tilde {s}\circ f|^{2}]=f^{*}D$ in the sense of currents; hence, $(ii)$ holds.

Let $D\in |L|,$ where $(L, h)$ is a Hermitian positive line bundle over $N.$ We define the proximity function of f with respect to D by

$$ \begin{align*} m_{f}(r,D)=\int_{S_{o}(r)}\log\frac{1}{\|s_{D}\circ f(x)\|}d\pi_{o}^{r}(x). \end{align*} $$

Write

$$ \begin{align*}\log\|s_{D}\circ f\|^{-2}=\log (h\circ f)^{-1}-\log|\tilde{s}_{D}\circ f|^{2}\end{align*} $$

as the difference of two pluri-subharmonic functions. It defines a Riesz charge $d\mu =d\mu _{1}-d\mu _{2},$ where $d\mu _{2}$ is a Riesz measure for $f^{*}D.$ The counting function of f with respect to D is defined by

$$ \begin{align*} N_{f}(r,D)= \frac{1}{4}\int_{B_{o}(r)}g_{r}(o,x)d\mu_{2}(x) =\frac{\pi^{m}}{(m-1)!}\int_{f^{*}D\cap B_{o}(r)}g_{r}(o,x)\alpha^{m-1}. \end{align*} $$

Similarly, we can define $\overline {N}_{f}(r,D):=N(r, {\text {Supp}}f^{*}D).$

3.2. Probabilistic expressions of Nevanlinna’s functions

Let us formulate Nevanlinna’s functions in terms of Brownian motion $X_{t}$ . Since I is a thin analytic subset contained in some pluripolar subset of $M, X_{t}$ hits I in probability 0, I will not affect the expectation of those quantities involving f treated with respect to probability measure $d\mathbb P_{o}.$ We define the stopping time

$$ \begin{align*}\tau_{r}=\inf\big{\{}t>0: X_{t}\not\in B_{o}(r)\big{\}}.\end{align*} $$

Set $\omega :=-dd^{c}\log h.$ By the co-area formula, we have

$$ \begin{align*}T_{f}(r,L) =\frac{1}{2}\mathbb E_{o}\left[\int_{0}^{\tau_{r}}e_{f^{*}\omega}(X_{t})dt\right].\end{align*} $$

By the relation between harmonic measures and hitting times, it gives that

$$ \begin{align*} m_{f}(r,D) =\mathbb E_{o}\left[\log\frac{1}{\|s_{D}\circ f(X_{\tau_{r}})\|}\right].\end{align*} $$

For the counting function $N_{f}(r,D),$ we use an alternative probabilistic expression (see [Reference Atsuji1, Reference Atsuji4, Reference Carne8, Reference Elworthy, Li and Yor12]) as follows:

(6) $$ \begin{align} N_{f}(r,D)=\lim_{\lambda\rightarrow\infty}\lambda\mathbb P_{o}\left(\sup_{0\leq t\leq\tau_{r}}\log\frac{1}{\|s_{D}\circ f(X_{t})\|}>\lambda\right). \end{align} $$

Remark 3.3. The definitions of Nevanlinna’s functions in the above are natural extensions of the classical ones. To see that, we recall the $\mathbb C^{m}$ -case:

$$ \begin{align*} T_{f}(r,L)&=\int_{0}^{r}\frac{dt}{t^{2m-1}}\int_{B_{o}(t)}f^{*}c_{1}(L,h)\wedge\alpha^{m-1}, \\ m_{f}(r,D)&=\int_{S_{o}(r)}\log\frac{1}{\|s_{D}\circ f\|}\gamma, \\ N_{f}(r,D)&=\int_{0}^{r}\frac{dt}{t^{2m-1}}\int_{f^{*}D\cap B_{o}(t)}\alpha^{m-1}, \end{align*} $$

where o is taken as the coordinate origin of $\mathbb C^{m},$ and

$$ \begin{align*}\alpha=dd^{c}\|z\|^{2},\ \ \ \gamma=d^{c}\log\|z\|^{2}\wedge \left(dd^{c}\log\|z\|^{2}\right)^{m-1}.\end{align*} $$

Notice the following facts:

$$ \begin{align*}\gamma=d\pi_{o}^{r}(z), \ \ \ g_{r}(o,z)=\left\{ \begin{array}{ll} \frac{\|z\|^{2-2m}-r^{2-2m}}{(m-1)\omega_{2m-1}}, & m\geq2; \\ \frac{1}{\pi}\log\frac{r}{|z|}, & m=1. \end{array} \right.,\end{align*} $$

where $\omega _{2m-1}$ is the volume of unit sphere in $\mathbb R^{2m}.$ By integration by part, it is not difficult to see that they are a match.

3.3. First main theorem

Let N be a complex projective algebraic manifold. There is a very ample holomorphic line bundle $L^{\prime }$ over $V.$ Equip $L^{\prime }$ with a Hermitian metric $h^{\prime }$ such that $\omega ^{\prime }:=-dd^{c}\log h^{\prime }>0.$ For an arbitrary holomorphic line bundle $L\rightarrow N$ equipped with a Hermitian metric $h,$ whose Chern form says $\omega :=-dd^{c}\log h,$ we can pick $k\in \mathbb N$ large enough so that $\omega +k\omega ^{\prime }>0.$ Take the natural product Hermitian metric $\|\cdot \|$ on $L\otimes L^{\prime \otimes k}$ ; then the Chern form is $\omega +k\omega ^{\prime }$ . Choose $\sigma \in H^{0}(M,L^{\prime })$ such that $f(M)\not \subset {\text {Supp}}(\sigma ).$ Due to $\omega +k\omega ^{\prime }>0$ and $\omega ^{\prime }>0,$ we see that $\log \|(s_{D}\otimes \sigma ^{k})\circ f\|^{2}$ and $\log \|\sigma \circ f\|^{2}$ are locally the difference of two pluri-subharmonic functions, where $D\in |L|$ . Thus,

$$ \begin{align*}\log\|s_{D}\circ f\|^{2}=\log\|(s_{D}\otimes \sigma^{k})\circ f\|^{2}-k\log\|\sigma\circ f\|^{2}\end{align*} $$

is locally the difference of two pluri-subharmonic functions. Hence, $m_{f}(r,D)$ can be defined.

We have the first main theorem (FMT).

Theorem 3.4 FMT

Assume that $f(o)\not \in D.$ Then

$$ \begin{align*}T_{f}(r,L)=m_{f}(r,D)+N_{f}(r,D)+O(1).\end{align*} $$

Proof. Since I is an indeterminacy set and $X_{t}$ meets I in probability 0, we may ignore $I.$ Set

$$ \begin{align*}T_{\lambda}=\inf\Big{\{}t>0: \sup_{s\in[0,t]}\log\frac{1}{\|s_{D}\circ f(X_{s})\|}>\lambda\Big{\}}.\end{align*} $$

Due to the definition of $T_{\lambda }, X_{t}$ does not hit ${\text {Supp}}f^{*}D$ when $0\leq t\leq \tau _{r}\wedge T_{\lambda }.$ By Dynkin’s formula, it follows that

(7) $$ \begin{align} & \mathbb E_{o}\left[\log\frac{1}{\|s_{D}\circ f(X_{\tau_{r}\wedge T_{\lambda}})\|}\right] \\ &= \frac{1}{2}\mathbb E_{o}\left[\int_{0}^{\tau_{r}\wedge T_{\lambda}}\Delta_{M} \log\frac{1}{\|s_{D}\circ f(X_{t})\|}dt\right]+\log\frac{1}{\|s_{D}\circ f(o)\|} \nonumber, \end{align} $$

where $\tau _{r}\wedge T_{\lambda }=\mathop {\mathrm {min}}\{\tau _{r}, T_{\lambda }\}.$ Note that $\Delta _{M}\log |\tilde {s}_{D}\circ f|=0$ outside $ f^{*}D$ . We see that

$$ \begin{align*}\Delta_{M} \log\frac{1}{\|s_{D}\circ f(X_{t})\|}=-\frac{1}{2}\Delta_{M}\log h\circ f(X_{t})\end{align*} $$

for $t\in [0,T_{\lambda }].$ Thus, (7) becomes

$$ \begin{align*} &\mathbb E_{o}\left[\log\frac{1}{\|s_{D}\circ f(X_{\tau_{r}\wedge T_{\lambda}})\|}\right] \\ &= -\frac{1}{4}\mathbb E_{o}\left[\int_{0}^{\tau_{r}\wedge T_{\lambda}}\Delta_{M}\log h\circ f(X_{t})dt\right] +O(1). \end{align*} $$

The monotone convergence theorem leads to

$$ \begin{align*} \frac{1}{4}\mathbb E_{o}\left[\int_{0}^{\tau_{r}\wedge T_{\lambda}}\Delta_{M}\log h\circ f(X_{t})dt\right] \rightarrow\frac{1}{2}\mathbb E_{o}\left[\int_{0}^{\tau_{r}}e_{f^{*}\omega}(X_{t})dt\right] =T_{f}(r,L) \end{align*} $$

as $\lambda \rightarrow \infty .$ We handle the first term in (7) and write it as two parts:

$$ \begin{align*} \mathrm{I}+\mathrm{II}&= \mathbb E_{o}\left[\log\frac{1}{\|s_{D}\circ f(X_{\tau_{r}})\|}: \tau_{r}<T_{\lambda}\right] + \mathbb E_{o}\left[\log\frac{1}{\|s_{D}\circ f(X_{T_{\lambda}})\|}: T_{\lambda}\leq\tau_{r} \right]. \end{align*} $$

Using the monotone convergence theorem again,

$$ \begin{align*} \mathrm{I}\rightarrow \mathbb E_{o}\left[\log\frac{1}{\|s_{D}\circ f(X_{\tau_{r}})\|}\right]= m_{f}(r,D) \end{align*} $$

as $\lambda \rightarrow \infty .$ Finally, we deal with $\mathrm {II}.$ By the definition of $T_{\lambda },$ we see that

$$ \begin{align*} \mathrm{II}&= \lambda\mathbb P_{o}\left(\sup_{t\in[0,\tau_{r}]}\log\frac{1}{\|s_{D}\circ f(X_{t})\|}>\lambda\right)\rightarrow N_{f}(r,D) \end{align*} $$

as $\lambda \rightarrow \infty .$ Putting the above together, we show the theorem.

4.1. Logarithmic derivative lemma

Let $(M,g)$ be an m-dimensional complete Kähler manifold and $\nabla _{M}$ be the gradient operator on M associated with $g.$ Let $X_{t}$ be the Brownian motion in M with generator $\Delta _{M}/2.$

Lemma 4.1 Calculus lemma, [Reference Atsuji1]

Let $k\geq 0$ be a locally integrable function on M such that it is locally bounded at $o\in M.$ Then for any $\delta>0,$ there exist a function $C(o, r, \delta )>0$ (independent of k) and a set $E_{\delta }\subset [0, \infty )$ of finite Lebesgue measure such that

(8) $$ \begin{align} \mathbb E_{o}\big{[}k(X_{\tau_{r}})\big{]}\leq C(o, r, \delta) \left(\mathbb E_{o}\left[\int_{0}^{\tau_{r}} k(X_{t})dt\right]\right)^{(1+\delta)^{2}} \end{align} $$

holds for $r>1$ outside $E_{\delta }.$

Let $\psi $ be a meromorphic function on $M.$ The norm of the gradient of $\psi $ is defined by

$$ \begin{align*}\|\nabla_{M}\psi\|^{2}=2\sum_{i,j=1}^{m}g^{i\overline j}\frac{\partial\psi}{\partial z_{i}}\overline{\frac{\partial \psi}{\partial z_{j}}},\end{align*} $$

where $(g^{i\overline {j}})$ is the inverse of $(g_{i\overline {j}}).$ Locally, we write $\psi =\psi _{1}/\psi _{0},$ where $\psi _{0},\psi _{1}$ are holomorphic functions so that ${\text {codim}}_{\mathbb C}(\psi _{0}=\psi _{1}=0)\geq 2$ if $\dim _{\mathbb C}M\geq 2.$ Identify $\psi $ with a meromorphic mapping into $\mathbb P^{1}(\mathbb C)$ by $x\mapsto [\psi _{0}(x):\psi _{1}(x)].$ The characteristic function of $\psi $ with respect to the Fubini–Study form $\omega _{FS}$ on $\mathbb P^{1}(\mathbb C)$ is defined by

$$ \begin{align*}T_{\psi}(r,\omega_{FS})=\frac{1}{4}\int_{B_{o}(r)}g_{r}(o,x)\Delta_{M}\log(|\psi_{0}(x)|^{2}+|\psi_{1}(x)|^{2})dV(x).\end{align*} $$

Let $i:\mathbb C\hookrightarrow \mathbb P^{1}(\mathbb C)$ be an inclusion defined by $z\mapsto [1:z].$ Via the pullback by $i,$ we have a (1,1)-form $i^{*}\omega _{FS}=dd^{c}\log (1+|\zeta |^{2})$ on $\mathbb C,$ where $\zeta :=w_{1}/w_{0}$ and $[w_{0}:w_{1}]$ is the homogeneous coordinate system of $\mathbb P^{1}(\mathbb C).$ The characteristic function of $\psi $ with respect to $i^{*}\omega _{FS}$ is defined by

$$ \begin{align*}\hat{T}_{\psi}(r,\omega_{FS}) = \frac{1}{4}\int_{B_{o}(r)}g_{r}(o,x)\Delta_{M}\log(1+|\psi(x)|^{2})dV(x).\end{align*} $$

Clearly,

$$ \begin{align*}\hat{T}_{\psi}(r,\omega_{FS})\leq T_{\psi}(r,\omega_{FS}).\end{align*} $$

We adopt the spherical distance $\|\cdot ,\cdot \|$ on $\mathbb P^{1}(\mathbb C)$ ; then the proximity function of $\psi $ with respect to $a\in \mathbb P^{1}(\mathbb C)=\mathbb C\cup \{\infty \}$ is defined by

$$ \begin{align*}\hat{m}_{\psi}(r,a)=\int_{S_{o}(r)}\log\frac{1}{\|\psi(x),a\|}d\pi_{o}^{r}(x).\end{align*} $$

Again, set

$$ \begin{align*}\hat{N}_{\psi}(r,a)=\frac{\pi^{m}}{(m-1)!}\int_{\psi^{-1}(a)\cap B_{o}(r)}g_{r}(o,x)\alpha^{m-1}.\end{align*} $$

Using similar arguments as in the proof of Theorem 3.4, we easily show that $\hat {T}_{\psi }(r,\omega _{FS})=\hat {m}_{\psi }(r,a)+\hat {N}_{\psi }(r,a)+O(1).$ We also define Nevanlinna’s characteristic function

$$ \begin{align*}T(r,\psi):=m(r,\psi)+N(r,\psi),\end{align*} $$

where

$$ \begin{align*} m(r,\psi)&=\int_{S_{o}(r)}\log^{+}|\psi(x)|d\pi^{r}_{o}(x), \\ N(r,\psi)&= \frac{\pi^{m}}{(m-1)!}\int_{\psi^{-1}(\infty)\cap B_{o}(r)}g_{r}(o,x)\alpha^{m-1}. \end{align*} $$

We have

(9) $$ \begin{align} T(r,\psi)=\hat{T}_{\psi}(r,\omega_{FS})+O(1), \ \ \ T\Big(r,\frac{1}{\psi-a}\Big)= T(r,\psi)+O(1). \end{align} $$

On $\mathbb P^{1}(\mathbb C),$ we take a singular metric

$$ \begin{align*}\Phi=\frac{1}{|\zeta|^{2}(1+\log^{2}|\zeta|)}\frac{\sqrt{-1}}{4\pi^{2}}d\zeta\wedge d\bar \zeta.\end{align*} $$

A direct computation shows that

(10) $$ \begin{align} \int_{\mathbb P^{1}(\mathbb C)}\Phi=1, \ \ \ 4m\pi\frac{\psi^{*}\Phi\wedge\alpha^{m-1}}{\alpha^{m}}=\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}. \end{align} $$

Set

$$ \begin{align*}T_{\psi}(r,\Phi)=\frac{1}{2}\int_{B_{o}(r)}g_{r}(o,x)e_{\psi^{*}\Phi}(x)dV(x).\end{align*} $$

Invoking (10), we obtain

(11) $$ \begin{align} T_{\psi}(r,\Phi)=\frac{1}{4\pi}\int_{B_{o}(r)}g_{r}(o,x)\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(x)dV(x). \end{align} $$

Lemma 4.2. We have

$$ \begin{align*}T_{\psi}(r,\Phi)\leq T(r,\psi)+O(1).\end{align*} $$

Proof. Using Fubini’s theorem,

$$ \begin{align*} T_{\psi}(r,\Phi)&=m\int_{B_{o}(r)}g_{r}(o,x)\frac{\psi^{*}\Phi\wedge\alpha^{m-1}}{\alpha^{m}}dV(x) \\ &=\frac{\pi^{m}}{(m-1)!}\int_{\mathbb P^{1}(\mathbb C)}\Phi(\zeta)\int_{\psi^{-1}(\zeta)\cap B_{o}(r)}g_{r}(o,x)\alpha^{m-1} \\ &=\int_{\zeta\in\mathbb P^{1}(\mathbb C)}N_{\psi}(r,\zeta)\Phi(\zeta) \\ &\leq\int_{\zeta\in\mathbb P^{1}(\mathbb C)}\big{(}T(r,\psi)+O(1)\big{)}\Phi(\zeta) \\ &= T(r,\psi)+O(1). \end{align*} $$

The proof is completed.

Lemma 4.3. Assume that $\psi (x)\not \equiv 0.$ For any $\delta>0,$ there are $C(o, r, \delta )>0$ independent of $\psi $ and $E_{\delta }\subset (1,\infty )$ of finite Lebesgue measure such that

$$ \begin{align*} \mathbb E_{o}\left[\log^{+}\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{\tau_{r}})\right] &\leq(1+\delta)^{2}\log T(r,\psi)+\log C(o, r, \delta) \end{align*} $$

holds for $r>1$ outside $E_{\delta }.$

Proof. By Jensen’s inequality, it is clear that

$$ \begin{align*} \mathbb E_{o}\left[\log^{+}\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{\tau_{r}})\right] &\leq \mathbb E_{o}\left[\log\Big{(}1+\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{\tau_{r}})\Big{)}\right] \nonumber \\ &\leq \log^{+}\mathbb E_{o}\left[\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{\tau_{r}})\right]+O(1). \nonumber \end{align*} $$

By Lemma 4.1 and the co-area formula, there is $C(o, r, \delta )>0$ such that

$$ \begin{align*} & \log^{+}\mathbb E_{o}\left[\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{\tau_{r}})\right] \\ &\leq (1+\delta)^{2}\log^{+}\mathbb E_{o}\left[\int_{0}^{\tau_{r}}\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{t})dt\right] +\log C(o, r, \delta) \nonumber \\ &\leq (1+\delta)^{2}\log T(r,\psi)+\log C(o, r, \delta)+O(1), \nonumber \end{align*} $$

where Lemma 4.2 and (11) are applied. Modify $C(o, r, \delta )$ such that the term $O(1)$ is removed; then we get the desired inequality.

Define

$$ \begin{align*}m\left(r,\frac{\|\nabla_{M}\psi\|}{|\psi|}\right)=\int_{S_{o}(r)}\log^{+}\frac{\|\nabla_{M}\psi\|}{|\psi|}(x)d\pi^{r}_{o}(x).\end{align*} $$

Now we prove Theorem 1.1.

Proof. On the one hand,

$$ \begin{align*} m\left(r,\frac{\|\nabla_{M}\psi\|}{|\psi|}\right) &\leq \frac{1}{2}\int_{S_{o}(r)}\log^{+}\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(x)d\pi^{r}_{o}(x) \\ & \quad +\frac{1}{2}\int_{S_{o}(r)}\log^{+}\big{(}1+\log^{2}|\psi(x)|\big{)}d\pi^{r}_{o}(x) \\ &= \frac{1}{2}\mathbb E_{o}\left[\log^{+}\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{\tau_{r}})\right] \\ & \quad +\frac{1}{2}\int_{S_{o}(r)}\log\big{(}1+\log^{2}|\psi(x)|\big{)}d\pi^{r}_{o}(x) \\ &\leq \frac{1}{2}\mathbb E_{o}\left[\log^{+}\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{\tau_{r}})\right] \\ & \quad +\frac{1}{2}\int_{S_{o}(r)}\log\Big{(}1+\big{(}\log^{+}|\psi(x)|+\log^{+}\frac{1}{|\psi(x)|}\big{)}^{2}\Big{)}d\pi^{r}_{o}(x) \\ &\leq \frac{1}{2}\mathbb E_{o}\left[\log^{+}\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{\tau_{r}})\right] \\ & \quad +\int_{S_{o}(r)}\log\Big{(}1+\log^{+}|\psi(x)|+\log^{+}\frac{1}{|\psi(x)|}\Big{)}d\pi^{r}_{o}(x). \end{align*} $$

Lemma 4.3 implies that

$$ \begin{align*} & \frac{1}{2}\mathbb E_{o}\left[\log^{+}\frac{\|\nabla_{M}\psi\|^{2}}{|\psi|^{2}(1+\log^{2}|\psi|)}(X_{\tau_{r}})\right] \\ &\leq \frac{(1+\delta)^{2}}{2}\log T(r,\psi)+\frac{1}{2}\log C(o, r, \delta)+O(1). \end{align*} $$

On the other hand, by Jensen’s inequality and (9),

$$ \begin{align*} & \int_{S_{o}(r)}\log\Big{(}1+\log^{+}|\psi(x)|+\log^{+}\frac{1}{|\psi(x)|}\Big{)}d\pi^{r}_{o}(x) \\ &\leq \log\int_{S_{o}(r)}\Big{(}1+\log^{+}|\psi(x)|+\log^{+}\frac{1}{|\psi(x)|}\Big{)}d\pi^{r}_{o}(x) \\ &\leq \log \big{(}m(r,\psi) +m(r,1/\psi)\big{)}+O(1) \\ &\leq \log T(r,\psi)+O(1). \end{align*} $$

Replacing $C(o, r, \delta )$ by $C^{2}(o, r, \delta )$ and combining the above, the theorem is proved.

4.2. Estimate of $C(o, r, \delta )$

Let M be a complete Kähler manifold of nonpositive sectional curvature. Indeed, we let $\kappa $ be defined by (1). Clearly, $\kappa $ is a nonpositive, nonincreasing and continuous function on $[0,\infty ).$ Treat the ordinary differential equation

(12) $$ \begin{align} G^{\prime\prime}(t)+\kappa(t)G(t)=0;\ \ \ G(0)=0, \ \ G^{\prime}(0)=1 \end{align} $$

on $[0,\infty ).$ Now compare (12) with $y^{\prime \prime }(t)+\kappa (0)y(t)=0$ under the same initial conditions; we see that G can be estimated simply as

$$ \begin{align*}G(t)=t \ \ \text{for} \ \kappa\equiv0; \ \ \ G(t)\geq t \ \ \text{for} \ \kappa\not\equiv0.\end{align*} $$

This follows that

(13) $$ \begin{align} G(r)\geq r \ \ \text{for} \ r\geq0; \ \ \ \int_{1}^{r}\frac{dt}{G(t)}\leq\log r \ \ \text{for} \ r\geq1. \end{align} $$

On the other hand, we rewrite (12) as the form

$$ \begin{align*}\log^{\prime}G(t)\cdot\log^{\prime}G^{\prime}(t)=-\kappa(t).\end{align*} $$

Since $G(t)\geq t$ is increasing, the decrease and nonpositivity of $\kappa $ imply that for each fixed $t, G$ must satisfy one of the following two inequalities:

$$ \begin{align*}\log^{\prime}G(t)\leq\sqrt{-\kappa(t)} \ \ \text{for} \ t>0; \ \ \ \log^{\prime}G^{\prime}(t)\leq\sqrt{-\kappa(t)} \ \ \text{for} \ t\geq0.\end{align*} $$

By virtue of $G(t)\rightarrow 0$ as $t\rightarrow 0,$ by integration, G is bounded from above by

(14) $$ \begin{align} G(r)\leq r\exp\big(r\sqrt{-\kappa(r)}\big) \ \ \text{for} \ r\geq0. \end{align} $$

In what follows, one assumes that M is simply connected. The purpose of this section is to show the following LDL by estimating $C(o, r, \delta ).$

Theorem 4.4 LDL

Let $\psi $ be a nonconstant meromorphic function on M. Then

$$ \begin{align*} m\left(r,\frac{\|\nabla_{M}\psi\|}{|\psi|}\right)&\leq \Big{(}1+\frac{(1+\delta)^{2}}{2}\Big{)}\log T(r,\psi) +O\Big{(}r\sqrt{-\kappa(r)}+\delta\log r\Big{)} \ \big{\|}, \end{align*} $$

where $\kappa $ is defined by $(1).$

We need some lemmas.

Lemma 4.5 [Reference Atsuji4]

Let $\eta>0$ be a number. Then there is a constant $C>0$ such that

$$ \begin{align*}g_{r}(o,x)\int_{\eta}^{r}G^{1-2m}(t)dt\geq C\int_{r(x)}^{r}G^{1-2m}(t)dt\end{align*} $$

holds for $r>\eta $ and $x\in B_{o}(r)\setminus \overline {B_{o}(\eta )},$ where G is defined by (12).

Lemma 4.6 [Reference Debiard, Gaveau and Mazet10, Reference Hsu16]

Let M be a simply connected, nonpositively curved and complete Hermitian manifold of complex dimension m. Then

$$ \begin{align*} (i) \, \ g_{r}(o,x)\leq\left\{ \begin{array}{l} \frac{1}{\pi}\log\frac{r}{r(x)},\kern100pt m=1 \\ \frac{1}{(m-1)\omega_{2m-1}}\big{(}r^{2-2m}(x)-r^{2-2m}\big{)},\ m\geq2 \\ \end{array} \right.; \ \ \ \ \ \ \ \ \ \ \ \end{align*} $$

$$ \begin{align*}(ii) \ \ \ d\pi^{r}_{o}(x)\leq\frac{1}{\omega_{2m-1}r^{2m-1}}d\sigma_{r}(x), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{align*} $$

where $g_{r}(o,x)$ denotes the Green function of $\Delta _{M}/2$ for $B_{o}(r)$ with Dirichlet boundary condition and a pole at $o,$ $d\pi _{o}^{r}(x)$ is the harmonic measure on $S_{o}(r)$ with respect to $o, \omega _{2m-1}$ is the Euclidean volume of a unit sphere in $\mathbb R^{2m}$ and $d\sigma _{r}(x)$ is the induced Riemannian volume element on $S_{o}(r).$

Lemma 4.7 Borel lemma, [Reference Ru23]

Let T be a strictly positive nondecreasing function of $\mathscr {C}^{1}$ -class on $(0,\infty ).$ Let $\gamma>0$ be a number such that $T(\gamma )\geq e$ and $\phi $ be a strictly positive nondecreasing function such that

$$ \begin{align*}c_{\phi}=\int_{e}^{\infty}\frac{1}{t\phi(t)}dt<\infty.\end{align*} $$

Then, the inequality

$$ \begin{align*}T^{\prime}(r)\leq T(r)\phi(T(r))\end{align*} $$

holds for $r\geq \gamma $ outside a set of Lebesgue measure not exceeding $c_{\phi }.$ Particularly, take $\phi (T)=T^{\delta }$ for a number $\delta>0$ ; then we have $T^{\prime }(r)\leq T^{1+\delta }(r)$ holds for $r>0$ outside a set $E_{\delta }\subset (0,\infty )$ of finite Lebesgue measure.

Now we prove the following so-called calculus lemma (see also [Reference Atsuji4]) which gives an estimate of $C(o,r,\delta ).$

Lemma 4.8 Calculus lemma

Let $k\geq 0$ be a locally integrable function on M such that it is locally bounded at $o\in M.$ Then for any $\delta>0,$ there is a constant $C>0$ independent of $k,\delta $ and a set $E_{\delta }\subset (1,\infty )$ of finite Lebesgue measure such that

$$ \begin{align*}\mathbb E_{o}[k(X_{\tau_{r}})]\leq \frac{C^{(1+\delta)^{2}} \log^{(1+\delta)^{2}}r}{r^{(1-2m)\delta}e^{(1-2m)(1+\delta)r\sqrt{-\kappa(r)}}}\left(\mathbb E_{o}\left[\int_{0}^{\tau_{r}}k(X_{t})dt\right]\right)^{(1+\delta)^{2}}\end{align*} $$

holds for $r>1$ outside $E_{\delta },$ where $\kappa $ is defined by $(1).$

Proof. By Lemma 4.5 and Lemma 4.6 with (13), we get

$$ \begin{align*} \mathbb E_{o}\left[\int_{0}^{\tau_{r}}k(X_{t})dt\right]&= \int_{B_{o}(r)}g_{r}(o,x)k(x)dV(x) \\ &=\int_{0}^{r}dt\int_{S_{o}(t)}g_{r}(o,x)k(x)d\sigma_{t}(x) \\ &\geq C_{0}\int_{0}^{r}\frac{\int_{t}^{r}G^{1-2m}(s)ds}{\int_{1}^{r}G^{1-2m}(s)ds}dt\int_{S_{o}(t)}k(x)d\sigma_{t}(x) \\ &= \frac{C_{0}}{\log r}\int_{0}^{r}dt\int_{t}^{r}G^{1-2m}(s)ds\int_{S_{o}(t)}k(x)d\sigma_{t}(x) \end{align*} $$

and

$$ \begin{align*}\mathbb E_{o}\big{[}k(X_{\tau_{r}})\big{]}=\int_{S_{o}(r)}k(x)d\pi_{o}^{r}(x)\leq\frac{1}{\omega_{2m-1}r^{2m-1}}\int_{S_{o}(r)}k(x)d\sigma_{r}(x),\end{align*} $$

where $\omega _{2m-1}$ denotes the Euclidean volume of a unit sphere in $\mathbb R^{2m}$ and $d\sigma _{r}$ is the induced volume measure on $S_{o}(r).$ Hence,

$$ \begin{align*}\mathbb E_{o}\left[\int_{0}^{\tau_{r}}k(X_{t})dt\right]\geq\frac{C_{0}}{\log r}\int_{0}^{r}dt\int_{t}^{r}G^{1-2m}(s)ds\int_{S_{o}(t)}k(x)d\sigma_{t}(x)\end{align*} $$

and

(15) $$ \begin{align} \mathbb E_{o}\big{[}k(X_{\tau_{r}})\big{]}\leq\frac{1}{\omega_{2m-1}r^{2m-1}}\int_{S_{o}(r)}k(x)d\sigma_{r}(x). \end{align} $$

Put

$$ \begin{align*}\Gamma(r)=\int_{0}^{r}dt\int_{t}^{r}G^{1-2m}(s)ds\int_{S_{o}(t)}k(x)d\sigma_{t}(x).\end{align*} $$

Then

(16) $$ \begin{align} \Gamma(r)\leq\frac{\log r}{C_{0}}\mathbb E_{o}\left[\int_{0}^{\tau_{r}}k(X_{t})dt\right]. \end{align} $$

A simple computation shows that

$$ \begin{align*}\Gamma^{\prime}(r)=G^{1-2m}(r)\int_{0}^{r}dt\int_{S_{o}(t)}k(x)d\sigma_{t}(x).\end{align*} $$

By this with (15),

(17) $$ \begin{align} \mathbb E_{o}\big{[}k(X_{\tau_{r}})\big{]}\leq\frac{1}{\omega_{2m-1}r^{2m-1}}\frac{d}{dr}\left(\frac{\Gamma^{\prime}(r)}{G^{1-2m}(r)}\right). \end{align} $$

Using Lemma 4.7 twice, for any $\delta>0$ we have

(18) $$ \begin{align} \frac{d}{dr}\left(\frac{\Gamma^{\prime}(r)}{G^{1-2m}(r)}\right)\leq\frac{\Gamma^{(1+\delta)^{2}}(r)}{G^{(1-2m)(1+\delta)}(r)} \end{align} $$

holds outside a set $E_{\delta }\subset (1,\infty )$ of finite Lebesgue measure. Using (16)–(18) and (14), it is not hard to conclude that

$$ \begin{align*}\mathbb E_{o}\big{[}k(X_{\tau_{r}})\big{]}\leq \frac{C^{(1+\delta)^{2}} \log^{(1+\delta)^{2}}r}{r^{(1-2m)\delta}e^{(1-2m)(1+\delta)r\sqrt{-\kappa(r)}}}\left(\mathbb E_{o}\left[\int_{0}^{\tau_{r}}k(X_{t})dt\right]\right)^{(1+\delta)^{2}}\end{align*} $$

with $C=1/C_{0}>0$ being a constant independent of $k,\delta .$

Lemma 4.8 implies an estimate

$$ \begin{align*} C(o, r, \delta)\leq \frac{C^{(1+\delta)^{2}} \log^{(1+\delta)^{2}}r}{r^{(1-2m)\delta}e^{(1-2m)(1+\delta)r\sqrt{-\kappa(r)}}}.\end{align*} $$

Thus, we get

(19) $$ \begin{align} \log C(o, r, \delta)\leq O\Big{(}r\sqrt{-\kappa(r)}+\delta\log r\Big{)}. \end{align} $$

We prove Theorem 4.4.

5.1. Meromorphic mappings into $\mathbb P^{n}(\mathbb C)$

Let $\psi :M\rightarrow \mathbb P^{n}(\mathbb C)$ be a meromorphic mapping from Kähler manifold M into $\mathbb P^{n}(\mathbb C)$ ; that is, there is an open covering $\{U_{\alpha }\}$ of M such that $\psi $ has a local representation $[\psi _{0}^{\alpha }:\cdots :\psi _{n}^{\alpha }]$ on each $U_{\alpha },$ where $\psi ^{\alpha }_{0},\cdots ,\psi ^{\alpha }_{n}$ are holomorphic functions on $U_{\alpha }$ satisfying

$$ \begin{align*}{\text{codim}}_{\mathbb C}(\psi^{\alpha}_{0}=\cdots=\psi^{\alpha}_{n}=0)\geq2.\end{align*} $$

Let $[w_{0}:\cdots :w_{n}]$ denote the homogeneous coordinate of $\mathbb P^{n}(\mathbb C).$ Assume that $w_{0}\circ \psi \not \equiv 0.$ Let $i:\mathbb C^{n}\hookrightarrow \mathbb P^{n}(\mathbb C)$ be an inclusion given by $(z_{1},\cdots ,z_{n})\mapsto [1:z_{1}:\cdots :z_{n}].$ Clearly, $\omega _{FS}$ induces a (1,1)-form $i^{*}\omega _{FS}=dd^{c}\log (|\zeta _{0}|^{2}+|\zeta _{1}|^{2}+\cdots +|\zeta _{n}|^{2})$ on $\mathbb C^{n},$ where $\zeta _{j}:=w_{j}/w_{0}$ for $0\leq j\leq n.$ The characteristic function of $\psi $ with respect to $i^{*}\omega _{FS}$ is well defined by

$$ \begin{align*} \hat{T}_{\psi}(r,\omega_{FS}) &= \frac{1}{4}\int_{B_{o}(r)}g_{r}(o,x)\Delta_{M}\log\Big{(}\sum_{j=0}^{n}|\zeta_{j}\circ \psi(x)|^{2}\Big{)}dV(x). \end{align*} $$

Clearly,

$$ \begin{align*} \hat{T}_{\psi}(r,\omega_{FS})\leq\frac{1}{4}\int_{B_{o}(r)}g_{r}(o,x)\Delta_{M}\log\|\psi(x)\|^{2}dV(x)=T_{\psi}(r,\omega_{FS}).\end{align*} $$

The co-area formula leads to

$$ \begin{align*} \hat{T}_{\psi}(r,\omega_{FS})=\frac{1}{4}\mathbb E_{o}\Big{[}\int_{0}^{\tau_{r}}\Delta_{M}\log\Big{(}\sum_{j=0}^{n}|\zeta_{j}\circ\psi(X_{t})|^{2}\Big{)}dt\Big{]}. \end{align*} $$

Note that the pole divisor of $\zeta _{j}\circ \psi $ is pluripolar. By Dynkin’s formula,

$$ \begin{align*} \hat{T}_{\psi}(r,\omega_{FS})&=\frac{1}{2}\int_{S_{o}(r)}\log\Big{(}\sum_{j=0}^{n}|\zeta_{j}\circ\psi(x)|^{2}\Big{)}d\pi^{r}_{o}(x) -\frac{1}{2}\log\Big{(}\sum_{j=0}^{n}|\zeta_{j}\circ\psi(o)|^{2}\Big{)}, \\ \hat{T}_{\zeta_{j}\circ\psi}(r, \omega_{FS}) &=\frac{1}{2}\int_{S_{o}(r)}\log\big{(}1+|\zeta_{j}\circ\psi(x)|^{2}\big{)}d\pi^{r}_{o}(x)-\frac{1}{2}\log\big{(}1+|\zeta_{j}\circ\psi(o)|^{2}\big{)}. \end{align*} $$

Theorem 5.1. We have

$$ \begin{align*}\max_{1\leq j\leq n}T(r,\zeta_{j}\circ\psi)+O(1)\leq \hat{T}_{\psi}(r,\omega_{FS})\leq\sum_{j=1}^{n}T(r,\zeta_{j}\circ\psi)+O(1).\end{align*} $$

Proof. On the one hand,

$$ \begin{align*} \hat{T}_{\psi}(r,\omega_{FS}) &\leq \frac{1}{2}\sum_{j=1}^{n}\Big{(}\int_{S_{o}(r)}\log\big{(}1+|\zeta_{j}\circ\psi(x)|^{2}\big{)}d\pi^{r}_{o}(x)-\log\big{(}1+|\zeta_{j}\circ\psi(o)|^{2}\big{)}\Big{)}+O(1)\\ &= \sum_{j=1}^{n}T(r,\zeta_{j}\circ\psi)+O(1). \end{align*} $$

On the other hand,

$$ \begin{align*} T(r,\zeta_{j}\circ \psi) &=\hat{T}_{\zeta_{j}\circ\psi}(r,\omega_{FS})+O(1) \\ &\leq \frac{1}{4}\int_{B_{o}(r)}g_{r}(o,x)\Delta_{M}\log\Big{(}\sum_{j=0}^{n}|\zeta_{j}\circ\psi(x)|^{2}\Big{)}dV(x)+O(1) \\ &= \hat{T}_{\psi}(r,\omega_{FS})+O(1). \end{align*} $$

We conclude the proof.

Corollary 5.2. We have

$$ \begin{align*}\max_{1\leq j\leq n}T(r,\zeta_{j}\circ\psi)\leq T_{\psi}(r,\omega_{FS})+O(1).\end{align*} $$

Let V be a complex projective algebraic variety and $\mathbb {C}(V)$ be the field of rational functions defined on V over $\mathbb C.$ Let $V\hookrightarrow \mathbb P^{N}(\mathbb C)$ be a holomorphic embedding and $H_{V}$ be the restriction of the hyperplane line bundle H over $\mathbb P^{N}(\mathbb C)$ to $V.$ Denote by $[w_{0}:\cdots :w_{N}]$ the homogeneous coordinate system of $\mathbb P^{N}(\mathbb C)$ and assume that $w_{0}\neq 0$ without loss of generality. Notice that the restriction $\{\zeta _{j}:=w_{j}/w_{0}\}$ to V gives a transcendental base of $\mathbb {C}(V).$ Hence, any $\phi \in \mathbb C(V)$ can be represented by a rational function in $\zeta _{1},\cdots ,\zeta _{N}$ ,

$$ \begin{align*} \phi=Q(\zeta_{1},\cdots,\zeta_{N}). \end{align*} $$

Theorem 5.3. Let $f:M\rightarrow V$ be an algebraically nondegenerate meromorphic mapping. Then for $\phi \in \mathbb C(V),$ there exists a constant $C>0$ such that

$$ \begin{align*} T(r,\phi\circ f)\leq CT_{f}(r,H_{V})+O(1).\end{align*} $$

Corollary 5.4. Let $f:M\rightarrow V$ be an algebraically nondegenerate meromorphic mapping. Fix a positive $(1,1)$ -form $\omega $ on $V.$ Then for any $\phi \in \mathbb C(V),$ there is a constant $C>0$ such that

$$ \begin{align*}T(r,\phi\circ f)\leq CT_{f}(r,\omega)+O(1).\end{align*} $$

5.2. Estimate of $\mathbb E_{o}[\tau_{r}]$

Now we assume M is a simply connected complete Kähler manifold of nonpositive sectional curvature, and let $X_{t}$ be the Brownian motion in M with generator $\Delta _{M}/2$ started at $o.$ Recall that $\dim _{\mathbb C}M=m, \tau _{r}=\inf \{t>0:X_{t}\not \in B_{o}(r)\}.$

Lemma 5.5. We have

$$ \begin{align*}\mathbb E_{o}\big{[}\tau_{r}\big{]}\leq\frac{2r^{2}}{2m-1}.\end{align*} $$

Proof. The argument follows essentially from Atsuji [Reference Atsuji4], but here we provide a simpler proof albeit a rougher estimate. We refer the reader to [Reference Atsuji4] for a better estimate that $\mathbb E_{o}[\tau _{r}]\leq r^{2}/2m.$ Let $X_{t}$ be the Brownian motion in M started at $o\not =o_{1},$ where $o_{1}\in B_{o}(r).$ Let $r_{1}(x)$ be the distance function of x from $o_{1}.$ Apply Itô’s formula to $r_{1}(x)$ ,

(20) $$ \begin{align} r_{1}(X_{t})-r_{1}(X_{0})=B_{t}-L_{t}+\frac{1}{2}\int_{0}^{t}\Delta_{M}r_{1}(X_{s})ds, \end{align} $$

where $B_{t}$ is the standard Brownian motion in $\mathbb R$ and $L_{t}$ is a local time on the cut locus of $o,$ an increasing process which increases only at the cut loci of $o.$ Since M is simply connected and nonpositively curved,

$$ \begin{align*}\Delta_{M}r_{1}(x)\geq\frac{2m-1}{r_{1}(x)}, \ \ L_{t}\equiv0.\end{align*} $$

By (20), we arrive at

$$ \begin{align*}r_{1}(X_{t})\geq B_{t}+\frac{2m-1}{2}\int_{0}^{t}\frac{ds}{r_{1}(X_{s})}.\end{align*} $$

Let $t=\tau _{r}$ and take expectation on both sides of the above inequality; then it yields that

$$ \begin{align*}\max_{x\in S_{o}(r)} r_{1}(x)\geq \frac{(2m-1)\mathbb E_{o}[\tau_{r}]}{2\max_{x\in S_{o}(r)} r_{1}(x)}.\end{align*} $$

Let $o^{\prime }\rightarrow o,$ and we are led to the conclusion.

5.3. Second main theorem

Let M be a complete Kähler manifold of nonpositive sectional curvature. Consider the (analytic) universal covering

$$ \begin{align*}\pi:\tilde{M}\rightarrow M.\end{align*} $$

Via the pullback by $\pi , \tilde {M}$ can be equipped with the induced metric from the metric of $M.$ So, under this metric, $\tilde {M}$ becomes a simply connected complete Kähler manifold of nonpositive sectional curvature. Take a diffusion process $\tilde {X}_{t}$ in $\tilde {M}$ such that $X_{t}=\pi (\tilde {X}_{t}),$ where $X_{t}$ is the Brownian motion started at $o\in M$ . Then $\tilde {X}_{t}$ is a Brownian motion generated by $\Delta _{\tilde {M}}/2$ induced from the pullback metric. Let $\tilde {X}_{t}$ start at $\tilde {o}\in \tilde {M}$ with $o=\pi (\tilde {o}).$ Then

$$ \begin{align*}\mathbb E_{o}[\phi(X_{t})]=\mathbb E_{\tilde{o}}\big{[}\phi\circ\pi(\tilde{X}_{t})\big{]}\end{align*} $$

for $\phi \in \mathscr {C}_{\flat }(M).$ Set

$$ \begin{align*}\tilde{\tau}_{r}=\inf\big{\{}t>0: \tilde{X}_{t}\not\in B_{\tilde{o}}(r)\big{\}},\end{align*} $$

where $B_{\tilde {o}}(r)$ is a geodesic ball centred at $\tilde {o}$ with radius r in $\tilde {M}.$ If necessary, one can extend the filtration in probability space where $(X_{t},\mathbb P_{o})$ are defined so that $\tilde {\tau }_{r}$ is a stopping time with respect to a filtration where the stochastic calculus of $X_{t}$ works. By the above arguments, we may assume M is simply connected by lifting f to the universal covering.

Let V be a complex projective algebraic manifold with complex dimension $n\leq m=\dim _{\mathbb C}M,$ and let $L\rightarrow V$ be a holomorphic line bundle. Let a divisor $D\in |L|$ be of simple normal crossing type; then one can express $D=\sum _{j=1}^{q}D_{j}$ as the union of irreducible components. Equip $L_{D_{j}}$ with a Hermitian metric which then induces a natural Hermitian metric h on $L=\otimes _{j=1}^{q}L_{D_{j}}.$ Fix a Hermitian metric form $\omega $ on V, which gives a (smooth) volume form $\Omega :=\omega ^{n}$ on $V.$ Pick $s_{j}\in H^{0}(V,L_{D_{j}})$ with $(s_{j})=D_{j}$ and $\|s_{j}\|<1.$ On V, one defines a singular volume form

(21) $$ \begin{align} \Phi=\frac{\Omega}{\prod_{j=1}^{q}\|s_{j}\|^{2}}. \end{align} $$

Set

$$ \begin{align*}\xi\alpha^{m}=f^{*}\Phi\wedge\alpha^{m-n}.\end{align*} $$

Note that

$$ \begin{align*}\alpha^{m}=m!\det(g_{i\bar j})\bigwedge_{j=1}^{m}\frac{\sqrt{-1}}{\pi}dz_{j}\wedge d\bar z_{j}.\end{align*} $$

A direct computation leads to

$$ \begin{align*}dd^{c}\big[\log\xi\big]\geq f^{*}c_{1}(L,h)-f^{*}{\text{Ric}} \Omega+\mathscr{R}_{M}-{\text{Supp}}f^{*}D\end{align*} $$

in the sense of currents, where $\mathscr {R}_{M}=-dd^{c}\log \det (g_{i\bar j}).$ This follows that

(22) $$ \begin{align} &\frac{1}{4}\int_{B_{o}(r)}g_{r}(o,x)\Delta_{M}\log\xi(x) dV(x) \\ &\geq T_{f}(r,L)+T_{f}(r,K_{V})+T(r,\mathscr{R}_{M})-\overline{N}_{f}(r,D)+O(1). \nonumber \end{align} $$

We now prove Theorem 1.2.

Proof. By Ru–Wong’s arguments (see [Reference Ru23], pp. 231–233), the simple normal crossing type of D implies that there exists a finite open covering $\{U_{\lambda }\}$ of V together with rational functions $w_{\lambda 1},\cdots ,w_{\lambda n}$ on V for $\lambda $ such that $w_{\lambda 1},\cdots $ are holomorphic on $U_{\lambda }$ as well as

$$ \begin{align*} dw_{\lambda1}\wedge\cdots\wedge dw_{\lambda n}(y)\neq0, & \ {}^{\forall} y\in U_{\lambda}, \\ D\cap U_{\lambda}=\big{\{}w_{\lambda1}\cdots w_{\lambda h_{\lambda}}=0\big{\}}, & \ {}^{\exists} h_{\lambda}\leq n. \end{align*} $$

In addition, we can require $L_{D_{j}}|_{U_{\lambda }}\cong U_{\lambda }\times \mathbb C$ for $\lambda ,j.$ On $U_{\lambda },$ we get

$$ \begin{align*}\Phi=\frac{e_{\lambda}}{|w_{\lambda1}|^{2}\cdots|w_{\lambda h_{\lambda}}|^{2}} \bigwedge_{k=1}^{n}\frac{\sqrt{-1}}{2\pi}dw_{\lambda k}\wedge d\bar w_{\lambda k},\end{align*} $$

where $\Phi $ is given by (21) and $e_{\lambda }$ is a smooth positive function. Let $\{\phi _{\lambda }\}$ be a partition of unity subordinate to $\{U_{\lambda }\}$ ; then $\phi _{\lambda } e_{\lambda }$ is bounded on $V.$ Set

$$ \begin{align*}\Phi_{\lambda}=\frac{\phi_{\lambda} e_{\lambda}}{|w_{\lambda1}|^{2}\cdots|w_{\lambda h_{\lambda}}|^{2}} \bigwedge_{k=1}^{n}\frac{\sqrt{-1}}{2\pi}dw_{\lambda k}\wedge d\bar w_{\lambda k}.\end{align*} $$

Put $f_{\lambda k}=w_{\lambda k}\circ f$ ; then on $f^{-1}(U_{\lambda })$ we obtain

(23) $$ \begin{align} f^{*}\Phi_{\lambda}= \frac{\phi_{\lambda}\circ f\cdot e_{\lambda}\circ f}{|f_{\lambda1}|^{2}\cdots|f_{\lambda h_{\lambda}}|^{2}} \bigwedge_{k=1}^{n}\frac{\sqrt{-1}}{2\pi}df_{\lambda k}\wedge d\bar f_{\lambda k}. \end{align} $$

Set

$$ \begin{align*}f^{*}\Phi\wedge\alpha^{m-n}=\xi\alpha^{m}, \ \ \ f^{*}\Phi_{\lambda}\wedge\alpha^{m-n}=\xi_{\lambda}\alpha^{m}\end{align*} $$

which arrives at (22). Clearly, we have $\xi =\sum _{\lambda } \xi _{\lambda }.$ Again, set

(24) $$ \begin{align} f^{*}\omega\wedge\alpha^{m-1}=\varrho\alpha^{m} \end{align} $$

which follows that

(25) $$ \begin{align} \varrho=\frac{1}{2m}e_{f^{*}\omega}. \end{align} $$

For each $\lambda $ and any $x\in f^{-1}(U_{\lambda }),$ take a local holomorphic coordinate system z around $x.$ Since $\phi _{\lambda }\circ f\cdot e_{\lambda }\circ f$ is bounded, it is not very hard to see from (23) and (24) that $\xi _{\lambda }$ is bounded from above by $P_{\lambda },$ where $P_{\lambda }$ is a polynomial in

$$ \begin{align*}\varrho, \ \ g^{i\bar{j}}\frac{\partial f_{\lambda k}}{\partial z_{i}}\overline{\frac{\partial f_{\lambda k}}{\partial z_{j}}}\Big{/}|f_{\lambda k}|^{2}, \ \ 1\leq i, j\leq m, \ 1\leq k\leq n.\end{align*} $$

This yields that

(26) $$ \begin{align} \log^{+}\xi_{\lambda}\leq O\Big{(}\log^{+}\varrho+\sum_{k}\log^{+}\frac{\|\nabla_{M} f_{\lambda k}\|}{|f_{\lambda k}|}\Big{)}+O(1). \end{align} $$

Thus, we conclude that

(27) $$ \begin{align} \log^{+}\xi \leq O\Big{(}\log^{+}\varrho+\sum_{k, \lambda}\log^{+}\frac{\|\nabla_{M} f_{\lambda k}\|}{|f_{\lambda k}|}\Big{)}+O(1) \end{align} $$

on $M.$ On the one hand,

$$ \begin{align*} \frac{1}{4}\int_{B_{o}(r)}g_{r}(o,x)\Delta_{M}\log\xi(x) dV(x) &=\frac{1}{2}\mathbb E_{o}\big{[}\log\xi(X_{\tau_{r}})\big{]}+O(1) \end{align*} $$

due to the co-area formula and Dynkin’s formula. Hence, by (22) we have

(28) $$ \begin{align} &\frac{1}{2}\mathbb E_{o}\big{[}\log\xi(X_{\tau_{r}})\big{]} \\ &\geq T_{f}(r,L)+T_{f}(r,K_{V})+T(r,\mathscr{R}_{M}) -\overline{N}_{f}(r,D) +O(1). \nonumber \end{align} $$

On the other hand, since $f_{\lambda k}$ is the pullback of rational function $w_{\lambda k}$ on V by f, Corollary 5.4 implies that

(29) $$ \begin{align} T(r,f_{\lambda k})\leq O(T_{f}(r,\omega))+O(1). \end{align} $$

Using (26) and (29) with Theorem 1.1,

$$ \begin{align*} & \frac{1}{2}\mathbb E_{o}\big{[}\log\xi(X_{\tau_{r}})\big{]} \\ &\leq O\Big{(}\sum_{k,\lambda}\mathbb E_{o}\left[\log^{+}\frac{\|\nabla_{M} f_{\lambda k}\|}{|f_{\lambda k}|}(X_{\tau_{r}})\right]\Big{)}+O\big{(}\mathbb E_{o}\left[\log^{+}\varrho(X_{\tau_{r}})\right]\big{)}+O(1) \\ &\leq O\Big{(}\sum_{k,\lambda}m\Big{(}r,\frac{\|\nabla_{M} f_{\lambda k}\|}{|f_{\lambda k}|}\Big{)}\Big{)}+O\big{(}\log^{+}\mathbb E_{o}\left[\varrho(X_{\tau_{r}})\right]\big{)}+O(1) \\ &\leq O\Big{(}\sum_{k,\lambda}\log T(r,f_{\lambda k})+\log C(o,r,\delta)\Big{)} +O\big{(}\log^{+}\mathbb E_{o}\big{[}\varrho(X_{\tau_{r}})\big{]}\big{)} \\ &\leq O\big{(}\log T_{f}(r,\omega)+\log C(o,r,\delta)\big{)}+O\big{(}\log^{+}\mathbb E_{o}\big{[}\varrho(X_{\tau_{r}})\big{]}\big{)}. \end{align*} $$

In the meanwhile, Lemma 4.1 and (25) imply

$$ \begin{align*} \log^{+}\mathbb E_{o}\big{[}\varrho(X_{\tau_{r}})\big{]} &\leq (1+\delta)^{2}\log^{+}\mathbb E_{o}\left[\int_{0}^{\tau_{r}}\varrho(X_{t})dt\right]+\log C(o,r,\delta) \\ &= \frac{(1+\delta)^{2}}{2m}\log^{+}\mathbb E_{o}\left[\int_{0}^{\tau_{r}}e_{f^{*}\omega}(X_{t})dt\right]+\log C(o,r,\delta) \\ &\leq \frac{(1+\delta)^{2}}{m}\log T_{f}(r,\omega)+\log C(o,r,\delta). \end{align*} $$

By this with (28), we prove the theorem.

We proceed to prove Theorem 1.3.

Lemma 5.6. Let $\kappa $ be defined by $(1).$ If M is nonpositively curved, then

$$ \begin{align*}T(r,\mathscr{R}_{M})\geq m\kappa(r)r^{2}.\end{align*} $$

Proof. Lemma 2.2 implies that $0\geq s_{M}\geq mR_{M}.$ By the co-area formula,

$$ \begin{align*} T(r,\mathscr{R}_{M})&=-{1\over4}\mathbb E_{o}\left[\int_{0}^{\tau_{r}}\Delta_{M}\log\det(g_{i\bar{j}}(X_{t}))dt\right] \\ &=\frac{1}{2}\mathbb E_{o}\left[\int_{0}^{\tau_{r}}s_{M}(X_{t})dt\right] \geq \frac{1}{2}m\mathbb E_{o}\left[\int_{0}^{\tau_{r}}R_{M}(X_{t})dt\right] \\ &\geq \frac{m(2m-1)}{2}\kappa(r)\mathbb E_{o}[\tau_{r}]. \end{align*} $$

Since $\mathbb E_{o}[\tau _{r}]\leq 2r^{2}/(2m-1)$ by Lemma 5.5, we prove the lemma.

Corollary 5.7 Carlson–Griffiths–King, [Reference Carlson and Griffiths7, Reference Griffiths and King14]; Noguchi, [Reference Noguchi19]

Let $f:\mathbb C^{m}\rightarrow V$ be a differentiably nondegenerate meromorphic mapping with $\dim _{\mathbb C} V\leq m.$ Let D be a divisor of simple normal crossing type, where L is a holomorphic line bundle over V. Fix a Hermitian metric $\omega $ on $V.$ Then

$$ \begin{align*} T_{f}(r,L)+T_{f}(r,K_{V}) \leq \overline{N}_{f}(r,D)+O\big{(}\log T_{f}(r,\omega)+\delta\log r\big{)} \ \big{\|}. \end{align*} $$