Lemma 10.6.1DC3C9650BBBE

Proof. We make the metric flat on $B$ and obtain the frame by parallel transport along the segments issuing from the center of $B$. Choose a smooth metric $h_0$ on $L$ (lemma 6.14.10(1)). By lemma 6.14.2 there is a smooth real function $p$ on $B$ with $\mathrm{dd}^{\mathrm{c}}p=c_1(L,h_0)$, and the metric $h_1=h_0\mathrm e^{p}$ on $L|_ B$ has curvature $c_1(L,h_0)-\mathrm{dd}^{\mathrm{c}}p=0$, by lemma 6.14.10(3). We call a holomorphic frame of $h_1$-length one over an open subset of $B$ a unit frame. Unit frames exist near every point of $B$. Indeed, every point of $B$ has a ball neighborhood $V\subset B$ over which $L$ has a holomorphic frame $e$, and $\log |e|_{h_1}^2$ is smooth and pluriharmonic, since $\mathrm{dd}^{\mathrm{c}}\log |e|_{h_1}^2=-c_1(L,h_1)=0$, so by lemma 6.14.2 it equals $\operatorname{Re}F$ for a holomorphic function $F$ on $V$, and $\mathrm e^{-F/2}e$ is a unit frame. Two unit frames on a connected open set differ by a holomorphic factor $g$ with $|g|=1$; then $\bar g=1/g$ is both holomorphic and antiholomorphic, so $\mathrm d\bar g=0$ and $g$ is constant.

Let $c$ be the center of $B$ and fix a unit vector $e_ c\in L_ c$. For $z\in B$, a flat lift along the segment from $c$ to $z$ is a continuous map $t\mapsto \varepsilon (t)\in L_{c+t(z-c)}$, $t\in [0,1]$, with $\varepsilon (0)=e_ c$, which near every $t$ is a constant multiple of a unit frame. There is at most one flat lift: if $\varepsilon '$ is another one, then $\varepsilon '=g\varepsilon $ for a continuous function $g$ on $[0,1]$ which is locally constant, by the preceding paragraph, and equals $1$ at $t=0$, so $g=1$.

It remains to show that flat lifts exist and that their values at $t=1$ depend holomorphically on $z$. Let $z_0\in B$, and choose $\delta _0>0$ with $\overline{B(z_0,\delta _0)}\subset B$. The set $\Gamma _0=\{ c+t(z-c):t\in [0,1],\ |z-z_0|\leq \delta _0\} $ is a compact subset of the convex set $B$. Cover it by finitely many balls $V_ i\subset B$ carrying unit frames $e_ i$, let $\lambda >0$ be a Lebesgue number of this cover, choose $\delta \leq \delta _0$ with $2\delta <\lambda /2$, and choose $0=t_0<t_1<\cdots <t_ N=1$ with $(t_ k-t_{k-1})(|z_0-c|+\delta _0)<\lambda /2$. For $W=B(z_0,\delta )$, each set $\Gamma _ k=\{ c+t(z-c):t\in [t_{k-1},t_ k],\ z\in W\} $ has diameter less than $\lambda $, because $|(c+t(z-c))-(c+t'(z'-c))|\leq |t-t'|\, |z-c|+|z-z'|$; so $\Gamma _ k$ lies in some $V_{i_ k}$. The convex set $V_{i_ k}\cap V_{i_{k+1}}$ is connected and contains $c+t_ k(z_0-c)$, so $e_{i_ k}=\gamma _ ke_{i_{k+1}}$ there for a constant $\gamma _ k$. Let $a$ be the number with $e_ c=a\, e_{i_1}(c)$. For $z\in W$, the map equal to $a\gamma _1\cdots \gamma _{k-1}e_{i_ k}(c+t(z-c))$ for $t\in [t_{k-1},t_ k]$ is well defined at the points $t_ k$ and is a flat lift along the segment from $c$ to $z$. Hence, on $W$, the value $E(z)$ of the unique flat lift at $t=1$ is $a\gamma _1\cdots \gamma _{N-1}e_{i_ N}(z)$, a constant multiple of the unit frame $e_{i_ N}$. Since $z_0$ was arbitrary, $E$ is a nowhere vanishing holomorphic section of $L$ on $B$. $\square$