PomeProposition 12.27.5.9877ACAAFD41 Let $D$ be a smooth nonempty compact divisor in a complex surface $S$, and suppose $\mathcal O_ S(D)$ is unitary flat near $D$. Then $\mathcal O_ S(D)$ has a smooth semipositive metric on $S$ whose curvature $B$ has rank at most one and vanishes both near $D$ and outside a relatively compact neighborhood of $D$. If $S$ is compact Kähler, then $\max _ S\operatorname {rank}B=1$.
Proposition 12.27.59877ACAAFD41
Proof. Let $s_ D$ be the defining section and $h_0$ a flat metric on a neighborhood $U$ of $D$. Put $F=\log |s_ D|_{h_0}^2$, which is pluriharmonic on $U\setminus D$. Choose a relatively compact tubular neighborhood $U_0\Subset U$ of $D$. Since $F$ tends uniformly to $-\infty $ along $D$ and is bounded on $\partial U_0$, after adding a constant to $F$ we may assume $F>0$ near $\partial U_0$. Choose a smooth concave function $\chi $ and a constant $c$ with $\chi (t)=t+c$ for $t\leq -1$ and $\chi (t)=0$ for $t\geq 0$. Define $\widetilde F=\chi (F)$ on $U_0$, extended by zero outside $U_0$, and define a metric $h$ on $S\setminus D$ by
\[ |s_ D|_ h^2=\mathrm e^{\widetilde F}. \]
Near $D$ this gives $h=\mathrm e^ c h_0$, so $h$ extends smoothly across $D$; it is also smooth near $\partial U_0$, where $\widetilde F=0$. By the Poincaré–Lelong formula its curvature is $B=[D]-\mathrm{dd}^{\mathrm{c}}\widetilde F$, and off $D$, since $F$ is pluriharmonic,
\[ B=-\chi ''(F)\, \mathrm dF\wedge \mathrm d^{\mathrm c}F\geq 0. \]
This smooth closed form has rank at most one and vanishes where asserted. On a compact Kähler surface with Kähler form $\kappa $,
\[ \int _ SB\wedge \kappa =\int _ D\kappa >0, \]
so $B$ is not identically zero and its maximal rank is one. $\square$
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