PomeCorollary 12.27.6.3CC0EFB19764 Let $S$ be a compact Kähler surface and $D$ a smooth nonempty divisor whose line bundle is unitary flat near $D$, and let $B$ be the curvature form of proposition 12.27.5. For every smooth function $f>0$, every Hermitian form $h$ and every smooth positive probability form $\mu $, there are constants $L,U>0$ such that
\[ L\epsilon \leq \mathcal V_\mu (fB+\epsilon h) \leq U\epsilon ,\qquad 0<\epsilon \leq 1. \]
For a Kähler form $\kappa $, the closed background $B$ itself satisfies
\[ \mathcal V_\mu (B+\epsilon \kappa ) =2\epsilon \int _ D\kappa +\epsilon ^2\int _ S\kappa ^2. \]
For suitable $f$, the background $fB$ is not closed.
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