Corollary 12.24.5.9860E4D04E6C On the ruled surface $X$ of lemma 16.29.16, keep its nef curvature form $\theta $, its section $D$ at infinity and its Kähler form $\kappa =\theta +\pi ^*\sigma _0$. For every smooth positive probability form $\mu $,
\[ \mathcal V_\mu (\theta +\epsilon \kappa )=2\epsilon (1+\epsilon ), \qquad \mathcal V_\mu (\epsilon \kappa )=2\epsilon ^2. \]
The only positive current of the form $\theta +\mathrm{dd}^{\mathrm{c}}u$ is $[D]$. It has minimal singularities, and removing its whole divisor leaves the zero background and changes the common order from one to two.
Proof.
Lemma 16.29.16 gives the uniqueness of $[D]$ and the nefness of $\theta $, and the calculation in the proof of proposition 16.29.18 gives $\int _ X\theta ^2=0$. Put $B=\pi ^*\sigma _0$. Then $B^2=0$, and
\[ \int _ X\theta \wedge B=\int _ D B=1, \]
by the Poincaré–Lelong formula, Stokes' theorem and the fact that the section $D$ maps isomorphically onto the elliptic base. The closed nef formula following theorem 12.11.1 now computes both volumes by expanding the squares, for every $\mu $. Every $\theta $-plurisubharmonic function $u$ satisfies $\theta +\mathrm{dd}^{\mathrm{c}}u=[D]$, so any two such functions differ by a pluriharmonic function, hence by a constant; in particular each has minimal singularities. Since $\theta $ is itself a curvature form of $\mathcal O_ X(D)$, removing the whole divisor leaves the zero background. The two volume polynomials give the orders one and two.
$\square$
Pome
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