Theorem 19.8.7.165328A043C0 Suppose that $V_ m\neq 0$ for arbitrarily large integers $m$. Let $\pi :Z\to Y$ be any smooth compact modification, let $E_1,\ldots ,E_ q$ be distinct prime divisors on $Z$, and let $\gamma $ be a Gauduchon metric on $Z$. Then
\[ \deg _\gamma (\pi ^*(L|_ Y)) \geq \sum _{i=1}^ q\tau _{Y,u}(E_ i)\int _{E_ i}\gamma ^{r-1}. \]
Moreover, the real closed $(1,1)$ class
\[ c_1(\pi ^*(L|_ Y))- \sum _{i=1}^ q\tau _{Y,u}(E_ i)c_1(\mathcal O_ Z(E_ i)) \]
is pseudoeffective: it contains a closed positive current.
Proof.
Choose an unbounded set of degrees $m$ and nonzero sections $s_ m\in V_ m$, and put $b_ i(m)=\operatorname {ord}_{E_ i}\mathfrak a_ m$. The pullback $\pi ^*s_ m$ is nonzero, and its local coefficients belong to the pulled-back ideal $\mathfrak a_ m\mathcal O_ Z$, so it vanishes to order at least $b_ i(m)$ along $E_ i$. Thus
\[ D_ m:=\operatorname {div}(\pi ^*s_ m)-\sum _{i=1}^ q b_ i(m)E_ i \geq 0. \]
Choose smooth Hermitian metrics on the line bundles $\mathcal O_ Z(E_ i)$, with curvature forms $\beta _ i$, and set
\[ \Theta _ m=\pi ^*(\beta |_ Y)-\sum _{i=1}^ q\frac{b_ i(m)}m\beta _ i, \qquad \Theta =\pi ^*(\beta |_ Y)-\sum _{i=1}^ q\tau _{Y,u}(E_ i)\beta _ i. \]
Dividing $\pi ^*s_ m$ by the corresponding powers of the canonical sections of the bundles $\mathcal O_ Z(E_ i)$ gives a nonzero holomorphic section of $\pi ^*(L^ m|_ Y)\otimes \mathcal O_ Z(-\sum _ i b_ i(m)E_ i)$ with divisor $D_ m$. Equip this line bundle with the metric induced by $h$ and the chosen metrics. One $m$th of the logarithm of the squared norm of this section is a quasi-plurisubharmonic function $w_ m$ on $Z$, and the Poincaré–Lelong formula gives
\[ \Theta _ m+\mathrm{dd}^{\mathrm{c}}w_ m=\frac1m[D_ m]\geq 0. \]
By theorem 20.3.2, $b_ i(m)/m\to \tau _{Y,u}(E_ i)$ as $m\to \infty $ through all integers, hence also along the chosen degrees, so $\Theta _ m\to \Theta $ smoothly. Consequently a single Hermitian form $\Lambda $ on $Z$ dominates all the forms $\Theta _ m$, and every $w_ m$ belongs to $\mathrm{PSH}(Z,\Lambda )$. Normalize $\sup _ Zw_ m=0$. By theorem 5.25.2, a subsequence converges in $L^1$, and its limit $w$ satisfies
\[ \Theta +\mathrm{dd}^{\mathrm{c}}w\geq 0. \]
Since $\Theta $ is closed and represents the class in the statement, this positive current proves the asserted pseudoeffectivity.
Finally, pair this current with $\gamma ^{r-1}$, where $\gamma ^{r-1}=1$ if $r=1$. Integration by parts removes $\mathrm{dd}^{\mathrm{c}}w$ because $\gamma $ is Gauduchon, and the Poincaré–Lelong formula for the canonical section of $\mathcal O_ Z(E_ i)$ gives $\int _ Z\beta _ i\wedge \gamma ^{r-1}=\int _{E_ i}\gamma ^{r-1}$. The nonnegativity of the pairing is exactly the stated degree inequality.
$\square$
Pome
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