Let $Y\subset X$ be a connected compact complex submanifold of dimension $r\geq 1$, and suppose that the generic Lelong number $\nu (u,Y)=\inf _{y\in Y}\nu (u,y)$ vanishes. Recall from subsection 20.3.1 the image ideals $\mathfrak a_ m=\mathfrak a_ m(u;Y)=\mathcal I(mu)\mathcal O_ Y$ and the divisorial trace data $\tau _{Y,u}(E)$ of a prime divisor $E$ on a smooth compact modification of $Y$, defined in (20.3.3) as the limit of $\operatorname {ord}_ E\mathfrak a_ m/m$. Put
\[ V_ m=\operatorname {im}\bigl (\mathcal H_ m(u)\longrightarrow H^0(Y,L^ m|_ Y)\bigr ). \]
The local coefficients of a restricted section lie in $\mathfrak a_ m$, so $V_ m\subseteq H^0(Y,L^ m|_ Y\otimes \mathfrak a_ m)$; the right-hand space can contain sections that do not extend to $X$.
On a modification of $Y$, the pullback of a nonzero restricted section vanishes along each prime divisor at least to the order imposed by the image ideal. Subtracting finitely many of these orders still leaves an effective divisor, which the Poincaré–Lelong formula turns into a positive current. If such sections exist in arbitrarily large degrees, normalized compactness produces a positive current in the class obtained by subtracting the trace data from the curvature class, and pairing it with a Gauduchon metric gives a degree inequality. Below, $c_1$ denotes the first Chern class in real Bott–Chern cohomology (subsection 20.4.9); it is represented by the curvature form of any smooth Hermitian metric.
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.
$\square$
Equivalently, if one of these inequalities fails for some choice of $Z$, $E_1,\ldots ,E_ q$ and $\gamma $, then $V_ m=0$ for all sufficiently large $m$.
The converse of the theorem fails in general. Let $E=\mathbb C/(\mathbb Z+\mathrm{i}\mathbb Z)$ be the square elliptic curve, choose an irrational real number $a$, and let $L$ be the flat line bundle on $E$ with multipliers $\mathrm{e}^{2\pi \mathrm{i} a}$ and $1$ along the two generators: its sections lift to holomorphic functions $f$ on $\mathbb C$ with $f(z+1)=\mathrm{e}^{2\pi \mathrm{i} a}f(z)$ and $f(z+\mathrm{i})=f(z)$. The constant metric on $L$ has curvature zero. Take $u=0$ and $Y=X=E$. Then every trace number is zero, and the class in the theorem is zero, hence pseudoeffective. Nevertheless no positive tensor power of $L$ has a nonzero holomorphic section. Indeed, a nonzero section of $L^ m$ would have a zero divisor of degree zero and would therefore be nowhere zero. The logarithm of its norm would be harmonic on $E$, hence constant, so its lift $f$ would be a holomorphic function of constant modulus on $\mathbb C$, and thus constant. Since $ma\notin \mathbb Z$, the relation $f(z+1)=\mathrm{e}^{2\pi \mathrm{i} ma}f(z)$ then forces $f=0$, a contradiction.
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