20.4 Divisorial currents on compact modificationsD53B61AD71E9

Proof. Set $w_ m=(u_ m|_ Y)\circ \pi $. By corollary 5.43.26, these functions are not identically $-\infty $. Moreover, $u_ m|_ Y$ has analytic type $(1/m,\mathfrak a_ m(u;Y))$ by proposition 5.34.1, as in the proof of that corollary, so its pullback $w_ m$ has generic Lelong number $b_ m(E)/m$ along $E$. Thus

Lemma 5.38.1 gives $D_{m,Z}\leq T_ m$. Integration by parts for currents and the Gauduchon condition therefore give

For any fixed finite set of prime divisors, let $m\to \infty $ in the coefficients of these divisors, using theorem 20.3.2, and then take the supremum over all finite sets. This proves (20.4.3) for every finite subsum.

Every prime divisor has strictly positive $\gamma $-area. Hence at most countably many coefficients are positive: for each positive integer $k$, only finitely many terms can have trace mass at least $1/k$, and every positive term is among these for some $k$. Enumerate the corresponding divisors. The finite partial sums are increasing positive closed currents with uniformly bounded trace mass. They therefore converge, in the sense of coefficient measures and weakly as currents, to a positive current of finite mass, which is closed because differentiation is continuous on currents. This constructs $S_ Z(u;Y)$ and proves the integral inequality.

The supremum formula for $\tau _{Y,u}(E)$ gives $b_ m(E)/m\leq \tau _{Y,u}(E)$ for every $m$ and $E$, and the limit along the integers gives coefficientwise convergence. Dominated convergence for the countable series with summable weights $\tau _{Y,u}(E)\int _ E\gamma ^{r-1}$ shows that the trace mass of $S_ Z-D_{m,Z}$ tends to zero. Since a fixed smooth test form is bounded in absolute value, on positive $(1,1)$-currents, by a constant times the trace mass, this gives weak convergence. Finally, $b_{2m}(E)\geq 2b_ m(E)$ gives $D_{2m,Z}\geq D_{m,Z}$. $\square$

Proof. Every prime divisor on $Z'$ is either the strict transform of a prime divisor on $Z$ or exceptional for $g$. If $E'$ is the strict transform of a prime divisor $E\subset Z$, then $g$ is biholomorphic near generic points of $E'$. Hence $E'$ and $E$ define the same vanishing order on the image ideals from $Y$, so $\tau _{Y,u}(E')=\tau _{Y,u}(E)$, and $g_*[E']=[E]$.

The image of an exceptional prime divisor $F$ has codimension at least two. The pullback of a smooth $(r-1,r-1)$ test form therefore restricts to zero on the regular locus of $F$, where the restriction of $g$ has complex rank at most $r-2$; thus $g_*[F]=0$. In dimension one, a modification between smooth compact curves is an isomorphism, and there are no exceptional divisors. Pushforward under the proper map $g$ is continuous on currents. Applying it to the increasing finite partial sums, or equivalently testing their summable positive coefficient measures, proves the displayed compatibility. Additivity and homogeneity follow coefficientwise from proposition 20.3.4.

For the inequality, fix a positive integer $m$ and let $\mathfrak a_{m,Z}=\mathcal I(mu)\mathcal O_ Z$ denote the image ideal on $Z$. Its divisorial vanishing gives

Indeed, on a manifold, a holomorphic function vanishing to the indicated generic orders is divisible by the corresponding powers of the local equations of the divisors. Pulling back this inclusion and taking the order along a prime divisor $F\subset Z'$ gives

Divide by $m$ and let $m\to \infty $; the sum is finite. Thus every coefficient of $g^*A_{\mathcal F}$ is at most the corresponding coefficient $\tau _{Y,u}(F)$ of $S_{Z'}(u;Y)$. Since both sides are positive divisorial currents, this proves the asserted inequality of currents. $\square$

Proof. Take for $v$ the trace envelope $\mathcal T_{\theta |_ Y}(u;Y)$ of (20.3.25). By theorem 20.3.27, it belongs to $\mathrm{PSH}(Y,\theta |_ Y)$ and satisfies every defining trace inequality. On a fixed model $Z$, lemma 5.38.1 therefore allows us to subtract every finite partial sum of $S_ Z(u;Y)$ from $\theta _ Z+\mathrm{dd}^{\mathrm{c}}(v\circ \pi )$ while preserving positivity. Passing to the limit of these partial sums, constructed in theorem 20.4.2, preserves positivity and proves the second assertion. $\square$

Proof. The decomposition of lemma 5.38.4 shows that the natural map from the smooth quotient to the current quotient is surjective, and it gives a quasi-plurisubharmonic potential for positive currents. The same lemma shows that two distributional potentials in such decompositions differ by a smooth function; equivalently, a distribution whose $\mathrm{dd}^{\mathrm{c}}$ is smooth is itself smooth. Hence two smooth forms that are identified in the current quotient are already identified in the smooth quotient, and the map is injective. $\square$

Proof. By proposition 20.4.8, there is one $v$ on $Y$ such that

on every model. Since $\theta $ and $S_ Z$ are closed, $R_ Z$ is closed, and its class is the class in the statement. This proves pseudoeffectivity. Compatibility with pushforward follows from theorem 20.4.5 together with $g_*g^*(\theta _ Z)=\theta _ Z$. Finally, the trace data do not change under ambient multiplier-ideal equivalence, and replacing $\theta |_ Y$ by $\theta |_ Y+\mathrm{dd}^{\mathrm{c}}f$ with $f$ smooth changes no class; this proves the last assertion. $\square$

Proof. Since $w$ is not identically $-\infty $, its Lelong number vanishes at almost every point; hence $\nu (w,Y)=0$, and the self-trace is defined. Fix a divisor $E$ on a compact modification and apply lemma 5.38.9 on a coordinate neighborhood of a point of its image in $Y$. With $\nu _ E=\nu (w\circ \pi ,E)$ and $A_ E$ the log discrepancy of $E$, it gives

The limit along the integers defining $\tau _{Y,w}(E)$ is therefore $\nu _ E$. Since all coefficients agree, the two divisorial currents are equal. $\square$

Proof. By the lemma, $N_ Z(T)=S_ Z(w;Y)$, so theorem 20.4.5 gives $g_*N_{Z'}(T)=N_ Z(T)$ for every modification $g:Z'\to Z$ over $Y$. Moreover, $g_*g^*(\pi ^*T)=\pi ^*T$. To see this, write the current as a smooth form plus $\mathrm{dd}^{\mathrm{c}}$ of a potential; the change of variables formula for the degree-one map $g$ applies both to the smooth form and to the locally integrable pulled-back potential, and pushforward commutes with $\mathrm{dd}^{\mathrm{c}}$. Therefore

Taking classes proves compatibility, and positivity of the regular parts proves pseudoeffectivity. Generic Lelong numbers are additive under addition of positive currents and scale under multiplication by nonnegative constants. Hence so are the divisorial parts and the regular parts, which proves the two identities. In the Kähler case, the regular parts and, by the $\partial \bar\partial $-lemma, their cohomology classes are exactly those of the construction cited in the statement. $\square$

Proof. Since $R$ is closed, so is $\beta $. Fix a Hermitian metric $\gamma $. By proposition 5.31.2, there are smooth functions $v_ j$ and numbers $\epsilon _ j\downarrow 0$ such that

These forms are closed representatives of $\{ R\} _{\mathrm{BC}}$. Given $\epsilon >0$, choosing $j$ with $\epsilon _ j\leq \epsilon $ gives the estimate required in the definition of nefness. $\square$

Proof. Resolve the analytic singularity ideal, using the log-resolution theorem in [section on analytic singularities, theorem on existence of log resolutions, SGPT]. On a suitable smooth compact model $\pi :Z\to Y$,

where $D$ is a finite effective rational divisor and $R$ is a closed positive current with locally bounded potentials. To obtain a global bounded potential, choose smooth metrics on the line bundles of the components of $D$ and subtract the corresponding logarithms of squared norms of the canonical sections from $w\circ \pi $. The remainder $v$ is globally bounded, and

for a smooth closed form $\beta _ Z$. By the lemma, the class of $R$ is nef.

On every further compact modification $g:Z'\to Z$, the current $g^*R$ still has bounded local potentials and therefore zero Lelong numbers. Hence the divisorial part of $g^*\pi ^*T$ is exactly $[g^*D]$, and its regular part is $g^*R$. It follows that $\mathbb D(T)_{Z'}=g^*\{ R\} _{\mathrm{BC}}$. Thus $(Z,\{ R\} _{\mathrm{BC}})$ is a nef Cartier realization. $\square$

Proof. Set $L_\gamma f=\sigma _\gamma (\mathrm{dd}^{\mathrm{c}}f)$. We use the inverse $G_\gamma $ of (11.31.3), normalized with the volume form $\gamma ^ n/V_\gamma $. It acts on top-degree distributions of total mass zero and is continuous by the elliptic estimates for the operator and its transpose.

Step 1: construction. Define

The current $B_\gamma (T)$ is closed and has scalar trace $c_ T$. We now prove its smoothness and the finite-dimensionality assertion without using any Kähler identity.

Step 2: an elliptic system. Let $J$ be the complex structure acting on real covectors. On real $(1,1)$-forms consider the first-order operator

Its principal symbol at a nonzero real covector $\xi $ is $B\mapsto (\xi \wedge B,\xi \, \sigma _\gamma (B))$, and we claim that it is injective. If $\xi \wedge B=0$, then the $(1,1)$ condition forces $B$ to be a real multiple of $\xi \wedge J\xi $. To check this, choose unitary real coordinates with $\xi =\mathrm{d}x_1$: every term of $B$ contains $\mathrm{d}x_1$, and invariance under $J$ forces every term to contain $\mathrm{d}y_1$ as well. If moreover $\xi \, \sigma _\gamma (B)=0$, then $\sigma _\gamma (B)=0$; since the scalar trace of $\xi \wedge J\xi $ is nonzero, the multiple is zero.

Let $\mathscr P^*$ be the formal adjoint of $\mathscr P$ for the fixed metric and volume. Since the symbol of $\mathscr P$ is injective, $\mathscr P^*\mathscr P$ is elliptic. By elliptic regularity and compactness, its distributional kernel is a finite-dimensional space of smooth forms. Since $\mathscr P B_\gamma (T)=0$, the form $B_\gamma (T)$ belongs to this space. Conversely, the kernel consists exactly of the closed forms with constant scalar trace: integrating $\langle \mathscr P^*\mathscr P B,B\rangle $ gives $\mathscr P B=0$, and connectedness makes the trace constant. Denote this kernel by $\mathscr H_\gamma $.

Step 3: Bott–Chern representatives. The construction is linear and continuous on distributions. Convergence of the components $B_\gamma (T)$ therefore takes place in the finite-dimensional space $\mathscr H_\gamma $, hence in every smooth norm. If $T=\mathrm{dd}^{\mathrm{c}}f$ for a distribution $f$, then $c_ T=0$ and the inverse gives $v_\gamma (T)=f-V_\gamma ^{-1}\int _ Yf\gamma ^ n$, so $B_\gamma (T)=0$. Applied to smooth closed forms, this shows that $\mathscr H_\gamma $ contains exactly one representative of each Bott–Chern class. Uniqueness of the decomposition can also be seen directly: if two decompositions are given, $L_\gamma $ of the difference of the two potentials is a constant, integration forces this constant to be zero, and the normalized kernel of $L_\gamma $ is zero.

Step 4: positive currents. Suppose now that $T\geq 0$. The inequality $\mathrm{dd}^{\mathrm{c}}v_\gamma (T)\geq -B_\gamma (T)$ gives a quasi-plurisubharmonic representative of the distribution $v_\gamma (T)$. Let positive closed currents $T_ j$ converge weakly to $T$. Their components $B_\gamma (T_ j)$ converge smoothly, so the potentials $v_\gamma (T_ j)$ have a common smooth lower curvature bound. By theorem 5.25.2, after subtracting their suprema they form an $L^1$-precompact family. If $a_ j=\sup _ Yv_\gamma (T_ j)$, the zero-mean normalization gives

and the right-hand side is uniformly bounded. Hence the potentials $v_\gamma (T_ j)$ themselves are $L^1$-precompact. Every cluster point equals their distributional limit $v_\gamma (T)$, so the whole sequence converges in $L^1$.

Finally, the positive closed currents with trace mass $\int _ YT\wedge \gamma ^{n-1}=1$ form a weakly compact set: positivity bounds their coefficient measures, weak limits remain positive and closed, and testing with $\gamma ^{n-1}$ preserves the trace mass. The smooth and $L^1$ continuity just proved bounds both norms on this set, and positive homogeneity gives (20.4.21). A positive current of trace mass zero is zero, so this case is included. $\square$

Proof. Write $T=B_\gamma (T)+\mathrm{dd}^{\mathrm{c}}v_\gamma (T)$. The pullback is

This current is positive and closed, since it is locally $\mathrm{dd}^{\mathrm{c}}$ of plurisubharmonic functions, and the composed potential is not identically $-\infty $ because $g$ is surjective. The formula agrees with the definition through local potentials, so it does not depend on the decomposition.

If $T_ j\to T$, the smooth parts converge in $C^\infty $, and the potentials converge in $L^1(Y)$ with a common lower curvature bound. By lemma 5.43.19, their compositions with $g$ converge in $L^1(Z)$, and taking $\mathrm{dd}^{\mathrm{c}}$ proves the weak convergence of the pullbacks.

Integration by parts on $Z$ gives

The absolute value of the right-hand side is at most a fixed constant times $\| B_\gamma (T)\| _{C^0}$. Since the left-hand side is nonnegative, (20.4.21) gives (20.4.23). $\square$

Proof. Apply the proposition to the increasing finite partial sums $S_ N$ and to their positive closed tails $S-S_ N$. Pullback is additive on positive closed currents, by addition of local potentials. Thus $g^*S-g^*S_ N=g^*(S-S_ N)\geq 0$, and its trace mass is at most $C_ g$ times the trace mass of $S-S_ N$, which tends to zero. The pullback of a finite divisor is the usual pullback of its local equations, which identifies $g^*S_ N$ with the corresponding partial sum of the series, and hence $g^*S$ with its limit. Its terms are integration currents of prime divisors with nonnegative coefficients, and regrouping this countable positive sum of finite mass does not change it. $\square$

Proof. Step 1: the exceptional divisor. Enumerate the prime divisors on $Z$ with positive trace coefficient. By theorem 20.4.5, the pullback of every finite partial sum of $S_ Z$ is at most $S_{Z'}$, and corollary 20.4.24 identifies the increasing limit of these pullbacks with $g^*S_ Z$. Thus $S_{Z'}-g^*S_ Z$ is a positive closed divisorial current. The two currents have equal coefficients along the strict transform of every prime divisor on $Z$, because these coefficients are computed by the same valuation. Every other prime divisor on $Z'$ is exceptional for $g$, and the exceptional analytic set has only finitely many irreducible components on the compact manifold $Z'$. Hence the difference is the integration current of a finite effective real divisor $E_ g$ supported on its divisorial components, and the coefficients of integration currents determine $E_ g$ uniquely.

Step 2: the cocycle identity. Pullback of positive closed $(1,1)$-currents is functorial, since it is defined by composing local plurisubharmonic potentials. Using (20.4.27) for $h$ and for $g$ gives

and uniqueness gives (20.4.28). Additivity and positive homogeneity follow from the same properties of the trace coefficients and the linearity of pullback.

Step 3: approximation. For fixed $m$, apply (20.4.6) to the finitely many prime divisors in the support of the image ideal on $Z$. After division by $m$, it gives $D_{m,Z'}-g^*D_{m,Z}\geq 0$. The coefficients along strict transforms vanish, so this difference is the integration current of an effective real divisor $E_{m,g}$ supported on the exceptional locus.

Choose also a Gauduchon metric $\gamma $ on $Z$. By theorem 20.4.2, the currents $S_ Z-D_{m,Z}$ and $S_{Z'}-D_{m,Z'}$ are positive, and their trace masses tend to zero. Subtracting the two identities for the exceptional currents gives

where both $S_{Z'}-D_{m,Z'}$ and $g^*(S_ Z-D_{m,Z})$ are positive. By (20.4.23), the total variation of the signed measure $([E_{m,g}]-[E_ g])\wedge (\gamma ')^{r-1}$ is therefore at most

Both terms tend to zero, which proves convergence in total variation. $\square$

Proof. Global logarithmic potentials exist: choose smooth Hermitian metrics on the line bundles of the prime components of $E_ g$, let $\alpha _{E_ g}$ be the corresponding weighted sum of their curvature forms, and let $\rho _{E_ g}$ be the weighted sum of the logarithms of the squared norms of the canonical sections. Then

and any two global logarithmic potentials differ by a smooth function, by lemma 5.38.4. We may therefore use this choice. The function $\rho _{E_ g}$ is quasi-plurisubharmonic and bounded above on the compact manifold $Z'$.

The chosen decompositions and (20.4.27) give

which is smooth. By lemma 5.38.4, this difference of locally integrable functions is represented by a smooth function. The resulting equality of quasi-plurisubharmonic functions extends from almost everywhere to everywhere by lemma 5.24.3. Since $\rho _{E_ g}$ is bounded above, this gives the ordinary domination.

If $E_ g=0$, the smooth difference gives ordinary equivalence. Conversely, ordinary equivalence would give equal generic Lelong numbers along every prime divisor of $Z'$, whereas the identity just proved shows that along a prime component of $E_ g$ these numbers differ by its positive coefficient. Hence ordinary equivalence forces $E_ g=0$. $\square$