10.4 Positivity cones and divisorial decomposition8E4902BB309B

Proof. Sums and positive multiples of positive currents, Kähler forms or nef representatives are of the same kind, which proves convexity of all four cones.

Let $\alpha _ j\to \alpha $ be pseudo-effective classes, with positive currents $T_ j\in \alpha _ j$. Since the $\mathrm{dd}^{\mathrm{c}}$ of a distribution pairs to zero with the closed form $\omega ^{n-1}$, the trace masses are

where $\theta _{\alpha _ j}$ are the representatives of (10.2.18); they are bounded because the classes converge. Let $K$ be a compact subset of a chart, $\chi $ a test function on the chart with $0\leq \chi \leq 1$ and $\chi =1$ on $K$, and $c'>0$ such that $\beta \leq c'\omega $ on $\operatorname{Supp}\chi $ (lemma 6.14.1), where $\beta $ is the Euclidean form (5.2.8) of the chart. By (10.2.10), applied with $\gamma _ j=c'\omega $ and $\gamma '_ j=\beta $, the form $(c'\omega )^{n-1}-\beta ^{n-1}$ is a sum of products of semipositive $(1,1)$-forms on $\operatorname{Supp}\chi $, hence strongly positive (proposition 5.3.2). Therefore $c'^{n-1}\omega ^{n-1}-\chi \beta ^{n-1}=(1-\chi )c'^{n-1}\omega ^{n-1}+\chi \bigl (c'^{n-1}\omega ^{n-1}-\beta ^{n-1}\bigr )$, with $\chi \beta ^{n-1}$ extended by zero outside the chart, is a smooth strongly positive form on $X$, so the trace measures satisfy $\mu _{T_ j}(K)\leq T_ j(\chi \beta ^{n-1})\leq c'^{n-1}\int _ XT_ j\wedge \omega ^{n-1}$, which is bounded independently of $j$. By the compactness assertion of theorem 5.3.7, a subsequence of $(T_ j)$ converges weakly to a closed positive current $T$, and $\{ T\} =\alpha $ because the class map is weakly continuous. Hence $\mathcal E$ is closed.

If $\omega _0\in \alpha $ is a Kähler form, then $\omega _0\geq c\omega $ for some $c>0$, and by (10.2.18) the form $\omega _0+\theta _{\alpha '-\alpha }\in \alpha '$ is Kähler whenever $C\| \alpha '-\alpha \| <c$; hence $\mathcal K$ is open. If $\alpha $ is nef and $\varepsilon >0$, a representative $\theta _\varepsilon \geq -\varepsilon \omega $ gives the Kähler form $\theta _\varepsilon +2\varepsilon \omega \in \alpha +2\varepsilon \{ \omega \} $, so $\alpha \in \overline{\mathcal K}$. Conversely, let Kähler classes $\alpha _ j$ with Kähler forms $\omega _ j\in \alpha _ j$ converge to $\alpha $. Then $\omega _ j+\theta _{\alpha -\alpha _ j}\in \alpha $ is at least $-C\| \alpha -\alpha _ j\| \omega $, so $\alpha $ is nef; thus $\mathcal N=\overline{\mathcal K}$, and $\mathcal N\subset \mathcal E$ because $\mathcal K\subset \mathcal E$ and $\mathcal E$ is closed.

Since $\mathcal K$ is open and contained in $\mathcal N$, it lies in the interior of $\mathcal N$. If $\alpha $ is nef and $t>0$, then $\alpha +t\{ \omega \} $ is Kähler: take a representative $\theta _\varepsilon \geq -\varepsilon \omega $ with $\varepsilon <t$. If $\alpha $ lies in the interior of $\mathcal N$, then $\alpha -t\{ \omega \} $ is nef for small $t>0$, and therefore $\alpha =(\alpha -t\{ \omega \} )+t\{ \omega \} $ is Kähler. Thus $\mathcal K$ is the interior of $\mathcal N$.

If $T\in \alpha $ and $T\geq \delta \omega $, then $T+\theta _{\alpha '-\alpha }\in \alpha '$ is positive whenever $C\| \alpha '-\alpha \| \leq \delta $, so a neighborhood of $\alpha $ lies in $\mathcal E$. Conversely, if $\alpha $ lies in the interior of $\mathcal E$, then $\alpha -\delta \{ \omega \} \in \mathcal E$ for some $\delta >0$, and adding $\delta \omega $ to a positive current in $\alpha -\delta \{ \omega \} $ gives a Kähler current in $\alpha $. Hence $\mathcal B$ is the interior of $\mathcal E$. $\square$

Proof. Let $\gamma $ be a Kähler class with a Kähler form $h\in \gamma $, and let $\alpha $ satisfy $Q(\alpha ,\gamma )=0$. Choose a smooth closed real form $a\in \alpha $; then $\int _ Xa\wedge h\wedge \Omega =0$. Apply lemma 10.2.9 with $A=h\wedge \Omega $, a product of $n-1$ Kähler forms, and $\eta =-a\wedge h\wedge \Omega $. The resulting $f$ makes $a'=a+\mathrm{dd}^{\mathrm{c}}f\in \alpha $ satisfy $a'\wedge h\wedge \Omega =0$ at every point. At a point, in holomorphic coordinates, the wedge products of real $(1,1)$-forms are expressed by mixed discriminants of their Hermitian matrices, so lemma 5.3.4 gives $a'\wedge a'\wedge \Omega \leq 0$, with equality only if $a'$ vanishes at that point. By Stokes' formula,

If this integral vanishes, the smooth nonpositive density $a'\wedge a'\wedge \Omega $ vanishes everywhere, so $a'=0$ and $\alpha =0$. Thus $Q$ is negative definite on the hyperplane $\{ Q(\cdot ,\gamma )=0\} $.

Since $Q(\gamma ,\gamma )=\int _ Xh^2\wedge \Omega >0$, every class splits uniquely as a multiple of $\gamma $ plus a class in that hyperplane. The space $H^{1,1}(X,\mathbb R)$ is finite dimensional by theorem 10.2.15, and $Q$ is positive on the line spanned by $\gamma $ and negative definite on its $Q$-orthogonal complement; this gives the signature. Finally, let $\alpha ,\beta $ be Kähler classes and write $\beta =c\alpha +\delta $ with $Q(\delta ,\alpha )=0$. Negative definiteness gives $Q(\beta ,\beta )=c^2Q(\alpha ,\alpha )+Q(\delta ,\delta )\leq c^2Q(\alpha ,\alpha )$, while $Q(\alpha ,\beta )=cQ(\alpha ,\alpha )$. Since $Q(\alpha ,\alpha )>0$, we obtain $Q(\alpha ,\beta )^2=c^2Q(\alpha ,\alpha )^2\geq Q(\alpha ,\alpha )Q(\beta ,\beta )$. $\square$

Proof. Assume first that all $\alpha _ i$ are Kähler. Then all the intersection numbers $\int _ X\prod _ i\alpha _ i^{k_ i}$ are positive, and we put $b(k)=\log \int _ X\prod _ i\alpha _ i^{k_ i}$ for $k=(k_1,\ldots ,k_ r)$ with $\sum _ ik_ i=n$. If $n=1$, only one $k_ i$ is positive and there is nothing to prove; let $n\geq 2$. Let $i\neq j$ with $k_ i,k_ j\geq 1$, let $\Omega $ be the wedge product of Kähler representatives of the remaining $n-2$ factors, counted with multiplicity, and let $e_ i$ be the $i$th coordinate vector. Then (10.4.4) for $\alpha _ i,\alpha _ j$ reads

Fix the coordinates other than $i,j$ and put $s=k_ i+k_ j$. The inequality says that the finite sequence $l\mapsto b(\ldots ,l,\ldots ,s-l,\ldots )$, with $l$ in the $i$th and $s-l$ in the $j$th place, $0\leq l\leq s$, is concave: its consecutive differences decrease. A concave sequence lies above the linear interpolation between its endpoints, since the average of its first $l$ differences is at least the average of all $s$ differences. Consequently

where $k^{(i)}$ is obtained from $k$ by putting all $s$ factors in the $i$th place and $k^{(j)}$ by putting them in the $j$th place.

We now prove $b(k)\geq \sum _ i(k_ i/n)\log \int _ X\alpha _ i^ n$ by induction on the number of positive coordinates of $k$. If only one coordinate is positive, this is an equality. Otherwise choose $i\ne j$ with $k_ i,k_ j\geq 1$. The vectors $k^{(i)}$ and $k^{(j)}$ have one positive coordinate fewer, and inserting the induction hypothesis for them into the last display gives the claim: for $m\ne i,j$ the coefficient of $\log \int _ X\alpha _ m^ n$ is $(k_ i/s+k_ j/s)\, k_ m/n=k_ m/n$, and the coefficients of $\log \int _ X\alpha _ i^ n$ and $\log \int _ X\alpha _ j^ n$ are $(k_ i/s)(s/n)=k_ i/n$ and $k_ j/n$. Exponentiating proves the corollary for Kähler classes.

For nef classes, $\alpha _ i+\varepsilon \{ \omega \} $ is Kähler for every $\varepsilon >0$ by proposition 10.4.2, so the inequality holds for these classes. Each intersection number involved is a polynomial in $\varepsilon $ and tends to the corresponding number for the $\alpha _ i$ as $\varepsilon \downarrow 0$; in particular the limits $\int _ X\alpha _ i^ n$ are nonnegative. Since $t\mapsto t^{k_ i/n}$ is continuous on $[0,\infty )$, the inequality passes to the limit. $\square$

Proof. Write the given current as $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ with $\varphi \in \mathrm{PSH}(X,\theta )$. If $\varphi $ is bounded, then every $\mathcal I(m\varphi )$ is the unit ideal, so the approximations $G_ m$ are smooth and satisfy $\theta +\mathrm{dd}^{\mathrm{c}}G_ m\geq -(c/m)\omega $; hence the class is nef. If moreover $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq \delta \omega $, then $\varphi \in \mathrm{PSH}(X,\theta -\delta \omega )$, and the approximations for the background $\theta -\delta \omega $ satisfy $\theta +\mathrm{dd}^{\mathrm{c}}G_ m\geq (\delta -c/m)\omega $; for large $m$ the smooth form $\theta +\mathrm{dd}^{\mathrm{c}}G_ m$ is a Kähler form in the class. For the last assertion, let $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq \delta \omega $ be a Kähler current in a big class, with $\varphi $ not necessarily bounded. The same inequality shows that $\theta +\mathrm{dd}^{\mathrm{c}}G_ m$ is a Kähler current for large $m$, and $G_ m$ has gentle analytic singularities of type $(m^{-1},\mathcal I(m\varphi ))$ by lemma 6.17.1. $\square$

Proof. Write the current as $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq \delta \omega $, where $\varphi $ has minimal singularities, and normalize $\varphi \leq 0$. Choose $C'\geq 0$ with $-\theta \leq C'\omega $ (lemma 6.14.1). For $0<t\leq \delta /(\delta +C')$ the current

is positive, so $(1-t)\varphi \in \mathrm{PSH}(X,\theta )$. By minimality, $(1-t)\varphi \leq \varphi +C$ for a constant $C$, so $\varphi \geq -C/t$ at every point where $\varphi $ is finite, hence almost everywhere. Since $\varphi $ is quasi-plurisubharmonic, the lower bound holds everywhere, as noted at the beginning of section 10.3. Thus $\varphi $ is bounded, and proposition 10.4.6 shows that the class is Kähler. For the second assertion, suppose that the class contains a Kähler form $\theta +\mathrm{dd}^{\mathrm{c}}u$. Then $u$ is smooth and $u\in \mathrm{PSH}(X,\theta )$, so a potential $\varphi $ with minimal singularities satisfies $u\leq \varphi +C$; hence $\varphi $ is bounded below, and it is bounded above as every quasi-plurisubharmonic function on $X$. $\square$

Proof. We first subtract one divisor, then show that only countably many divisors occur and that the series converges, and finally identify the residual current and prove uniqueness.

Step 1: one divisor. Let $D$ be a prime divisor and $c=\nu (T,D)$; we show that $T-c[D]\geq 0$. Near a regular point of $D$ choose coordinates with $D=\{ z_1=0\} $ on a coordinate polydisc $W$ with $|z_1|<1$, and a plurisubharmonic potential $u$ of $T$ on $W$. Since the Lelong numbers of $u$ at the points of $D\cap W$ are at least $c$, proposition 6.9.8 gives $u\leq c\log |z_1|^2+C$ near the point. Off $D$ the function $u-c\log |z_1|^2$ is plurisubharmonic, and it is bounded above near $D$; by theorem 5.11.1 it extends to a plurisubharmonic function $v$. Since $v=u-c\log |z_1|^2$ almost everywhere and $[D]=\mathrm{dd}^{\mathrm{c}}\log |z_1|^2$ near the point (theorem 5.5.6), we get $\mathrm{dd}^{\mathrm{c}}v=T-c[D]$ there. Hence $T-c[D]$ is positive on a neighborhood of $D_{\mathrm{reg}}$, and it equals $T$ on $X\setminus D$; so it is positive on the open set $X\setminus D_{\mathrm{sing}}$.

To extend the positivity across $D_{\mathrm{sing}}$, let $\phi $ be a strongly positive test form of bidegree $(n-1,n-1)$. Since $T$ and $[D]$ have measure coefficients (theorem 5.3.7), there is a real measure $m_\phi $ on $X$, a combination of the coefficient measures with the coefficients of $\phi $ as densities, such that $(T-c[D])(\chi \phi )=\int \chi \, \mathrm dm_\phi $ for test functions $\chi $. The restriction of $m_\phi $ to the open set $X\setminus D_{\mathrm{sing}}$ is a positive measure, because $(T-c[D])(\chi \phi )\geq 0$ for test functions $\chi \geq 0$ supported there; and its restriction to $D_{\mathrm{sing}}$ is that of the positive measure $\chi \mapsto T(\chi \phi )$, because $[D]$ gives no mass to $D_{\mathrm{sing}}$. Hence $(T-c[D])(\phi )=m_\phi (X)\geq 0$, that is, $T-c[D]\geq 0$.

Step 2: the series. By theorem 6.9.5, each set $E_ j=\{ x:\nu (T,x)\geq 1/j\} $ is an analytic subset of $X$ different from $X$. Being compact, $E_ j$ has finitely many irreducible components $Z$ by corollary 9.3.16. We first show:

An analytic subset of the connected manifold $X$ with an interior point is all of $X$, since the set of its interior points is open and, by the identity theorem applied to local defining functions, closed. Hence $E$ has no interior points, so its germs, and their branches, have dimension at most $n-1$, and $E_{\mathrm{sing}}$ has dimension at most $n-2$ at every point (proposition 9.3.9, corollary 9.3.13). At a point $y\in D_{\mathrm{reg}}$ the germ of $D$ has dimension $n-1$, so by monotonicity of dimension (see proposition 9.3.9 and the remarks following it) it is not contained in the germ of $E_{\mathrm{sing}}$; thus there are points $y'\in D_{\mathrm{reg}}\cap E_{\mathrm{reg}}$. Near such a point $E$ is a complex submanifold of dimension at most $n-1$ containing the $(n-1)$-dimensional submanifold $D$, so $D$ and $E$ coincide near $y'$.

Let $F$ be the connected component of $E_{\mathrm{reg}}$ containing $y'$. The analytic subset $D\cap F$ of the connected manifold $F$ has an interior point, so it is all of $F$, and $D$ contains the closure $Z$ of $F$, an irreducible component of $E$ (corollary 9.3.16). Since $D$ is compact and the components of $E$ form a locally finite family, $D$ meets only finitely many of them and is the union of its intersections with these. By induction on the number of sets, an irreducible set is not a finite union of analytic subsets different from itself; hence $D=D\cap Z'$ for a component $Z'$. Then $Z\subset D\subset Z'$, and since no component is contained in another, $D=Z$. This proves (10.4.10). In particular, two distinct prime divisors are not contained in one another, and every prime divisor $D$ with $\nu (T,D)\geq 1/j$, being contained in $E_ j$, is one of its finitely many components. Therefore at most countably many prime divisors have $\nu (T,D)>0$. Let $D_1,D_2,\ldots $ be an enumeration of them and put $S_ N=\sum _{k\leq N}\nu (T,D_ k)[D_ k]$.

We claim that $T-S_ N\geq 0$ for every $N$, and argue by induction; suppose that $T-S_{N-1}\geq 0$. Choose a point $x\in D_ N$ outside the proper analytic subsets $D_ N\cap D_ k$, $k<N$, and outside the exceptional sets of proposition 6.9.7 for $T$ and $T-S_{N-1}$ along $D_ N$. Such points exist, because by that proposition the complement in $D_ N$ of a countable union of proper analytic subsets of $D_ N$ is dense in $D_ N$. The currents $[D_ k]$, $k<N$, vanish near $x$, so $T-S_{N-1}=T$ near $x$, and therefore $\nu (T-S_{N-1},D_ N)=\nu (T-S_{N-1},x)=\nu (T,x)=\nu (T,D_ N)$. Step 1, applied to $T-S_{N-1}$, gives $T-S_ N\geq 0$. Consequently the trace measures of the $S_ N$ in every chart are bounded by that of $T$, and the masses $\int _ XS_ N\wedge \omega ^{n-1}$ are bounded by $\int _ XT\wedge \omega ^{n-1}$.

We now show that the sequence $(S_ N)$ converges weakly. Every test form is a finite linear combination of test forms $g\gamma $, where $g\geq 0$ is a test function supported in a chart and $\gamma $ is a strongly positive form with constant coefficients in that chart. To see this, use a partition of unity (proposition 5.2.6), the fact that strongly positive forms span the real forms (proposition 5.3.2), and write a real test function $g$ as $(g+M\psi )-M\psi $ with $\psi \geq 0$ a test function equal to one on $\operatorname{Supp}g$ and $M\geq \sup |g|$. For such $g\gamma $ the numbers $S_ N(g\gamma )$ increase with $N$, since $S_{N+1}-S_ N\geq 0$, and they are bounded by $T(g\gamma )$; hence they converge, and so does $S_ N(\phi )$ for every test form $\phi $. By the compactness assertion of theorem 5.3.7, a subsequence converges weakly to a closed positive current $S$, so $S_ N(\phi )\to S(\phi )$ for every $\phi $. Finally $R=T-S$ is closed and positive, as a weak limit of the closed positive currents $T-S_ N$.

Step 3: properties of $R$ and uniqueness. Fix a prime divisor $D$ and write $T=R+\nu (T,D)[D]+S_ D$, where $S_ D$ is the sum of the terms of the series for the other divisors. Lelong numbers are additive and nonnegative (proposition 6.7.9), and $\nu ([D],x)=\nu (\log |f|^2,x)=1$ at regular points $x$ of $D$, where $f$ is a local coordinate defining $D$. Hence, at a point of $D_{\mathrm{reg}}$ outside the countable union of proper analytic subsets where one of the currents $T$, $R$, $S_ D$ has a nongeneric Lelong number (such points exist by proposition 6.9.7, since $D_{\mathrm{sing}}$ is also a proper analytic subset of $D$),

Hence $\nu (R,D)=0$.

Now consider any decomposition $T=R'+\sum _ Db_ D[D]$ as in the statement, fix $D$, and let $S'_ D$ be the sum of the terms for the other divisors. We first show that the coefficient measures of $S'_ D$ give no mass to $D$. If $D'\neq D$ is another prime divisor, then $D\cap D'_{\mathrm{reg}}$ is an analytic subset of the connected manifold $D'_{\mathrm{reg}}$ (corollary 9.3.16) which is not all of it, since otherwise $D'\subset D$, and then $D'=D$ by (10.4.10). Hence $D\cap D'_{\mathrm{reg}}$ is a null set for the surface measure of $D'_{\mathrm{reg}}$ (lemma 5.5.1, in local coordinates of $D'_{\mathrm{reg}}$), and $[D']$ gives no mass to $D$. Now let $K\subset D$ be a compact subset of a chart, and for large $m$ let $\chi _ m$ be test functions with $0\leq \chi _ m\leq 1$, equal to one on $K$ and supported in the $1/m$-neighborhood of $K$ (lemma 2.5.1), all supported in a fixed compact subset of the chart, on which a test function $\psi $ equals one. For the trace measures in the chart, the weak convergence of the series gives $\mu _{S'_ D}(\chi _ m)=\sum _{D'}b_{D'}\mu _{[D']}(\chi _ m)$. As $m\to \infty $, $\chi _ m\to \mathbf1_ K$ pointwise, and dominated convergence, applied to the measure $\mu _{S'_ D}$ and to the series, whose terms are bounded by $b_{D'}\mu _{[D']}(\psi )$ with $\sum _{D'}b_{D'}\mu _{[D']}(\psi )=\mu _{S'_ D}(\psi )<\infty $, gives $\mu _{S'_ D}(K)=\sum _{D'}b_{D'}\mu _{[D']}(K)=0$. Since $D$ is covered by countably many such compact sets, the trace measures of $S'_ D$, and hence its coefficient measures (theorem 5.3.7), vanish on $D$.

By Step 1, $S'_ D-\nu (S'_ D,D)[D]\geq 0$; comparing the trace measures of both sides on $D$, where $[D]$ has positive trace mass, forces $\nu (S'_ D,D)=0$. Additivity of Lelong numbers at generic points of $D$, applied to $T=R'+b_ D[D]+S'_ D$, gives $\nu (T,D)=0+b_ D+0$. Thus $b_ D=\nu (T,D)$ for every $D$, and $R'=R$. $\square$

Proof. Near a point, let $h$ be smooth with $\mathrm{dd}^{\mathrm{c}}h=\theta $. Off the support of $D$ the function $v+h=(\varphi +h)-\sum _ ia_ i\log |s_ i|^2$ is plurisubharmonic, since $\log |s_ i|^2$ is pluriharmonic there, and it is bounded above. On a connected coordinate neighborhood of the point, the support of $D$ lies in the zero set of the holomorphic function $\prod _ is_ i\not\equiv 0$, so theorem 5.11.1 extends $v+h$ to a plurisubharmonic function $\widetilde v$. The values of $\widetilde v$ on the support of $D$ are upper limits of values of $v+h$, so $\widetilde v$ is bounded, and $\widetilde v=v+h$ outside the null set $\operatorname{Supp}D$. Since $\mathrm{dd}^{\mathrm{c}}\log |s_ i|^2=[D_ i]$ (lemma 9.3.19), the current $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ satisfies $T-[D]=\mathrm{dd}^{\mathrm{c}}\widetilde v\geq 0$ near the point, where $[D]=\sum _ ia_ i[D_ i]$. Thus $T-[D]$ is a positive current in $\{ \theta \} -\{ D\} $ with locally bounded potentials. If $T\geq \delta \omega $, apply the same argument to $T-\delta \omega $, whose potential has the same logarithmic singularities; then $T-[D]\geq \delta \omega $. In both cases proposition 10.4.6 gives the assertion. $\square$