10.5 Minimal multiplicities and positivity loci272951B4998D

Proof. The positive current $T_0=\theta +\mathrm{dd}^{\mathrm{c}}V_\theta \in \alpha $ gives the inequalities, by the bound preceding the proposition. Suppose that $\alpha $ is big, and choose $S\in \alpha $ with $S\geq \delta \omega $, $\delta >0$. For $\varepsilon >0$ put $T_\varepsilon =\theta +\mathrm{dd}^{\mathrm{c}}V_{\theta +\varepsilon \omega }\geq -\varepsilon \omega $ and $t=\varepsilon /(\delta +\varepsilon )$. Then $(1-t)T_\varepsilon +tS\geq \bigl (-(1-t)\varepsilon +t\delta \bigr )\omega =0$ is a positive current in $\alpha $, whose potential $(1-t)V_{\theta +\varepsilon \omega }+t\varphi _ S$, with $S=\theta +\mathrm{dd}^{\mathrm{c}}\varphi _ S$, is more singular than $V_\theta $. Since Lelong numbers are additive and positively homogeneous (proposition 6.7.9) and do not decrease when a function is replaced by a more singular one, we obtain

As $\varepsilon \downarrow 0$ we have $t\to 0$, and since $\nu (S,x)$ is finite we obtain $\nu (V_\theta ,x)\leq \nu (\alpha ,x)$. For a prime divisor $D$, evaluate the same inequality at a point $x\in D$ at which the Lelong numbers of $V_{\theta +\varepsilon \omega }$ and of $S$ attain their generic values along $D$ (proposition 6.9.7); it gives $\nu (V_\theta ,D)\leq \nu (V_\theta ,x)\leq (1-t)\nu (\alpha ,D)+t\nu (S,D)$, and letting $\varepsilon \downarrow 0$ proves the equality for $D$.

If $\alpha $ is modified nef and $T_\varepsilon $ is as in definition 10.4.11, then $T_\varepsilon =\theta +\mathrm{dd}^{\mathrm{c}}\varphi _\varepsilon $ with $\varphi _\varepsilon \in \mathrm{PSH}(X,\theta +\varepsilon \omega )$, so $\nu (V_{\theta +\varepsilon \omega },D)\leq \nu (\varphi _\varepsilon ,D)=0$ for every $\varepsilon >0$; hence $\nu (\alpha ,D)=0$. Conversely, if $\nu (\alpha ,D)=0$ for every $D$, then the currents $\theta +\mathrm{dd}^{\mathrm{c}}V_{\theta +\varepsilon \omega }$ satisfy the conditions of definition 10.4.11. $\square$

Proof. We treat $\nu (\cdot ,x)$. The proof for $D$ is the same, except that the additivity of Lelong numbers along $D$ is applied at a point of $D$ at which all the functions involved attain their generic Lelong numbers along $D$ (proposition 6.9.7). For $t>0$ we have $\mathrm{PSH}(X,t\theta +\varepsilon \omega )=t\mathrm{PSH}(X,\theta +(\varepsilon /t)\omega )$, hence $V_{t\theta +\varepsilon \omega }=tV_{\theta +(\varepsilon /t)\omega }$, and taking suprema over $\varepsilon $ gives $\nu (t\alpha ,x)=t\nu (\alpha ,x)$. If $\alpha _1,\alpha _2\in \mathcal E$ have representatives $\theta _1,\theta _2$, the sum of the minimal currents with allowance $\varepsilon $ in the two classes is a current $\theta _1+\theta _2+\mathrm{dd}^{\mathrm{c}}(V_{\theta _1+\varepsilon \omega }+V_{\theta _2+\varepsilon \omega })\geq -2\varepsilon \omega $ in $\alpha _1+\alpha _2$. By minimality and additivity of Lelong numbers,

and taking the supremum over $\varepsilon $ gives subadditivity. Together with homogeneity this proves convexity.

Let $\alpha _ j\to \alpha $ in $\mathcal E$, with representatives $\theta _ j$, and fix $\varepsilon >0$. For large $j$, by (10.2.18), the class $\alpha -\alpha _ j$ has a smooth representative $\eta _ j$ with $-\varepsilon \omega \leq \eta _ j\leq \varepsilon \omega $. The current $\theta _ j+\mathrm{dd}^{\mathrm{c}}V_{\theta _ j+\varepsilon \omega }+\eta _ j\in \alpha $ is at least $-2\varepsilon \omega $, and its potential relative to $\theta $ differs from $V_{\theta _ j+\varepsilon \omega }$ by a smooth function. By minimality,

for large $j$, so $\nu (V_{\theta +2\varepsilon \omega },x)\leq \varliminf _ j\nu (\alpha _ j,x)$. Taking the supremum over $\varepsilon $ proves lower semicontinuity.

Finally, a convex function $f$ with finite values on an open convex subset $U$ of a finite-dimensional real vector space is continuous. Indeed, let $Q\subset U$ be a small closed cube; by convexity the function is bounded above on $Q$ by the maximum $M$ of its values at the vertices. If $c$ is the center of $Q$ and $c\pm v\in Q$, then $f(c+v)\geq 2f(c)-f(c-v)\geq 2f(c)-M$, so $f$ is also bounded below on $Q$. If $y,z$ lie in the concentric cube of half the size, extend the segment from $z$ to $y$ beyond $y$ to the point $w$ at distance $s$ from $y$, where $s$ is half the side length of the smaller cube, so that $w\in Q$. The slope inequality for convex functions gives $f(y)-f(z)\leq |y-z|(f(w)-f(y))/|w-y|\leq |y-z|(M-\inf _ Qf)/s$, and by symmetry $f$ is Lipschitz on the smaller cube. Applied to the open convex cone $\mathcal B$, this proves continuity there. $\square$

Proof. The non-nef locus. By the monotonicity in $\varepsilon $, $E_{\mathrm{nn}}(\alpha )$ is the union over positive integers $j,k$ of the sets $\{ x:\nu (V_{\theta +\omega /j},x)\geq 1/k\} $. Each of them is analytic by theorem 6.9.5, and proper, because it is contained in the pole set of $V_{\theta +\omega /j}$. If $\alpha $ is nef, then $\alpha +\varepsilon \{ \omega \} $ is Kähler for every $\varepsilon >0$, so $V_{\theta +\varepsilon \omega }$ is bounded by corollary 10.4.7 and all these Lelong numbers vanish. Conversely, suppose that $\nu (V_{\theta +\varepsilon \omega },x)=0$ for all $x\in X$ and $\varepsilon >0$. By corollary 6.9.3, $\mathcal I(mV_{\theta +\varepsilon \omega })$ is the unit ideal at every point, so the approximations $G_ m$ of lemma 6.17.1 for the background $\theta +\varepsilon \omega $ are smooth and satisfy $\theta +\mathrm{dd}^{\mathrm{c}}G_ m\geq -(\varepsilon +c/m)\omega $. Hence $\alpha $ is nef.

Realization of the non-Kähler locus. Let $\alpha $ be big. Kähler currents with analytic singularities exist in $\alpha $ by proposition 10.4.6, and their pole sets are closed analytic subsets of $X$. Since $X$ has a countable basis, countably many of the open complements of these pole sets already have the same union as all of them: for every element of a countable basis that is contained in one of these complements, choose one such complement. The intersection of the corresponding countably many pole sets $Z_1,Z_2,\ldots $ is therefore $E_{\mathrm{nK}}(\alpha )$. Let $\mathcal I_{Z_ k}$ be the ideal sheaf of $Z_ k$, which is coherent by theorem 9.3.10. The increasing sequence of coherent ideals $\mathcal I_{Z_1}+\cdots +\mathcal I_{Z_ k}$, whose zero sets are $Z_1\cap \cdots \cap Z_ k$, is stationary on the compact manifold $X$ by lemma 6.8.4. Hence $E_{\mathrm{nK}}(\alpha )=Z_1\cap \cdots \cap Z_ N$ for some $N$.

Let $T_ k=\theta +\mathrm{dd}^{\mathrm{c}}\varphi _ k$, $k\leq N$, be the corresponding Kähler currents, and choose $\delta >0$ with $T_ k\geq \delta \omega $ for all $k\leq N$. Then $\varphi =\max _ k\varphi _ k$ belongs to $\mathrm{PSH}(X,\theta -\delta \omega )$, it has analytic singularities by proposition 6.14.8, and its pole set is exactly $Z_1\cap \cdots \cap Z_ N=E_{\mathrm{nK}}(\alpha )$. Thus $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ is a Kähler current realizing $E_{\mathrm{nK}}(\alpha )$, which is a proper analytic subset because $\varphi $ is finite almost everywhere. If $\nu (\alpha ,x)>0$, then $\nu (\varphi ,x)\geq \nu (V_\theta ,x)=\nu (\alpha ,x)>0$ by minimality of $V_\theta $ and proposition 10.5.2, so $\varphi (x)=-\infty $; hence $E_{\mathrm{nn}}(\alpha )\subset E_{\mathrm{nK}}(\alpha )$.

If $E_{\mathrm{nK}}(\alpha )=\varnothing $, the potential $\varphi $ just constructed has analytic singularities and no poles, so it is bounded, and proposition 10.4.6 shows that $\alpha $ is Kähler. Conversely, if $\alpha $ contains a Kähler form $\theta +\mathrm{dd}^{\mathrm{c}}u$, then $u$ is smooth and has analytic singularities without poles, so $E_{\mathrm{nK}}(\alpha )=\varnothing $.

The identity (10.5.5). Suppose first that $x\notin E_{\mathrm{nK}}(\alpha )$. Then $\alpha $ is big, and there is a Kähler current $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq \delta \omega $ with analytic singularities and $\varphi (x)>-\infty $; in particular $\varphi $ is bounded near $x$ and $\nu (\varphi ,x)=0$. For $0<\varepsilon <\delta $, the current $\theta -\varepsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi $ is positive and lies in $\alpha -\varepsilon \{ \omega \} $, which is therefore pseudo-effective, and the bound $\nu (\alpha -\varepsilon \{ \omega \} ,x)\leq \nu (\varphi ,x)=0$ shows that $x\notin E_{\mathrm{nn}}(\alpha -\varepsilon \{ \omega \} )$.

Conversely, suppose that $x\notin E_{\mathrm{nn}}(\alpha -\varepsilon \{ \omega \} )$ for some $\varepsilon >0$. By the convention in definition 10.5.1, the class $\alpha -\varepsilon \{ \omega \} $ is pseudo-effective. For $0<\eta <\varepsilon /2$ put $u=V_{\theta -(\varepsilon -\eta )\omega }$; then $\nu (u,x)\leq \nu (\alpha -\varepsilon \{ \omega \} ,x)=0$. By corollary 6.9.3, $\mathcal I(mu)$ is the unit ideal near $x$ for every $m$, and the approximation $G_ m$ of lemma 6.17.1 for $u$ and the background $\theta -(\varepsilon -\eta )\omega $ is smooth near $x$ and satisfies

For large $m$ this is a Kähler current in $\alpha $ with analytic singularities whose potential is finite at $x$. Hence $\alpha $ is big and $x\notin E_{\mathrm{nK}}(\alpha )$. This proves (10.5.5); the argument also shows that $\alpha $ is big as soon as the right-hand side of (10.5.5) is not all of $X$, so the formula is consistent with the convention $E_{\mathrm{nK}}(\alpha )=X$ for classes that are not big. $\square$

Proof. By lemma 6.14.1 on the compact manifold $Y$, $\pi ^*\omega \leq C\omega _ Y$ for a constant $C$. If $\theta _\varepsilon \geq -\varepsilon \omega $ are nef representatives of a class on $X$, then $\pi ^*\theta _\varepsilon \geq -C\varepsilon \omega _ Y$, so the pullback class is nef. Let $S$ be a non-divisorial closed positive current on $Y$ and $D\subset X$ a prime divisor, with strict transform $D'$. Over $X\setminus A$ the map $\pi $ is a biholomorphism, which identifies $\pi _*S$ with $S$ and $D\setminus A$ with $D'\setminus \pi ^{-1}(A)$, and Lelong numbers are invariant under biholomorphisms (proposition 6.7.1). The generic value of the Lelong numbers of $S$ along $D'$ is attained on a dense subset of $D'$ (proposition 6.9.7), hence at some point $y$ of the nonempty open subset $D'\setminus \pi ^{-1}(A)$. Therefore $\nu (\pi _*S,D)\leq \nu (\pi _*S,\pi (y))=\nu (S,y)=\nu (S,D')=0$. Hence $\pi _*S$ is non-divisorial.

Let $\beta $ be a modified nef class on $Y$ and $t>0$. The class $\beta +t\{ \omega _ Y\} $ is big, and it is modified nef because $\mathcal{MN}$ is a convex cone containing the Kähler classes. By proposition 10.5.2, a positive current with minimal singularities in this class is non-divisorial. Its pushforward is a non-divisorial positive current in $\pi _*\beta +t\pi _*\{ \omega _ Y\} $, so this class is modified nef: the definition is satisfied with $T_\varepsilon $ equal to this current for every $\varepsilon $. These classes tend to $\pi _*\beta $ as $t\downarrow 0$, and $\mathcal{MN}$ is closed, so $\pi _*\beta $ is modified nef. $\square$

Proof. Let $A$ be as above. The positive current $\pi ^*\pi _*S$, defined through local potentials as above, agrees with $S$ on $Y\setminus \pi ^{-1}(A)$. Their difference is a closed real current with measure coefficients supported on the proper analytic subset $\pi ^{-1}(A)$ of $Y$. By theorem 9.3.20, it is a finite real linear combination of the currents of integration along the irreducible components of codimension one of the compact analytic set $\pi ^{-1}(A)$, which are exactly the exceptional prime divisors, as shown above; there are finitely many of them by corollary 9.3.16.

It remains to show that the coefficients $c_ E$ are nonnegative. Near a regular point of an exceptional prime divisor $E$ that lies on no other component, the difference is $c_ E[E]=\mathrm{dd}^{\mathrm{c}}(c_ E\log |s_ E|^2)$, where $s_ E$ is a local equation of $E$ (lemma 9.3.19), so local potentials $u'$ of $\pi ^*\pi _*S$ and $u$ of $S$ satisfy $u'=u+c_ E\log |s_ E|^2+p$ with $p$ pluriharmonic. The additivity of Lelong numbers (proposition 6.7.9), applied to this identity if $c_ E\geq 0$ and to $u=u'+|c_ E|\log |s_ E|^2-p$ otherwise, together with $\nu (\log |s_ E|^2,y)=1$, gives $\nu (u',y)=\nu (u,y)+c_ E$ at such points $y$. At points that are moreover generic for both currents, which exist because the points of $E$ excluded by these conditions form a countable union of proper analytic subsets of $E$ (proposition 6.9.7), this reads $\nu (\pi ^*\pi _*S,E)=\nu (S,E)+c_ E$; hence $c_ E=\nu (\pi ^*\pi _*S,E)\geq 0$, since $S$ is non-divisorial.

Since $\pi $ is a biholomorphism over $X\setminus A$, the positive current $\pi _*\pi ^*T$ agrees with $T$ on $X\setminus A$. Hence $\pi _*\pi ^*T-T$ is a closed real current with measure coefficients (theorem 5.3.7) supported on $A$, which has codimension at least two, so it vanishes by theorem 9.3.20. For a prime divisor $E$ on $Y$, the current $\pi _*[E]$ is positive, closed and supported on the analytic set $\pi (E)$. If $E$ is exceptional, $\pi (E)$ has codimension at least two and $\pi _*[E]=0$ by the same theorem. Otherwise $\pi $ maps $E\setminus \pi ^{-1}(A)$ isomorphically onto $\pi (E)\setminus A$, so $\pi _*[E]=[\pi (E)]$ on $X\setminus A$, and the closed current $\pi _*[E]-[\pi (E)]$ of order zero is supported on $\pi (E)\cap A$, of codimension at least two in $X$; it vanishes by the same theorem. $\square$

Proof. Choose $C$ with $\pi ^*\omega \leq C\omega _ Y$. Since $\pi _*$ preserves positivity, lemma 10.5.7 gives $\pi _*\omega _ Y\geq C^{-1}\pi _*\pi ^*\omega =C^{-1}\omega $. Hence a current $S\geq \delta \omega _ Y$ on $Y$ has $\pi _*S\geq \delta C^{-1}\omega $, which proves the assertion for $\pi _*$.

Let $\alpha $ be big on $X$ and $T\in \alpha $ with $T\geq \delta \omega $. Choose a smooth closed representative $\eta $ of the class $\{ \pi _*\omega _ Y\} $, a constant $B>0$ with $\eta \leq B\omega $ (lemma 6.14.1), and $t>0$ with $tB\leq \delta $, so that $T-t\eta \geq 0$. The positive current $T-t\eta +t\pi _*\omega _ Y$ lies in $\alpha $. By lemma 10.5.7, applied to the smooth, hence non-divisorial, current $\omega _ Y$, its pullback is

This is a Kähler current in $\pi ^*\alpha $, so $\pi ^*\alpha $ is big. $\square$