Pome

5.19 The space of currentsEB85809E6CA8

EB85809E6CA8

Let $X$ be a connected compact Kähler manifold of dimension $n$ and $\alpha \in \mathrm{H}^{1,1}(X,\mathbb {R})$.

Definition 5.19.1.DF4FCB370781 Let $Y$ be a complex manifold of dimension $N$ and $m\in \mathbb {N}$. We say an $(m,m)$-current $T$ on $Y$ is positive1 if either $m>N$ or for any smooth $(1,0)$-forms $\beta _1,\ldots ,\beta _{N-m}$ on $Y$, the measure

\[ T\wedge \mathrm{i}\beta _1\wedge \overline{\beta _1}\wedge \cdots \wedge \mathrm{i}\beta _{N-m}\wedge \overline{\beta _{N-m}} \]

is positive.

The basic properties of positive currents can be found in [Section III.1, DemBook]. Recall that a positive current is necessarily real.

Definition 5.19.2.AE4F30A1999E We say $\alpha $ is pseudo-effective if there is a closed positive $(1,1)$-current in $\alpha $.

We say $\alpha $ is big if there is a closed positive $(1,1)$-current $T$ in $\alpha $ dominating a Kähler form. Such currents are called Kähler currents.

Given classes $\alpha ,\beta \in \mathrm{H}^{1,1}(X,\mathbb {R})$, we say $\alpha \leq \beta $ if $\beta -\alpha $ is pseudo-effective.

Definition 5.19.3.49CF4A9C5660 We introduce the following notations:

  1. 9DEEC889C347

    $\mathcal{Z}_+(X)$ denotes the space of closed positive $(1,1)$-currents on $X$;

  2. 146392D01DAE

    given a pseudo-effective $(1,1)$-class $\alpha $ on $X$, we write $\mathcal{Z}_+(X,\alpha )$ for the set of $T\in \mathcal{Z}_+(X)$ such that $\{ T\} =\alpha $.

Here $\{ T\} $ denotes the cohomology class represented by $T$.

definition 16.7.1 has a natural analogue for currents.

Definition 5.19.4.0909012E7F31 Given $T,T'\in \mathcal{Z}_+(X)$, we write $T\preceq T'$ and say $T$ is more singular than $T'$ if when we write $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $, $T'=\theta '+\mathrm{dd}^{\mathrm{c}}\varphi '$, we have $\varphi \preceq \varphi '$. We write $T\sim T'$ if $T\preceq T'$ and $T'\preceq T$. In this case, we say $T$ and $T'$ have the same singularity type.

Remark 5.19.5.877B9D0BDD26 Observe that

\[ \mathcal{Z}_+(X)/\sim \cong \mathrm{QPSH}(X)/\sim \]

canonically. The correspondence sends the class of a closed positive current $\theta _{\varphi }=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ to the class of $\varphi $.

We will adopt the following convention: Whenever we have a notion for quasi-plurisubharmonic functions which depends only on the singularity type, we use the same notation and the same definition for closed positive $(1,1)$-currents.

Example 5.19.6.AAD0C8775B34 As an important example of 5.19.5, given $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi \in \mathcal{Z}_+(X)$ and $x\in X$, we define

3E19B5D1CC32 5.19.7
\begin{equation} \nu (T,x)=\nu (\varphi ,x). \end{equation}

Again, as in 5.16.3, this differs from the definitions in some literature by a factor of $2$. But given our normalization

\[ \mathrm{dd}^{\mathrm{c}}=\frac{\mathrm{i}}{2\pi }\partial \overline{\partial }, \]

(5.19.7) seems to be the most natural choice.

The key example to keep in mind is the following:

\[ \nu ([0],0)=1, \]

where $[0]$ is the current of integration at $0\in \mathbb {P}^1$. In fact, as a simple application of the Green's second identity, one can verify that

\[ \frac{\mathrm{i}}{2\pi }\partial \overline{\partial }\log |z|^2=\delta _0, \]

where the right-hand side is the Dirac delta distribution at $0\in \mathbb {C}$.

Definition 5.19.8.4A951ED7EEE8 Given $T\in \mathcal{Z}_+(X)$, represent $T$ as $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ for some closed smooth real $(1,1)$-form $\theta $ on $X$ and $\varphi \in \mathrm{PSH}(X,\theta )$. The polar locus of $T$ is defined as the set $\{ \varphi =-\infty \} $.

It is clear that the polar locus of $T$ is independent of the choices of $\theta $ and $\varphi $. We also call the set $\{ -\infty \} $ the polar locus of $\varphi $.

Definition 5.19.9.AEE3AF4068CD Assume that $\alpha $ is big. The non-Kähler locus $\operatorname{nK}(\alpha )$ of $\alpha $ is the intersection of the polar loci of all Kähler currents with analytic singularities in $\alpha $.

Theorem 5.19.10 (Boucksom).847815EFB4FE Assume that $\alpha $ is big. There is a Kähler current $T\in \alpha $ with analytic singularities, such that the polar locus of $T$ is exactly $\operatorname{nK}(\alpha )$. In particular, $\operatorname{nK}(\alpha )$ is a proper Zariski closed subset of $X$.

Proof. The big-class part of the theorem on positivity loci constructs a single Kähler current with analytic singularities whose pole set is $E_{\mathrm{nK}}(\alpha)$. This set is the same as $\operatorname{nK}(\alpha)$ here, by the defining intersection of singular loci of Kähler currents in $\alpha$. The same theorem proves that this pole set is a proper analytic subset. Thus its constructed current has all the required properties, including equality, rather than just containment, of the polar locus.

See [Théorème 2.1.20, Bou02].

Definition 5.19.11.1AA67767FB84 Assume that $\alpha $ is big. The non-nef locus $\operatorname{nn}(\alpha )$ of $\alpha $ is the following set:

\[ \operatorname{nn}(\alpha )\coloneqq \{ x\in X :\nu (T_{\min },x)>0\} , \]

where $T_{\min }$ is a current with minimal singularities in $\alpha $.

Note that $\operatorname{nn}(\alpha )\subseteq \operatorname{nK}(\alpha )$. Thanks to theorem 5.16.7, $\operatorname{nn}(\alpha )$ is a countable union of proper Zariski closed subsets of $X$. The non-Kähler locus and non-nef locus are studied in detail in [Bou02].

Lemma 5.19.12 (Siu's decomposition).F956E74CD8C5 Let $E$ be a prime divisor on $X$. Then for any closed positive $(1,1)$-current $T$ on $X$, the difference $T-\nu (T,E)[E]$ is a closed positive $(1,1)$-current.

Proof. By the proved divisorial decomposition, \[T=R+\sum_D\nu(T,D)[D],\qquad R\geq0,\] where the series converges weakly and every summand is positive. Remove the summand with $D=E$; the remaining weakly convergent sum together with $R$ is positive and closed. It is exactly $T-\nu(T,E)[E]$. If $\nu(T,E)=0$, the assertion reduces to positivity of $T$.

Here $[E]$ is the current of integration associated with $E$, while $\{ E\} $ denotes the cohomology class of $E$. See [Page 386, Example 1, GH14] for the precise definition. See [Lemma 2.17, Dem12] for the proof.

As a consequence, for each closed positive $(1,1)$-current $T$ on $X$, we can write

6720366F8D5B 5.19.13
\begin{equation} T=\operatorname{Reg}T+\sum _{i} c_ i [E_ i], \end{equation}

where $\{ E_ i\} $ is a countable collection of prime divisors on $X$, $c_ i>0$.

Definition 5.19.14.5545837BA9C1 A closed positive $(1,1)$-current $T$ on $X$ is non-divisorial (resp. divisorial) if $T=\operatorname{Reg}T$ (resp. $\operatorname{Reg}T=0$).

It is helpful to check that our conventions are always consistent: There is no extra factor of $2$ or $1/2$ anywhere. One can verify this using our favorite example as in example 5.19.6.

Next we recall the notion of modified nef classes.

Definition 5.19.15.4DFBEC43A4B0 A class $\alpha \in \mathrm{H}^{1,1}(X,\mathbb {R})$ is modified nef if the following condition holds: after fixing a reference Kähler metric $\omega $ on $X$, for any $\epsilon >0$, we can find a closed $(1,1)$-current $T\in \alpha $ such that

  1. EEF77CF757EA

    $T+\epsilon \omega \geq 0$;

  2. AEF6FAE47DAC

    $\nu (T+\epsilon \omega ,D)=0$ for any prime divisor $D$ on $X$.

This definition is independent of the choice of $\omega $.

These classes are called nef en codimension $1$ in Boucksom's thesis [Bou02], where they were introduced for the first time. They are also known as movable classes in other references. Modified nef classes form a closed convex cone in $\mathrm{H}^{1,1}(X,\mathbb {R})$. Note that a modified nef class is necessarily pseudo-effective. A nef class is obviously modified nef. Recall the multiplicity of a cohomology class as defined in [Section 2.1.3, Bou02]:

Definition 5.19.16.0EB97717F408 Let $\alpha \in \mathrm{H}^{1,1}(X,\mathbb {R})$ be a pseudo-effective class and $D$ be a prime divisor on $X$. We define the Lelong number $\nu (\alpha ,D)$ as follows:

  1. C6833F83A5C1

    When $\alpha $ is big, define $\nu (\alpha ,D)=\nu (T,D)$ for any closed positive $(1,1)$-current $T\in \alpha $ with minimal singularities.

  2. 3CE6AD34390C

    In general, define

    \[ \nu (\alpha ,D)\coloneqq \lim _{\epsilon \to 0+}\nu (\alpha +\epsilon \{ \omega \} ,D). \]
When $\alpha $ is big, 2 is compatible with 1 and the definition is independent of the choice of $\omega $.

By definition, a pseudo-effective class $\alpha $ is modified nef if and only if $\nu (\alpha ,D)=0$ for all prime divisors $D$ on $X$.

Let us recall the behavior of several cones under modifications.

Proposition 5.19.17.191D96316EC3 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$.

  1. 4F00658A644B

    For any nef class $\alpha \in \mathrm{H}^{1,1}(X,\mathbb {R})$, $\pi ^*\alpha $ is nef.

  2. 98B07470D933

    For any modified nef class $\beta \in \mathrm{H}^{1,1}(Y,\mathbb {R})$, $\pi _*\beta $ is modified nef.

Proof. Only 2 requires a proof. Fix a Kähler class $\gamma $ on $Y$. Since the modified nef cone is closed, replacing $\beta $ by $\beta +\epsilon \gamma $ and then letting $\epsilon \to 0+$ reduces the proof to the case where $\beta $ is big as well. Let $T$ (resp. $S$) be a current with minimal singularities in $\pi _*\beta $ (resp. in $\beta $), and let $D$ be a prime divisor on $X$. It suffices to show that

\[ \nu (T,D)=0. \]

Since $\beta $ is modified nef, $S$ is non-divisorial. Hence by 5.19.18 below, $\nu (\pi _*S,D)=0$. Since $T$ has minimal singularities in $\pi _*\beta $, we have $T\succeq \pi _*S$, and the desired equality follows. $\square$

Recall that non-divisorial currents are introduced in definition 5.19.14.

Lemma 5.19.18.9F062B47A2B4 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$. Let $T$ be a non-divisorial current on $Y$. Then $\pi _*T$ is non-divisorial.

Conversely, if $S$ is a non-divisorial current on $X$, $\pi ^*S$ could have divisorial part. As a simple example, consider $S$ on $\mathbb {P}^2$, whose local potential near $0\in \mathbb {C}^2_{z,w}$ looks like $\log (|z|^2+|w|^2)$.

Proof. Let $D$ be a prime divisor on $X$. It follows from Zariski's main theorem theorem 9.10.5 that $D$ is not contained in the exceptional locus of $\pi $. Let $D'$ be the strict transform of $D$. Thanks to Siu's semicontinuity theorem theorem 5.16.7, we have

\[ \nu (\pi _*T,D)=\nu (T,D')=0. \]

Hence $\pi _*T$ is non-divisorial. $\square$

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub