5.3 Currents and positivityACB622806098

ACB622806098

Let $X$ be a complex manifold of dimension $n$. A test form on $X$ is a smooth complex differential form with compact support; the test forms of degree zero are the test functions. In a chart with real coordinates $t_1,\ldots ,t_{2n}$ we write test forms as $\varphi =\sum _ J\varphi _ J\, \mathrm dt_ J$ and measure them by the seminorms $p_{K,m}$ of section 2.4 applied to their coefficients.

For an integer $0\leq k\leq 2n$, a current of degree $k$ on $X$ is a complex-linear functional $T$ on the test forms of degree $2n-k$ with the following continuity property: for every coordinate chart and every compact set $K$ in it, there are $C>0$ and $m\in \mathbb N$ such that $|T(\varphi )|\leq C\sum _ Jp_{K,m}(\varphi _ J)$ for all test forms $\varphi $ of degree $2n-k$ supported in $K$. Equivalently, in every chart and for every multi-index $J$ with $|J|=2n-k$, the functional $\chi \mapsto T(\chi \, \mathrm dt_ J)$ is a distribution. In a chart there are therefore unique distributions $T_ J$, $|J|=k$, such that

for all test forms $\varphi $ supported in the chart. Namely, if $J'$ is the increasing multi-index complementary to $J$ and $\mathrm dt_ J\wedge \mathrm dt_{J'}=\varepsilon _ J\, \mathrm dt_1\wedge \cdots \wedge \mathrm dt_{2n}$ with $\varepsilon _ J=\pm 1$, then $\langle T_ J,\chi \rangle =\varepsilon _ JT(\chi \, \mathrm dt_{J'})$, because $g_ J=\varepsilon _ J\varphi _{J'}$ for $\varphi =\sum _ L\varphi _ L\, \mathrm dt_ L$. We write $T=\sum _ JT_ J\, \mathrm dt_ J$ and call the $T_ J$ the coefficients of $T$; more generally, for multi-indices $M$ that are not increasing, $S\, \mathrm dt_ M$ denotes the current $\varphi \mapsto \langle S,g\rangle $, where $\mathrm dt_ M\wedge \varphi =g\, \mathrm dt_1\wedge \cdots \wedge \mathrm dt_{2n}$. In holomorphic coordinates we use the basis $\mathrm dz_ I\wedge \mathrm d\bar z_ J$ in the same way. Thus locally a current is a differential form with distribution coefficients. A locally integrable $k$-form $\alpha $, that is, a form with locally integrable coefficients, defines the current $T_\alpha (\varphi )=\int _ X\alpha \wedge \varphi $ of degree $k$, which we usually denote again by $\alpha $.

A current of degree $k$ has bidegree $(p,q)$, where $p+q=k$, if it vanishes on all test forms of bidegree other than $(n-p,n-q)$; in holomorphic coordinates it is then $T=\sum _{|I|=p,|J|=q}T_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$. Its bidimension is $(n-p,n-q)$. Every current is the sum of its components of pure bidegree. Forms and currents of a degree or bidegree outside the admissible range are taken to be zero. The conjugate of a current $T$ is $\overline T(\varphi )=\overline{T(\bar\varphi )}$, and $T$ is real if $\overline T=T$, that is, $T(\bar\varphi )=\overline{T(\varphi )}$ for every test form $\varphi $.

A sequence, or more generally a net, of currents $T_ i$ of degree $k$ converges weakly to $T$ if $T_ i(\varphi )\to T(\varphi )$ for every test form $\varphi $ of degree $2n-k$. For currents of degree $2n$ given by positive measures, weak convergence is the vague convergence of these measures; see the discussion following definition 5.3.5.

As for distributions (section 2.4), a current $T$ vanishes on an open set $W\subset X$ if $T(\varphi )=0$ for every test form $\varphi $ with $\operatorname{Supp}\varphi \subset W$, and the support $\operatorname{Supp}T$ of $T$ is the complement of the union of all open sets on which $T$ vanishes. If $T$ vanishes on each member of a family of open sets, it vanishes on their union, because a partition of unity subordinate to the cover formed by these sets and by $X\setminus \operatorname{Supp}\varphi $ (proposition 5.2.6) writes every test form $\varphi $ supported in the union as a finite sum of test forms supported in single members. Hence $X\setminus \operatorname{Supp}T$ is the largest open set on which $T$ vanishes, and $T(\varphi )=0$ whenever $\operatorname{Supp}\varphi \cap \operatorname{Supp}T=\varnothing $.

Let $T$ be a current of degree $k$ and $\alpha $ a smooth form. We define

and, in the same way, $(\partial T)(\varphi )=(-1)^{k+1}T(\partial \varphi )$ and $(\bar\partial T)(\varphi )=(-1)^{k+1}T(\bar\partial \varphi )$. These are currents, because $\mathrm d$, $\partial $, $\bar\partial $ and multiplication by $\alpha $ map test forms supported in a compact set $K$ of a chart to test forms supported in $K$, and the seminorms $p_{K,m}$ of the new coefficients are bounded by a constant times the seminorms $p_{K,m+1}$ of the old ones. For a smooth form $T$ these definitions agree with the exterior derivative and the exterior product: by the Leibniz rule and proposition 5.2.6, $\int _ X\mathrm dT\wedge \varphi =(-1)^{k+1}\int _ XT\wedge \mathrm d\varphi $, and applying this identity to the components of pure bidegree of $T$ and $\varphi $, of which only the products of bidegree $(n,n)$ have nonzero integrals, we obtain the analogous identities for $\partial $ and $\bar\partial $.

In a chart, these operations act on the distribution coefficients exactly as on the coefficients of smooth forms. Indeed, let $t=(x_1,y_1,\ldots ,x_ n,y_ n)$ be the real coordinates of a holomorphic chart, let $\varphi $ be a test form of degree $2n-k-1$ supported in the chart, and write $\mathrm dt_ l\wedge \mathrm dt_ J\wedge \varphi =g_{l,J}\, \mathrm dt_1\wedge \cdots \wedge \mathrm dt_{2n}$ and $\mathrm dt_ J\wedge \mathrm d\varphi =h_ J\, \mathrm dt_1\wedge \cdots \wedge \mathrm dt_{2n}$. For every smooth function $a$ on the chart, the identity just proved for the smooth form $a\, \mathrm dt_ J$ reads $\sum _ l\int \partial _{t_ l}a\, g_{l,J}\, \mathrm dV_{2n}=(-1)^{k+1}\int ah_ J\, \mathrm dV_{2n}$. Integrating by parts on the left, we find $(-1)^{k+1}h_ J=-\sum _ l\partial _{t_ l}g_{l,J}$. Hence, for a current $T=\sum _ JT_ J\, \mathrm dt_ J$,

that is, $\mathrm dT=\sum _{J,l}\partial _{t_ l}T_ J\, \mathrm dt_ l\wedge \mathrm dt_ J$ with distributional derivatives. The same computation applies to $\partial $ and $\bar\partial $ in the basis $\mathrm dz_ I\wedge \mathrm d\bar z_ J$, and directly from the definitions $T\wedge \alpha =\sum _{J,L}\alpha _ LT_ J\, \mathrm dt_ J\wedge \mathrm dt_ L$ for $\alpha =\sum _ L\alpha _ L\, \mathrm dt_ L$. If $\alpha $ has degree $r$, then, by the Leibniz rule for $\alpha \wedge \varphi $,

so that $\mathrm d(T\wedge \alpha )=\mathrm dT\wedge \alpha +(-1)^ kT\wedge \mathrm d\alpha $, and similarly for $\partial $ and $\bar\partial $. In general, the product of two currents is not defined. We put $\mathrm{dd}^{\mathrm{c}}T=(\mathrm{i}/2\pi )\partial \bar\partial T$ and $\mathrm d^{\mathrm c}T=(\mathrm{i}/4\pi )(\bar\partial T-\partial T)$. Since $(\partial \bar\partial T)(\varphi )=(-1)^{k+2}(\bar\partial T)(\partial \varphi )=-T(\bar\partial \partial \varphi )=T(\partial \bar\partial \varphi )$, and since the identities $\mathrm d=\partial +\bar\partial $, $\partial ^2=\bar\partial ^2=0$ and $\bar\partial \partial =-\partial \bar\partial $ pass from test forms to currents by duality, we obtain

All these operations are continuous for weak convergence, directly from the definitions. A current $T$ is closed if $\mathrm{d}T=0$. For a current of pure bidegree this is equivalent to $\partial T=\bar\partial T=0$, since $\partial T$ and $\bar\partial T$ are the components of $\mathrm dT$ of two different bidegrees.

We now turn to positivity of forms, which is a question of linear algebra. Let $V$ be a complex vector space of dimension $n$, identified with $\mathbb C^ n$ by linear coordinates $z_1,\ldots ,z_ n$, and let $\beta $ be the form (5.2.8) in these coordinates; the forms below are constant forms on $V$. For an integer $0\leq p\leq n$, a real $(p,p)$-form is strongly positive if it is a finite sum, with nonnegative coefficients, of the forms

where $\theta _1,\ldots ,\theta _ p\in V^*$ are complex-linear forms, that is, forms of type $(1,0)$. We call the forms (5.3.1) generators. The identity in (5.3.1) holds because rearranging the factors of the right-hand side produces the sign $(-1)^{p(p-1)/2}=\mathrm{i}^{p^2-p}$, and the generators are real because each factor $\mathrm i\theta _ j\wedge \bar\theta _ j$ is real. A real $(p,p)$-form is weakly positive if its exterior product with every strongly positive $(n-p,n-p)$-form is a nonnegative multiple of the volume form $\beta ^ n/n!$. In degree zero both notions mean a nonnegative real number. A form on a complex manifold is strongly or weakly positive if it is so at every point, as a form on the tangent space. Finally, the restriction of a $(p,p)$-form to a complex subspace $W$ of dimension $p$ is a form of top degree on $W$; we call it nonnegative if it is a nonnegative multiple of the volume form of $W$ in its complex orientation.

Proposition 5.3.2.E58104E974B0 Let $0\leq p\leq n$. The strongly positive $(p,p)$-forms on $V$ form a closed convex cone, which spans the real vector space of real $(p,p)$-forms and contains $\beta ^ p$ in its interior. A real $(p,p)$-form is weakly positive if and only if its restriction to every complex $p$-dimensional subspace of $V$ is nonnegative. Pullback by complex-linear maps, and hence by holomorphic maps between complex manifolds, preserves strong and weak positivity. The product of strongly positive forms is strongly positive, and the product of a weakly positive form with a strongly positive form is weakly positive when its bidegree is at most $(n,n)$. Moreover, strongly positive forms are weakly positive.

Proof. We use the Hermitian inner product on $V^*$ for which $\mathrm dz_1,\ldots ,\mathrm dz_ n$ are orthonormal. The form $\beta $ is invariant under unitary changes of coordinates: if $\mathrm dw_ j=\sum _ kU_{jk}\, \mathrm dz_ k$ with a unitary matrix $U$, then $\sum _ j\mathrm dw_ j\wedge \mathrm d\bar w_ j=\sum _{k,l}\big (\sum _ jU_{jk}\overline{U_{jl}}\big )\mathrm dz_ k\wedge \mathrm d\bar z_ l=\sum _ k\mathrm dz_ k\wedge \mathrm d\bar z_ k$. Let $\Lambda ^{p,0}V^*$ be the space of $(p,0)$-forms. For $\theta \in \Lambda ^{p,0}V^*$ put $g(\theta )=\mathrm i^{p^2}\theta \wedge \bar\theta $, so that $g(c\theta )=|c|^2g(\theta )$ and the generators are the forms $g(\theta )$ with $\theta $ decomposable.

Closedness. Let $G$ be the set of the generators $g(e_1\wedge \cdots \wedge e_ p)$ with $e_1,\ldots ,e_ p$ orthonormal in $V^*$. It is the continuous image of the compact set of orthonormal $p$-tuples, so $G$ is compact. Every nonzero generator $g(\theta _1\wedge \cdots \wedge \theta _ p)$ is a positive multiple of an element of $G$. Indeed, the $\theta _ j$ are linearly independent, and the Gram–Schmidt process gives an orthonormal family $e_1,\ldots ,e_ p$ such that each $\theta _ j$ is a linear combination of $e_1,\ldots ,e_ j$; hence $\theta _1\wedge \cdots \wedge \theta _ p=c\, e_1\wedge \cdots \wedge e_ p$, where $c\neq 0$ is the product of the diagonal entries of the triangular transition matrix, and $g(\theta _1\wedge \cdots \wedge \theta _ p)=|c|^2g(e_1\wedge \cdots \wedge e_ p)$.

Define a linear functional $\tau $ on real $(p,p)$-forms by $u\wedge \beta ^{n-p}=\tau (u)\beta ^ n/n!$. If $e_1,\ldots ,e_ p$ is orthonormal, complete it to an orthonormal basis $e_1,\ldots ,e_ n$ of $V^*$ and use the unitary coordinates $w_ j$ with $\mathrm dw_ j=e_ j$. Then $\beta =(\mathrm{i}/2)\sum _ j\mathrm dw_ j\wedge \mathrm d\bar w_ j$, only one term of $\beta ^{n-p}$ contributes, and $\tau (g(e_1\wedge \cdots \wedge e_ p))=2^ p(n-p)!$. Thus $\tau $ is a positive constant $c_0$ on $G$. Let $H$ be the convex hull of $G$ in the real vector space of real $(p,p)$-forms, of dimension $N$ say. Then $H$ is compact, by Carathéodory's argument: if a point is a convex combination of more than $N+1$ points of $G$, these points are affinely dependent, and subtracting a suitable multiple of an affine dependence relation makes one coefficient zero while the others stay nonnegative. Repeating this, we see that $H$ is the continuous image of the compact set $G^{N+1}\times \Delta _ N$, where $\Delta _ N$ is the standard simplex of dimension $N$. Every strongly positive form can be written as $u=sh$ with $h\in H$ and $s=\tau (u)/c_0\geq 0$. If strongly positive forms $u_ k=s_ kh_ k$ converge to $u$, then $s_ k=\tau (u_ k)/c_0$ converges to some $s\geq 0$, and a subsequence of $(h_ k)$ converges to some $h\in H$; hence $u=sh$ is strongly positive. The cone is convex by definition.

Spanning. Let $\lambda $ be a real-linear functional on the real $(p,p)$-forms that vanishes on all generators; we show that $\lambda =0$. Every $(p,p)$-form is uniquely $u+\mathrm iv$ with $u,v$ real, so $\lambda $ extends complex-linearly to all $(p,p)$-forms. The form $B(a,b)=\lambda (\mathrm i^{p^2}a\wedge \bar b)$ on $\Lambda ^{p,0}V^*$ is sesquilinear, and $B(a,a)=0$ for every decomposable $a$. We prove by induction on $r=0,\ldots ,p$ that $B(a_1\wedge \cdots \wedge a_ p,b_1\wedge \cdots \wedge b_ p)=0$ whenever $a_ j=b_ j$ for all $j>r$. For $r=0$ this is the hypothesis. Assume it for $r-1$, and fix all vectors except the $r$th ones. The resulting form $B_ r(x,y)=B(\cdots \wedge x\wedge \cdots ,\cdots \wedge y\wedge \cdots )$, with $x$ and $y$ in the $r$th slots, is sesquilinear and vanishes for $x=y$ by the induction hypothesis. By the polarization identity $B_ r(x,y)=\frac14\sum _{k=0}^3\mathrm{i}^ kB_ r(x+\mathrm{i}^ ky,x+\mathrm{i}^ ky)$, valid for every sesquilinear form, $B_ r$ vanishes identically. For $r=p$ we obtain $B=0$ on decomposable vectors, which span $\Lambda ^{p,0}V^*$, so $B=0$. Since the forms $a\wedge \bar b$ with $a,b\in \Lambda ^{p,0}V^*$ span all $(p,p)$-forms, $\lambda =0$. Hence the generators span the real $(p,p)$-forms over $\mathbb R$.

Interior. Since the generators span the real $(p,p)$-forms and every nonzero generator is a positive multiple of an element of $G$, we can choose finitely many elements $q_1,\ldots ,q_ M$ of $G$ spanning the real $(p,p)$-forms. For a permutation $\sigma $ of $\{ 1,\ldots ,n\} $ and signs $\varepsilon \in \{ \pm 1\} ^ n$, consider the linear map $z\mapsto (\varepsilon _1z_{\sigma (1)},\ldots ,\varepsilon _ nz_{\sigma (n)})$; its pullback maps generators to generators. Let $\widehat q_ r$ be the average of the pullbacks of $q_ r$ under all these maps.

To compute $\widehat q_ r$, write $q_ r=\mathrm i^{p^2}\sum _{I,J}c_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$. The sign changes multiply $c_{IJ}$ by $\prod _{i\in I}\varepsilon _ i\prod _{j\in J}\varepsilon _ j$, whose average over $\varepsilon $ vanishes unless $I=J$. A permutation maps $\mathrm dz_ I\wedge \mathrm d\bar z_ I$ to $\mathrm dz_{I'}\wedge \mathrm d\bar z_{I'}$, where $I'$ is the permuted index set in increasing order, because reordering the factors produces the same sign for $\mathrm dz_{I'}$ and for $\mathrm d\bar z_{I'}$; since every $p$-element set is mapped to every other one by the same number of permutations, the average makes all diagonal coefficients equal to $\binom np^{-1}\sum _ Ic_{II}$. Since $q_ r=g(\theta )$ with $\theta =\sum _ I\theta _ I\, \mathrm dz_ I\neq 0$, we have $c_{IJ}=\theta _ I\overline{\theta _ J}$ and $s_ r=\sum _ Ic_{II}=\sum _ I|\theta _ I|^2>0$. Therefore, by (5.3.1) and the expansion $\beta ^ p=p!\sum _{|I|=p}\bigwedge _{i\in I}(\frac{\mathrm i}2\mathrm dz_ i\wedge \mathrm d\bar z_ i)$,

\[ \widehat q_ r=s_ r\binom np^{-1}\mathrm{i}^{p^2}\sum _{|I|=p}\mathrm{d}z_ I\wedge \mathrm{d}\bar z_ I =\frac{2^ ps_ r}{p!\binom np}\beta ^ p. \]

Consequently $\beta ^ p$ is a positive multiple of $\sum _ r\widehat q_ r$, which is a combination of generators with strictly positive coefficients in which every $q_ r$ occurs. A real $(p,p)$-form $v$ of small norm can be written as $\sum _ rt_ r(v)q_ r$ with small coefficients depending linearly on $v$. Adding these coefficients to the positive coefficients of the $q_ r$ keeps all of them positive, so $\beta ^ p+v$ is strongly positive. Thus $\beta ^ p$ is an interior point.

Restriction criterion. Let $W$ be a complex subspace of dimension $p$, and choose linear coordinates in which $W=\{ z_{p+1}=\cdots =z_ n=0\} $. For a real $(p,p)$-form $u$, the product $u\wedge \bigwedge _{j>p}(\mathrm{i}\mathrm{d}z_ j\wedge \mathrm{d}\bar z_ j)$ involves only the coefficient of $\mathrm dz_{(1,\ldots ,p)}\wedge \mathrm d\bar z_{(1,\ldots ,p)}$ in $u$, which is also the coefficient of the restriction $u|_ W$ in the coordinates $z_1,\ldots ,z_ p$ of $W$. Hence this product is a nonnegative multiple of the volume form if and only if $u|_ W$ is nonnegative. Conversely, every generator $\bigwedge _{j=1}^{n-p}(\mathrm{i}\theta _ j\wedge \bar\theta _ j)$ of bidegree $(n-p,n-p)$ is zero if the $\theta _ j$ are linearly dependent, and otherwise it is of the above form, with $z_{p+j}=\theta _ j$ and $W=\{ \theta _1=\cdots =\theta _{n-p}=0\} $. Since weak positivity need only be tested on generators, the restriction criterion follows.

Pullbacks and products. Let $F\colon V'\to V$ be complex linear. The pullback of the generator $\bigwedge _ j(\mathrm i\theta _ j\wedge \bar\theta _ j)$ is the generator $\bigwedge _ j(\mathrm iF^*\theta _ j\wedge \overline{F^*\theta _ j})$, so $F^*$ preserves strong positivity. Let $u$ be weakly positive on $V$ and let $W'\subset V'$ be a complex subspace of dimension $p$. If $F|_{W'}$ is injective, it is a complex-linear isomorphism onto the $p$-dimensional subspace $F(W')$ and preserves the complex orientations, so $(F^*u)|_{W'}=(F|_{W'})^*(u|_{F(W')})$ is nonnegative. Otherwise $F(W')$ has dimension less than $p$, and $(F^*u)|_{W'}=0$ because it factors through the $(p,p)$-forms on $F(W')$. By the restriction criterion, $F^*u$ is weakly positive. For a holomorphic map, apply this to its derivative at each point.

A product of generators is a generator, so products of strongly positive forms are strongly positive. If $u$ is a weakly positive $(p,p)$-form and $s$ a strongly positive $(r,r)$-form with $p+r\leq n$, then $s\wedge t$ is strongly positive for every strongly positive $(n-p-r,n-p-r)$-form $t$, so $u\wedge s\wedge t$ is a nonnegative multiple of the volume form; hence $u\wedge s$ is weakly positive. Finally, an $(n,n)$-generator equals $|\det (\theta _1,\ldots ,\theta _ n)|^2\bigwedge _ j(\mathrm i\mathrm dz_ j\wedge \mathrm d\bar z_ j)$, a nonnegative multiple of the volume form, so strongly positive forms are weakly positive. $\square$

In bidegree $(1,1)$ the two notions coincide. Write a real $(1,1)$-form as $u=\mathrm{i}\sum a_{j\bar k}\mathrm{d}z_ j\wedge \mathrm{d}\bar z_ k$, where reality means that the matrix $A=(a_{j\bar k})$ is Hermitian. The restriction of $u$ to the complex line spanned by a vector $\xi $ is $(\sum _{j,k}a_{j\bar k}\xi _ j\bar\xi _ k)\, \mathrm i\mathrm dt\wedge \mathrm d\bar t$ in the coordinate $t$ of the line $\{ t\xi :t\in \mathbb C\} $. Hence, by the restriction criterion of proposition 5.3.2, $u$ is weakly positive if and only if $A$ is positive semidefinite. In that case a unitary diagonalization of $A$ writes $u$ as $\sum _ r\lambda _ r\mathrm{i}\theta _ r\wedge \bar\theta _ r$ with $\lambda _ r\geq 0$, a sum of generators.

By duality, the two notions also coincide in bidegree $(n-1,n-1)$. Let $u$ be a real $(n-1,n-1)$-form. For $\theta ,\eta \in V^*$ define $B_ u(\theta ,\eta )\in \mathbb C$ by $u\wedge \mathrm i\theta \wedge \bar\eta =B_ u(\theta ,\eta )\, \beta ^ n/n!$. Then $B_ u$ is sesquilinear, and $B_ u(\eta ,\theta )=\overline{B_ u(\theta ,\eta )}$ because $u$ is real; thus $H_ u(\theta )=B_ u(\theta ,\theta )$ is a Hermitian quadratic form on $V^*$, and it depends linearly on $u$. Since the strongly positive $(1,1)$-forms are the finite sums of the forms $\mathrm i\theta \wedge \bar\theta $, the form $u$ is weakly positive if and only if $H_ u\geq 0$.

Moreover, $H_ u$ determines $u$. Indeed, $H_ u$ determines $B_ u$ by the polarization identity $B_ u(\theta ,\eta )=\frac14\sum _{k=0}^3\mathrm i^ kH_ u(\theta +\mathrm i^ k\eta )$, which holds for every form that is linear in the first and conjugate-linear in the second variable (expand $H_ u(\theta +\mathrm i^ k\eta )=H_ u(\theta )+\mathrm i^{-k}B_ u(\theta ,\eta )+\mathrm i^ kB_ u(\eta ,\theta )+H_ u(\eta )$ and use $\sum _{k=0}^3\mathrm i^ k=\sum _{k=0}^3\mathrm i^{2k}=0$). Hence $H_ u$ determines the products $u\wedge \mathrm dz_ j\wedge \mathrm d\bar z_ k$; and if $\hat\jmath $ denotes the increasing multi-index of length $n-1$ omitting $j$, then $u\wedge \mathrm dz_ j\wedge \mathrm d\bar z_ k=c_{jk}u_{\hat\jmath \hat k}\, \beta ^ n/n!$, where $u_{\hat\jmath \hat k}$ is the coefficient of $\mathrm dz_{\hat\jmath }\wedge \mathrm d\bar z_{\hat k}$ in $u$ and $c_{jk}\neq 0$ is a constant.

Now let $u$ be weakly positive. Diagonalizing $H_ u$ with respect to the inner product on $V^*$ used in the proof of proposition 5.3.2, we find unitary coordinates $w_1,\ldots ,w_ n$ and numbers $\lambda _ j\geq 0$ such that $H_ u(\sum _ j\xi _ j\mathrm dw_ j)=\sum _ j\lambda _ j|\xi _ j|^2$. In these coordinates $\beta $ keeps its form, and for $\theta =\sum _ l\xi _ l\, \mathrm dw_ l$ we have $\bigwedge _{l\neq j}(\mathrm i\mathrm dw_ l\wedge \mathrm d\bar w_ l)\wedge \mathrm i\theta \wedge \bar\theta =2^ n|\xi _ j|^2\beta ^ n/n!$. Hence the strongly positive form $s=2^{-n}\sum _ j\lambda _ j\bigwedge _{l\neq j}(\mathrm i\mathrm dw_ l\wedge \mathrm d\bar w_ l)$ satisfies $H_ s=H_ u$, and therefore $u=s$ is strongly positive.

In general the two notions differ. Let $n=4$ and $\theta =\mathrm{d}z_1\wedge \mathrm{d}z_2+\mathrm{d}z_3\wedge \mathrm{d}z_4$. The $(2,2)$-form $\theta \wedge \bar\theta =\mathrm i^4\theta \wedge \bar\theta $ is real, since $2$-forms commute, and it is weakly positive by the restriction criterion of proposition 5.3.2: if $w_1,w_2$ are linear coordinates on a complex $2$-plane $W$, the restriction of $\theta $ to $W$ is $c\, \mathrm dw_1\wedge \mathrm dw_2$ for some $c\in \mathbb C$, and then the restriction of $\theta \wedge \bar\theta $ is $|c|^2\bigwedge _{j=1}^2(\mathrm i\mathrm dw_ j\wedge \mathrm d\bar w_ j)$, by (5.3.1).

However, $\theta \wedge \bar\theta $ is not strongly positive. To see this, identify a real $(p,p)$-form $\mathrm i^{p^2}\sum _{I,J}c_{IJ}\mathrm dz_ I\wedge \mathrm d\bar z_ J$ with the Hermitian matrix $(c_{IJ})$. A generator $\mathrm i^{p^2}\eta \wedge \bar\eta $, with $\eta =\sum _ I\eta _ I\mathrm dz_ I$ decomposable, corresponds to the rank-one matrix $(\eta _ I\bar\eta _ J)$. If $\theta \wedge \bar\theta $ were a sum of generators $\mathrm i^4\eta _ r\wedge \bar\eta _ r$ with nonzero decomposable $\eta _ r$, the corresponding matrices would satisfy $(\theta _ I\bar\theta _ J)=\sum _ r(\eta _{r,I}\bar\eta _{r,J})$. Let $\langle a,b\rangle =\sum _ Ia_ I\overline{b_ I}$ for $a=\sum _ Ia_ I\, \mathrm dz_ I$ and $b=\sum _ Ib_ I\, \mathrm dz_ I$ in $\Lambda ^{2,0}V^*$. For every $v\in \Lambda ^{2,0}V^*$ with $\langle \theta ,v\rangle =0$, evaluating the Hermitian forms of both matrices at the vector $(v_ I)$ gives $0=|\langle \theta ,v\rangle |^2=\sum _ r|\langle \eta _ r,v\rangle |^2$, so $\langle \eta _ r,v\rangle =0$ for every $r$; hence every $\eta _ r$ lies in the orthogonal complement of the orthogonal complement of $\theta $, that is, it is a multiple of $\theta $. This is impossible: $\theta \wedge \theta =2\, \mathrm dz_1\wedge \mathrm dz_2\wedge \mathrm dz_3\wedge \mathrm dz_4\ne 0$, whereas $\eta \wedge \eta =0$ for every decomposable $2$-form $\eta $.

Since the two notions agree in bidegrees $(1,1)$ and $(n-1,n-1)$, we speak simply of positive forms in these bidegrees. Such a form is strictly positive if it lies in the interior of the cone of positive forms, and a form on a complex manifold is strictly positive if it is so at every point. For example, $\beta $ and $\beta ^{n-1}$ are strictly positive by proposition 5.3.2. A real $(1,1)$-form $\mathrm i\sum a_{j\bar k}\mathrm dz_ j\wedge \mathrm d\bar z_ k$ is strictly positive if and only if the matrix $A=(a_{j\bar k})$ is positive definite. Indeed, if $\xi ^*A\xi \geq c|\xi |^2$ for all $\xi \in \mathbb C^ n$, with $c>0$, then $A+E$ is positive semidefinite for every Hermitian matrix $E$ with $|\xi ^*E\xi |\leq c|\xi |^2$ for all $\xi $; conversely, if $A$ is an interior point of the cone of positive semidefinite matrices, then $A-\varepsilon I$ is positive semidefinite for some $\varepsilon >0$. Positive $(1,1)$-forms are also called semipositive, to stress that they need not be strictly positive.

If $\omega _3,\ldots ,\omega _ n$ are fixed positive $(1,1)$-forms, the exterior product $\omega \wedge \omega '\wedge \omega _3\wedge \cdots \wedge \omega _ n$ is a multiple of the volume form that depends symmetrically and bilinearly on the real $(1,1)$-forms $\omega $ and $\omega '$. We now describe this bilinear form in terms of matrices and determine its signature. For Hermitian $n\times n$ matrices $A_1,\ldots ,A_ n$, the mixed discriminant $D(A_1,\ldots ,A_ n)$ is $1/n!$ times the coefficient of $t_1\cdots t_ n$ in the polynomial $\det (\sum _ jt_ jA_ j)$. For a Hermitian matrix $A$ put $\omega _ A=\mathrm{i}\sum _{j,k}A_{j\bar k}\mathrm{d}z_ j\wedge \mathrm{d}\bar z_ k$; thus $\omega _ I=2\beta $ for the identity matrix $I$.

Lemma 5.3.3.3D25BAD0C21E Let $n\geq 1$, and let $A,A_1,\ldots ,A_ n$ be Hermitian $n\times n$ matrices.

1. The mixed discriminant is real, symmetric and real-multilinear in its arguments, and $D(A,\ldots ,A)=\det A$. More precisely,

   \[ n!\, D(A_1,\ldots ,A_ n)=\sum _\sigma \det \big (A_{\sigma (1)}e_1,\ldots ,A_{\sigma (n)}e_ n\big ), \]

   the sum running over all permutations $\sigma $ of $\{ 1,\ldots ,n\} $, where $Ae_ l$ is the $l$th column of $A$.

2. We have

   \[ \omega _ A^ n=\det A\, \omega _ I^ n,\qquad \omega _{A_1}\wedge \cdots \wedge \omega _{A_ n}=D(A_1,\ldots ,A_ n)\, \omega _ I^ n, \]

   where $\omega _ I^ n=n!\bigwedge _ l(\mathrm i\mathrm dz_ l\wedge \mathrm d\bar z_ l)=2^ n\beta ^ n$ is a positive multiple of the volume form.

3. If $A_1,\ldots ,A_ n$ are positive semidefinite, then $D(A_1,\ldots ,A_ n)\geq 0$. If they are positive definite, then $D(A_1,\ldots ,A_ n)>0$.

Proof. (1) The $l$th column of $\sum _ jt_ jA_ j$ is $\sum _ jt_ jA_ je_ l$. Expanding the determinant by multilinearity in its columns, we obtain

\[ \det \Bigl (\sum _ jt_ jA_ j\Bigr )=\sum _\tau t_{\tau (1)}\cdots t_{\tau (n)}\det \big (A_{\tau (1)}e_1,\ldots ,A_{\tau (n)}e_ n\big ), \]

the sum running over all maps $\tau \colon \{ 1,\ldots ,n\} \to \{ 1,\ldots ,n\} $. The monomial $t_{\tau (1)}\cdots t_{\tau (n)}$ equals $t_1\cdots t_ n$ exactly when $\tau $ is a permutation, which gives the displayed formula. A permutation of $A_1,\ldots ,A_ n$ permutes the terms of this sum, so $D$ is symmetric. Each term is multilinear in $(A_1,\ldots ,A_ n)$, since each $A_ j$ occurs in exactly one column, so $D$ is multilinear. For $A_1=\cdots =A_ n=A$ all $n!$ terms equal $\det A$, so $D(A,\ldots ,A)=\det A$. Finally, $D$ is real: for real $t_ j$ the matrix $\sum _ jt_ jA_ j$ is Hermitian, so its determinant is real, being the product of its real eigenvalues; hence the polynomial $\det (\sum _ jt_ jA_ j)$ is real on $\mathbb R^ n$, and so are its coefficients, which are its partial derivatives at $t=0$ divided by factorials.

(2) In the expansion of $\omega _ A^ n$, the products $\bigwedge _ l(\mathrm dz_{j_ l}\wedge \mathrm d\bar z_{k_ l})$ vanish unless $(j_1,\ldots ,j_ n)$ and $(k_1,\ldots ,k_ n)$ are permutations of $(1,\ldots ,n)$. Since the $2$-forms $\mathrm dz_ j\wedge \mathrm d\bar z_ k$ commute, such a product equals $\bigwedge _ l(\mathrm dz_ l\wedge \mathrm d\bar z_{\pi (l)})$ for a permutation $\pi $, which arises from $n!$ ordered products, and $\bigwedge _ l(\mathrm dz_ l\wedge \mathrm d\bar z_{\pi (l)})=\operatorname {sign}(\pi )\bigwedge _ l(\mathrm dz_ l\wedge \mathrm d\bar z_ l)$, because the two products differ by a permutation of the factors $\mathrm d\bar z_ k$. Therefore

\[ \omega _ A^ n=n!\sum _\pi \operatorname {sign}(\pi )\prod _ lA_{l\overline{\pi (l)}}\bigwedge _ l(\mathrm i\mathrm dz_ l\wedge \mathrm d\bar z_ l)=n!\det A\bigwedge _ l(\mathrm i\mathrm dz_ l\wedge \mathrm d\bar z_ l)=\det A\, \omega _ I^ n, \]

where the last equality is the case $A=I$ of the preceding one. Since $\mathrm i\mathrm dz_ l\wedge \mathrm d\bar z_ l=2\, \mathrm dx_ l\wedge \mathrm dy_ l$, we get $\omega _ I^ n=2^ nn!\, \mathrm dV_{2n}=2^ n\beta ^ n$ by (5.2.8). Now let $t_1,\ldots ,t_ n$ be real, and apply the first identity to the Hermitian matrix $\sum _ jt_ jA_ j$, for which $\omega _{\sum _ jt_ jA_ j}=\sum _ jt_ j\omega _{A_ j}$. Since $2$-forms commute, the coefficient of $t_1\cdots t_ n$ in $(\sum _ jt_ j\omega _{A_ j})^ n$ is $n!\, \omega _{A_1}\wedge \cdots \wedge \omega _{A_ n}$, while in $\det (\sum _ jt_ jA_ j)\, \omega _ I^ n$ it is $n!\, D(A_1,\ldots ,A_ n)\, \omega _ I^ n$. Two polynomials that agree on $\mathbb R^ n$ have the same coefficients, and we obtain the second identity.

(3) A real $(1,1)$-form $\omega _ A$ with $A$ positive semidefinite is strongly positive, by the discussion following proposition 5.3.2: a unitary diagonalization of $A$ writes it as a sum of generators with nonnegative coefficients. By proposition 5.3.2, the product $\omega _{A_1}\wedge \cdots \wedge \omega _{A_ n}$ is then strongly positive, hence weakly positive, that is, a nonnegative multiple of the volume form. By (2) and since $\omega _ I^ n$ is a positive multiple of the volume form, $D(A_1,\ldots ,A_ n)\geq 0$. If all $A_ j$ are positive definite, choose $\varepsilon >0$ smaller than the least eigenvalue of every $A_ j$, so that every $A_ j-\varepsilon I$ is positive semidefinite, and expand $D(A_1,\ldots ,A_ n)=D\bigl ((A_1-\varepsilon I)+\varepsilon I,\ldots ,(A_ n-\varepsilon I)+\varepsilon I\bigr )$ by multilinearity. Every term is the mixed discriminant of positive semidefinite matrices, hence nonnegative, and one of them is $D(\varepsilon I,\ldots ,\varepsilon I)=\varepsilon ^ n\det I=\varepsilon ^ n>0$. $\square$

Lemma 5.3.4.8907196C3513 Let $n\geq 2$, and let $A_3,\ldots ,A_ n$ be positive definite Hermitian $n\times n$ matrices. The real symmetric bilinear form

\[ q(M,N)=D(M,N,A_3,\ldots ,A_ n) \]

on the real vector space of Hermitian $n\times n$ matrices, which has dimension $n^2$, has signature $(1,n^2-1)$. For every positive definite $H$, the restriction of $q$ to the hyperplane $\{ M:q(M,H)=0\} $ is negative definite.

Proof. We first show that, for a given tuple $A_3,\ldots ,A_ n$, the second assertion follows from the first. Let $H$ be positive definite. Then $q(H,H)>0$ by lemma 5.3.3(3), so the space of Hermitian matrices is the $q$-orthogonal direct sum of $\mathbb RH$ and $H^\perp =\{ M:q(M,H)=0\} $. The restriction of $q$ to $H^\perp $ is nondegenerate, since a vector of $H^\perp $ orthogonal to $H^\perp $ would be orthogonal to the whole space. By Sylvester's law of inertia it has signature $(0,n^2-1)$; that is, it is negative definite.

We prove the signature assertion by induction on $n$. For $n=2$ there are no fixed matrices, and expanding $\det (t_1M+t_2N)$ gives $q(M,N)=(\operatorname {tr}M\operatorname {tr}N-\operatorname {tr}(MN))/2$. The decomposition of a matrix into a scalar matrix and a trace-free matrix is $q$-orthogonal, $q(I,I)=1$, and $q(M,M)=-\operatorname {tr}(M^2)/2<0$ for a nonzero trace-free $M$, since $\operatorname {tr}(M^2)$ is the sum of the squares of the real eigenvalues of $M$. Hence the signature is $(1,3)$.

Let $n\geq 3$, and assume both assertions in dimension $n-1$. We first show that $q$ is nondegenerate for every tuple of positive definite matrices. Replacing every matrix $X$ by $G^*XG$ multiplies $D$ by $|\det G|^2$, by the definition of $D$ through the determinant, and preserves positive definiteness; since $X\mapsto G^*XG$ is a linear automorphism of the Hermitian matrices, we may take $G=A_ n^{-1/2}$ and assume that $A_ n=I$. For a Hermitian matrix $X$, let $X^{(i)}$ be the $(n-1)\times (n-1)$ matrix obtained by deleting the $i$th row and column, and let $P_ i$ be the diagonal matrix with entry $1$ in position $(i,i)$ and zeros elsewhere. Expanding along the $i$th row, $\det (tP_ i+Y)=\det Y+t\det Y^{(i)}$; comparing coefficients of $t_1\cdots t_ n$ in $\det (t_1P_ i+\sum _{j\geq 2}t_ jB_ j)$ gives, for Hermitian matrices $B_2,\ldots ,B_ n$,

\[ D(P_ i,B_2,\ldots ,B_ n)=\frac1nD_{n-1}(B_2^{(i)},\ldots ,B_ n^{(i)}), \]

where the factor $1/n=(n-1)!/n!$ comes from the normalizations of $D$ and of the mixed discriminant $D_{n-1}$ of size $n-1$. Put

\[ q_ i(B,C)=D_{n-1}(B,C,A_3^{(i)},\ldots ,A_{n-1}^{(i)}); \]

the matrices $A_ j^{(i)}$ and $I^{(i)}$ are positive definite, as principal submatrices of positive definite matrices. Using the last formula, and writing $I=\sum _ iP_ i$ in the last slot of $q(M,M)$, we obtain

\[ q(M,P_ i)=\frac1nq_ i(M^{(i)},I^{(i)}),\qquad q(M,M)=\frac1n\sum _{i=1}^ nq_ i(M^{(i)},M^{(i)}). \]

Now let $M$ lie in the kernel of $q$. Then $q(M,P_ i)=0$, so $M^{(i)}$ lies in the hyperplane $\{ q_ i(\, \cdot \, ,I^{(i)})=0\} $, on which $q_ i$ is negative definite by the induction hypothesis. Since also $q(M,M)=0$, every term of the sum vanishes, and $M^{(i)}=0$ for every $i$. As $n\geq 3$, every entry of $M$ lies in some $M^{(i)}$, so $M=0$.

Finally, the set of tuples $A=(A_3,\ldots ,A_ n)$ of positive definite matrices is convex, hence connected, and we write $q_ A$ for the corresponding form. By Sylvester's law of inertia, the number $k(A)$ of positive squares of $q_ A$ is the largest dimension of a subspace on which $q_ A$ is positive definite. We show that $k$ is locally constant. Fix a tuple $A^0$. Since $q_{A^0}$ is nondegenerate, the space of Hermitian matrices is a direct sum $E_+\oplus E_-$, where $q_{A^0}$ is positive definite on $E_+$, negative definite on $E_-$, and $\dim E_+=k(A^0)$. The function $(A,M)\mapsto q_ A(M,M)$ is continuous, and it is positive on $\{ A^0\} \times S_+$ and negative on $\{ A^0\} \times S_-$, where $S_\pm $ is the unit sphere of $E_\pm $ for some norm. These spheres are compact, so for all tuples $A$ near $A^0$ the form $q_ A$ is still positive definite on $E_+$ and negative definite on $E_-$. Then $k(A)\geq \dim E_+$, and $k(A)\leq \dim E_+$ because a subspace of larger dimension meets $E_-$ in a nonzero vector. Hence $k(A)=k(A^0)$ near $A^0$, and $k$ is constant on the connected set of tuples.

It therefore suffices to compute the signature at $A_3=\cdots =A_ n=I$. With $s=t_3+\cdots +t_ n$, the coefficient of $s^{n-2}$ in $\det (t_1M+t_2N+sI)$ is the sum of the principal $2\times 2$ minors of $t_1M+t_2N$, that is, $((\operatorname {tr}X)^2-\operatorname {tr}(X^2))/2$ for $X=t_1M+t_2N$, in which $t_1t_2$ has the coefficient $\operatorname {tr}M\operatorname {tr}N-\operatorname {tr}(MN)$; moreover, the coefficient of $t_3\cdots t_ n$ in $s^{n-2}$ is $(n-2)!$. Hence

\[ q(M,N)=\frac{\operatorname {tr}M\operatorname {tr}N-\operatorname {tr}(MN)}{n(n-1)}, \]

and, as in the case $n=2$, the decomposition into scalar and trace-free matrices shows that the signature is $(1,n^2-1)$. $\square$

Definition 5.3.5.A86EC2473CB6 Let $0\leq p\leq n$. A current $T$ of bidegree $(p,p)$ on $X$ is positive if $T(\varphi )\ge 0$ for every test form $\varphi $ of bidegree $(n-p,n-p)$ that is strongly positive at every point. We then write $T\geq 0$, and $T\geq S$ if $T-S\geq 0$. If $T$ is positive and $U$ is a coordinate chart, the trace measure of $T$ in $U$ is the positive Radon measure

\[ \mu _ T=T\wedge \beta ^{n-p},\qquad \text{that is,}\qquad \mu _ T(\chi )=T(\chi \beta ^{n-p})\quad \text{for test functions }\chi \text{ on }U, \]

where $\beta $ is the form (5.2.8) in the coordinates of $U$.

The functional $\chi \mapsto T(\chi \beta ^{n-p})$ is indeed a Radon measure. It is nonnegative on nonnegative test functions, because $\chi \beta ^{n-p}$ is then strongly positive, and such a distribution is real: a real test function $\chi $ is the difference of the nonnegative test functions $\chi +\| \chi \| _\infty \eta $ and $\| \chi \| _\infty \eta $, where $\eta $ is a smooth cutoff function for $\operatorname{Supp}\chi $ in $U$ (lemma 2.5.1). Hence it is a positive distribution, and theorem 2.5.6 applies. The trace measure depends on the chart; a version defined by a Hermitian metric is introduced at the end of section 5.6.

A current $S$ of degree $2n$ acts on test functions, and by definition 5.3.5 with $p=n$ it is positive if $S(\chi )\geq 0$ for every nonnegative test function $\chi $. In a chart, its only coefficient is the distribution $\chi \mapsto S(\chi )$ itself. Positive currents of degree $2n$ are measures on $X$. We call a Borel measure $\mu $ on $X$ a positive Radon measure if it is finite on compact sets and inner regular, that is, $\mu (E)=\sup \{ \mu (K):K\subset E\text{ compact}\} $ for every Borel set $E\subset X$; on open subsets of $\mathbb C^ n$ these are the Radon measures of definition 2.2.1, by proposition 2.2.2. The restriction of $\mu $ to the Borel subsets of a chart is a Radon measure in the coordinates, by the same proposition, and $\chi \mapsto \int \chi \, \mathrm d\mu $ is a positive current of degree $2n$. Conversely, every positive current $S$ of degree $2n$ is given in this way by a unique positive Radon measure, which we denote again by $S$.

For uniqueness, note that a compact set $K\subset X$ is the disjoint union of finitely many Borel sets, each contained in a compact subset of a chart. On each of them $\mu $ is determined by $S$, by the uniqueness in theorem 2.5.6. Hence $\mu (K)$ is determined by $S$, and so is $\mu $, by inner regularity.

For existence, let $\mu _ U$ be the Radon measure representing $S$ in a chart $U$ (theorem 2.5.6); by the same uniqueness, $\mu _ U$ and $\mu _{U'}$ agree on the Borel subsets of $U\cap U'$. Choose a partition of unity $(\chi _ a)$ with compact supports subordinate to a cover of $X$ by charts $U_ a$ (proposition 5.2.6), and put $\mu (E)=\sum _ a\int _{E\cap U_ a}\chi _ a\, \mathrm d\mu _{U_ a}$, a sum of nonnegative terms, defined as the supremum of its finite partial sums. Since such sums can be rearranged, $\mu $ is countably additive. If $E$ is a Borel subset of a chart $U$, then only countably many $\chi _ a$ are not identically zero on $U$, because $U$ is a countable union of compact sets, each meeting only finitely many supports; since $\mu _{U_ a}=\mu _ U$ on $U_ a\cap U$, monotone convergence gives $\mu (E)=\int _ E\sum _ a\chi _ a\, \mathrm d\mu _ U=\mu _ U(E)$. Hence $\mu $ is finite on compact sets, which are finite unions of compact subsets of charts, and $\int \chi \, \mathrm d\mu =S(\chi )$ for test functions $\chi $ supported in a chart, and then for all test functions, by a partition of unity.

Finally, $\mu $ is inner regular: given a Borel set $E$, finitely many indices $a$ and $\varepsilon >0$, inner regularity of the measures $\mu _{U_ a}$ gives compact sets $K_ a\subset E\cap \operatorname{Supp}\chi _ a$ with $\int _{K_ a}\chi _ a\, \mathrm d\mu _{U_ a}>\int _{E\cap U_ a}\chi _ a\, \mathrm d\mu _{U_ a}-\varepsilon $, and the compact set $K=\bigcup _ aK_ a\subset E$ satisfies $\mu (K)\geq \sum _ a\int _{K_ a}\chi _ a\, \mathrm d\mu _{U_ a}$; since $\mu (E)$ is the supremum of the finite partial sums of $\sum _ a\int _{E\cap U_ a}\chi _ a\, \mathrm d\mu _{U_ a}$, it is the supremum of the numbers $\mu (K)$.

A net of positive Radon measures $\mu _ i$ on $X$ converges vaguely to $\mu $ if $\int f\, \mathrm d\mu _ i\to \int f\, \mathrm d\mu $ for every $f\in C_ c(X)$. Applying proposition 2.6.1 in charts and writing $f\in C_ c(X)$ as a finite sum $\sum _ a\chi _ af$ with a partition of unity, we see that this is the same as weak convergence of the corresponding currents; for this reason, vague convergence of measures is called weak convergence in this part of the book.

A smooth real $(p,p)$-form $\alpha $, regarded as a current, is positive if and only if it is weakly positive at every point. Indeed, if $\alpha $ is weakly positive, then $\alpha \wedge \varphi $ is a nonnegative multiple of the volume form for every strongly positive test form $\varphi $, so that $\int _ X\alpha \wedge \varphi \geq 0$. Conversely, let $\alpha $ be positive as a current, let $s$ be a constant strongly positive $(n-p,n-p)$-form in a chart, and write $\alpha \wedge s=f\, \mathrm dV_{2n}$ with $f$ continuous and real. For every nonnegative test function $\chi $ in the chart, $\int f\chi \, \mathrm dV_{2n}=\alpha (\chi s)\geq 0$, hence $f\geq 0$. Since every strongly positive form on the tangent space at a point of the chart is the value of such a constant form $s$, the form $\alpha $ is weakly positive at every point. (By theorem 5.3.7 below, a smooth form that is positive as a current is automatically real.)

Positive currents have measure coefficients. To state this precisely we need measures with complex values. Let $\Omega \subset \mathbb R^ d$ be open. A complex Radon measure on $\Omega $ is a linear functional on $C_ c(\Omega )$ of the form $\sigma =\sigma _1-\sigma _2+\mathrm{i}(\sigma _3-\sigma _4)$ with positive Radon measures $\sigma _1,\ldots ,\sigma _4$; it assigns the complex number $\sigma (A)=\sigma _1(A)-\sigma _2(A)+\mathrm i(\sigma _3(A)-\sigma _4(A))$ to every relatively compact Borel set $A\subset \Omega $. This number does not depend on the decomposition: if $\sigma =\sigma '_1-\sigma '_2+\mathrm i(\sigma '_3-\sigma '_4)$ is another one, then, evaluating on real functions and taking real and imaginary parts, we find $\sigma _1+\sigma '_2=\sigma '_1+\sigma _2$ and $\sigma _3+\sigma '_4=\sigma '_3+\sigma _4$ as positive linear functionals on $C_ c(\Omega ,\mathbb R)$, hence as measures, by the uniqueness in theorem 2.3.3. Its total variation $|\sigma |$ is defined, as for real signed measures in section 2.2, by

the supremum being taken over all finite Borel partitions of the relatively compact Borel set $A$. The argument given there for real signed measures shows that $|\sigma |$ is countably additive on relatively compact Borel sets, and $|\sigma |\leq \sigma _1+\sigma _2+\sigma _3+\sigma _4$. Exhausting $\Omega $ by relatively compact open sets, we obtain a Borel measure $|\sigma |$ on $\Omega $ that is finite on compact sets, hence a positive Radon measure by proposition 2.2.2. Approximation by simple functions gives $|\int f\, \mathrm d\sigma |\leq \int |f|\, \mathrm d|\sigma |$ for bounded Borel functions $f$ with compact support.

A current $T$ on $X$ has order zero if, in every chart, its coefficients are complex Radon measures; we call them the coefficient measures of $T$. To compare two charts, let $t$ and $t'$ be the real coordinates of holomorphic charts $U$ and $U'$, regard functions on $U\cap U'$ as functions of either set of coordinates, write $\mathrm dt_ J=\sum _{J'}a_{JJ'}\, \mathrm dt'_{J'}$ on $U\cap U'$ with smooth functions $a_{JJ'}$, and let $\delta >0$ be the real Jacobian determinant of the coordinate change, so that $\mathrm dt'_1\wedge \cdots \wedge \mathrm dt'_{2n}=\delta \, \mathrm dt_1\wedge \cdots \wedge \mathrm dt_{2n}$. If $\mathrm dt'_{J'}\wedge \varphi =g'_{J'}\, \mathrm dt'_1\wedge \cdots \wedge \mathrm dt'_{2n}$, then $\mathrm dt_ J\wedge \varphi =\delta \sum _{J'}a_{JJ'}g'_{J'}\, \mathrm dt_1\wedge \cdots \wedge \mathrm dt_{2n}$. Comparing the two expressions for $T(\varphi )$, in which the functions $g'_{J'}$ are arbitrary test functions on $U\cap U'$, we obtain

for test functions $h$ on $U\cap U'$, where the subscripts indicate the coordinates in which the pairings are computed. Hence, if the $T_ J$ are complex Radon measures on $U\cap U'$, then so are the $T'_{J'}$, namely the measures $\sum _ J\delta a_{JJ'}T_ J$ transported to the coordinates $t'$; and for every compact set $K\subset U\cap U'$ we have $|T'_{J'}|(K)\leq C_ K\sum _ J|T_ J|(K)$, where $C_ K$ depends only on the charts and on $K$. Moreover, a distribution that agrees near every point with a complex Radon measure is a complex Radon measure. Indeed, take a partition of unity $(\chi _ a)$ subordinate to a cover by open sets on which the distribution agrees with complex Radon measures $\sigma _ a$. Then the distribution is the locally finite sum of the complex measures $\chi _ a\sigma _ a$. Each of these is a combination of four positive Radon measures with compact support, and the locally finite sums of such measures are Borel measures finite on compact sets, hence Radon by proposition 2.2.2. Consequently, it suffices to check the condition of order zero in the charts of an atlas.

A family $(T_ i)$ of currents of order zero has locally uniformly bounded total variation if the total variations of the coefficient measures of the $T_ i$ are uniformly bounded on every compact subset of every chart, and $T_ i\to T$ in total variation on compact sets if the total variations of the coefficient measures of $T_ i-T$ tend to zero on every such compact set.

A current of order zero extends, by the formula $T(\varphi )=\sum _ J\int g_ J\, \mathrm dT_ J$ in charts and a partition of unity, to forms $\varphi $ with bounded Borel coefficients and compact support. The result does not depend on the choices: (5.3.6) persists for bounded Borel functions $h$ with compact support in $U\cap U'$, because both sides are integrals against complex Radon measures which agree on test functions, hence on $C_ c(U\cap U')$ by lemma 2.5.1, hence as measures; and two partitions of unity give the same value by the computation used for integrals of forms. If $f$ is a locally bounded Borel function on $X$, multiplying all coefficient measures by $f$ defines a current $fT$ of order zero; this is compatible with coordinate changes. In particular $\mathbf1_ ET$ is defined for every Borel set $E\subset X$. We say that $T$ gives $E$ zero mass, or does not charge $E$, if $\mathbf1_ ET=0$, that is, if all coefficient measures of $T$ vanish on $E$ in every chart.

Theorem 5.3.7.1C212E45A497 Let $T$ be a positive current of bidegree $(p,p)$ on a complex manifold $X$ of dimension $n$. Then $T$ is real and of order zero. More precisely, in every coordinate chart the coefficient measures of $T=\sum _{|I|=|J|=p}T_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$ satisfy

\[ |T_{IJ}|\leq C_{n,p}\, \mu _ T, \]

where $\mu _ T$ is the trace measure and $C_{n,p}$ depends only on $n$ and $p$. Moreover, suppose that $X$ has a countable basis, and let $(T_ j)$ be a sequence of positive currents of bidegree $(p,p)$ whose trace measures are locally uniformly bounded: $\sup _ j\mu _{T_ j}(K)<\infty $ for every compact subset $K$ of every chart. Then a subsequence converges weakly to a positive current, which is closed if all $T_ j$ are closed.

Proof. Fix a chart and put $q=n-p$. By proposition 5.3.2, $\beta ^ q$ is an interior point of the cone of strongly positive $(q,q)$-forms. Hence for every constant real $(q,q)$-form $\tau $ there is $C_\tau >0$ such that the forms $C_\tau \beta ^ q\pm \tau $ are strongly positive. For a nonnegative test function $\chi $, the forms $\chi (C_\tau \beta ^ q\pm \tau )$ are strongly positive test forms, so positivity of $T$ gives $T(\chi (C_\tau \beta ^ q\pm \tau ))\geq 0$. Therefore $T(\chi \tau )$ is real and

\[ |T(\chi \tau )|\le C_\tau \mu _ T(\chi ). \]

The distributions $C_\tau \mu _ T\pm T(\, \cdot \, \tau )$ are nonnegative on nonnegative test functions, hence positive Radon measures by the remark after definition 5.3.5 and theorem 2.5.6. Consequently $S_\tau =T(\, \cdot \, \tau )$ is a real signed Radon measure, the difference of the positive measures $(C_\tau \mu _ T\pm S_\tau )/2$, and $|S_\tau (A)|\leq C_\tau \mu _ T(A)$ for every relatively compact Borel set $A$, so that $|S_\tau |\leq C_\tau \mu _ T$.

Choose real constant $(q,q)$-forms $\tau _1,\ldots ,\tau _ N$ forming a basis of the real $(q,q)$-forms; they also form a complex basis of all constant $(q,q)$-forms, and they can be chosen depending only on $n$ and $p$. Up to a constant factor depending only on $n$, $I$ and $J$, the coefficient $T_{IJ}$ is the distribution $\chi \mapsto T(\chi \, \mathrm dz_{I^ c}\wedge \mathrm d\bar z_{J^ c})$, where $I^ c$ and $J^ c$ are the complementary multi-indices. Expanding $\mathrm dz_{I^ c}\wedge \mathrm d\bar z_{J^ c}=\sum _ rc_ r\tau _ r$, we see that $T_{IJ}$ is a fixed complex linear combination of the measures $S_{\tau _ r}$, and the coefficient bound follows from $|S_{\tau _ r}|\leq C_{\tau _ r}\mu _ T$. To prove that $T$ is real, let $\varphi $ be a test form supported in the chart; only its $(q,q)$-component contributes, and we write it as $\sum _ rf_ r\tau _ r$ with complex test functions $f_ r$. Since $T(f\tau _ r)=\int f\, \mathrm dS_{\tau _ r}$ is real for real $f$, we obtain $T(\bar\varphi )=\overline{T(\varphi )}$, and a partition of unity extends this identity to all test forms.

Now let $X$ have a countable basis, and let $(T_ j)$ be as in the statement. Then $X$ is covered by countably many charts $U_1,U_2,\ldots $: for each basic open set contained in some chart, choose one such chart. On $U_1$ the positive Radon measures $C_{\tau _ r}\mu _{T_ j}\pm S^ j_{\tau _ r}$, where $S^ j_\tau =T_ j(\, \cdot \, \tau )$, have mass at most $2C_{\tau _ r}\sup _ j\mu _{T_ j}(K)$ on every compact $K\subset U_1$. Applying theorem 2.6.5 successively to these finitely many sequences, we find a subsequence along which all of them converge vaguely on $U_1$; then each $S^ j_{\tau _ r}$ converges vaguely, and $T_ j(\varphi )$ converges for every test form $\varphi $ supported in $U_1$. Repeating the extraction for $U_2,U_3,\ldots $ and passing to the diagonal subsequence, we may assume that $T_ j(\varphi )$ converges for every test form $\varphi $ supported in some $U_ k$. A partition of unity subordinate to $(U_ k)$ writes every test form as a finite sum of such forms, so $T(\varphi )=\lim _ jT_ j(\varphi )$ exists for every test form $\varphi $.

The functional $T$ is linear, and it is a current. Indeed, if $\varphi =\sum _ rf_ r\tau _ r$ is supported in a compact set $K\subset U_ k$, then $|T_ j(\varphi )|\leq \sum _ rC_{\tau _ r}\| f_ r\| _\infty \sup _ i\mu _{T_ i}(K)$ for all $j$, and this bound passes to the limit. On compact subsets of the intersection of two charts, the coefficients of a form in the two charts are related by a matrix of continuous functions. Hence, covering a compact subset of an arbitrary chart by finitely many $U_ k$ and using a partition of unity, we obtain the estimate in the definition of a current, with $m=0$. Thus $T_ j\to T$ weakly. If $\varphi $ is a strongly positive test form, then $T(\varphi )=\lim _ jT_ j(\varphi )\geq 0$; and if all $T_ j$ are closed, then $T(\mathrm d\psi )=\lim _ jT_ j(\mathrm d\psi )=0$ for every test form $\psi $. $\square$

Currents can be regularized in coordinates, and the regularization preserves positivity and closedness. Let $T$ be a current on an open set $\Omega \subset \mathbb C^ n$ with coefficients $T_{IJ}$, and let $\rho _\varepsilon $ and $\Omega _\varepsilon $ be as in section 2.5. The smooth form $T_\varepsilon =\sum _{I,J}(T_{IJ})_\varepsilon \, \mathrm dz_ I\wedge \mathrm d\bar z_ J$ on $\Omega _\varepsilon $ satisfies $T_\varepsilon (\varphi )=T(\varphi *\rho _\varepsilon )$ for test forms $\varphi $ supported in $\Omega _\varepsilon $, where $\varphi *\rho _\varepsilon $ is obtained by mollifying the coefficients of $\varphi $; this is (2.5.4), applied to each coefficient. By proposition 2.5.3, $T_\varepsilon \to T$ weakly as $\varepsilon \to 0$, in the sense that $T_\varepsilon (\varphi )\to T(\varphi )$ for every test form $\varphi $ on $\Omega $; here $T_\varepsilon (\varphi )$ is defined as soon as $\operatorname{Supp}\varphi \subset \Omega _\varepsilon $.

If $T$ is positive, so is $T_\varepsilon $ on $\Omega _\varepsilon $. Indeed, let $\varphi $ be a strongly positive test form supported in $\Omega _\varepsilon $. Since the frame $\mathrm dz_ I\wedge \mathrm d\bar z_ J$ is invariant under translations, the translates $x\mapsto \varphi (x-h)$ are strongly positive, and $(\varphi *\rho _\varepsilon )(x)=\int \rho _\varepsilon (h)\varphi (x-h)\, \mathrm dV_{2n}(h)$ is a limit of Riemann sums, which are nonnegative combinations of strongly positive forms; it is strongly positive because the cone is closed (proposition 5.3.2). Moreover, $\varphi *\rho _\varepsilon $ is a test form on $\Omega $, supported in the set of points at distance at most $\varepsilon $ from $\operatorname{Supp}\varphi $. Hence $T_\varepsilon (\varphi )=T(\varphi *\rho _\varepsilon )\geq 0$. Finally, if $T$ is closed, so is $T_\varepsilon $, since $\mathrm d$ acts on the coefficients by differentiation, and derivatives commute with mollification (proposition 2.5.3).

If $T$ is positive of bidegree $(p,p)$ and $\alpha $ is a smooth strongly positive $(s,s)$-form with $p+s\leq n$, then $T\wedge \alpha $ is positive, because $\alpha \wedge \varphi $ is a strongly positive test form for every strongly positive test form $\varphi $ (proposition 5.3.2). The trace measure $T\wedge \beta ^{n-p}$ is an example.

Currents on a complex manifold $X$ can be glued from local pieces; we use this in section 5.6 to define pullbacks of currents. For an open set $W\subset X$, the restriction of $T$ to $W$ is the current on $W$ given by the values of $T$ on test forms supported in $W$; two currents agree on $W$ if their restrictions to $W$ coincide.

Lemma 5.3.8.376F5C171BED Let $(W_ a)_{a\in A}$ be an open cover of $X$, and for each $a$ let $T_ a$ be a current of degree $k$ on $W_ a$, such that $T_ a$ and $T_ b$ agree on $W_ a\cap W_ b$ for all $a,b\in A$. Then there is a unique current $T$ of degree $k$ on $X$ which agrees with $T_ a$ on $W_ a$ for every $a$. If all $T_ a$ have bidegree $(p,q)$, then so has $T$; if all $T_ a$ are closed, then so is $T$; and if $p=q$ and all $T_ a$ are positive, then so is $T$.

Proof. If two currents on $X$ agree with $T_ a$ on $W_ a$ for every $a$, their difference vanishes on every $W_ a$, hence on $X$, since a current that vanishes on each member of a family of open sets vanishes on their union, as shown at the beginning of this section. For existence, choose a partition of unity $(\chi _ c)_{c\in C}$ with compact supports subordinate to the cover (proposition 5.2.6), and for each $c$ an index $a(c)$ with $\operatorname{Supp}\chi _ c\subset W_{a(c)}$. For a test form $\varphi $ of degree $2n-k$ on $X$, only finitely many $\chi _ c$ are not identically zero on the compact set $\operatorname{Supp}\varphi $, and $\chi _ c\varphi $ is a test form supported in $W_{a(c)}$; we put

\[ T(\varphi )=\sum _ cT_{a(c)}(\chi _ c\varphi ). \]

This is a linear functional, and it is a current. Indeed, let $K$ be a compact subset of a chart domain $U$. Only finitely many indices $c_1,\ldots ,c_ N$ have $\chi _{c_ i}\not\equiv 0$ on $K$, and for test forms $\varphi $ supported in $K$ the form $\chi _{c_ i}\varphi $ is supported in the compact subset $K\cap \operatorname{Supp}\chi _{c_ i}$ of the chart domain $U\cap W_{a(c_ i)}$ of $W_{a(c_ i)}$. By the Leibniz rule, its seminorms $p_{K,m}$ are bounded by a constant times those of $\varphi $, so the estimates for the currents $T_{a(c_ i)}$ give the estimate in the definition of a current for $T$. If $\varphi $ is supported in some $W_ b$, then each $\chi _ c\varphi $ is supported in $W_{a(c)}\cap W_ b$, where $T_{a(c)}$ and $T_ b$ agree, so that $T(\varphi )=\sum _ cT_ b(\chi _ c\varphi )=T_ b(\varphi )$. Thus $T$ agrees with $T_ b$ on $W_ b$.

Multiplication by $\chi _ c$ preserves bidegrees, and $\chi _ c\varphi $ is strongly positive at every point if $\varphi $ is, because $\chi _ c\geq 0$; so the assertions about bidegrees and positivity follow from the formula for $T(\varphi )$. Finally, if all $T_ a$ are closed and $\psi $ is a test form supported in $W_ a$, then so is $\mathrm d\psi $, and $(\mathrm dT)(\psi )=(-1)^{k+1}T(\mathrm d\psi )=(-1)^{k+1}T_ a(\mathrm d\psi )=(\mathrm dT_ a)(\psi )=0$; hence $\mathrm dT$ vanishes on every $W_ a$, and therefore $\mathrm dT=0$. $\square$

Example 5.3.9.1176DC9B7704 Let $Z$ be a complex submanifold of $X$ of dimension $r$, with its complex orientation. Then

\[ [Z](\varphi )=\int _ Z\varphi |_ Z \]

defines a closed positive current of bidegree $(n-r,n-r)$ on $X$, the current of integration over $Z$. Since $Z$ is closed, $Z\cap \operatorname{Supp}\varphi $ is compact for every test form $\varphi $, so the integral is defined. In a chart in which $Z=\{ z_{r+1}=\cdots =z_ n=0\} $, the value $[Z](\varphi )$ is a constant times the integral over this coordinate plane of the coefficient of $\mathrm dz_ I\wedge \mathrm d\bar z_ I$ in $\varphi $, where $I=(1,\ldots ,r)$, because the pullbacks of $\mathrm dz_ j$ and $\mathrm d\bar z_ j$ to $Z$ vanish for $j>r$. Hence $[Z]$ is a current, and its coefficients in such a chart are constant multiples of Lebesgue measure on the plane. Since $[Z]$ vanishes on $X\setminus Z$, the charts of this kind and the charts contained in $X\setminus Z$ form an atlas in which $[Z]$ has measure coefficients, and $[Z]$ is a current of order zero. If $\varphi $ is strongly positive, so is its pullback $\varphi |_ Z$ to $Z$, which is therefore a nonnegative multiple of the volume form of $Z$ at each point (proposition 5.3.2); thus $[Z](\varphi )\geq 0$. Finally, since pullback commutes with $\mathrm d$ by (5.2.5), $[Z](\mathrm d\psi )=\int _ Z\mathrm d(\psi |_ Z)=0$ by proposition 5.2.6 applied on $Z$ when $r\geq 1$; when $r=0$ the current $[Z]$ has degree $2n$, and $\mathrm d[Z]=0$ because currents of degree $2n+1$ are zero.