Proposition 5.3.2E58104E974B0

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 $\beta ^ p=p!\sum _{|I|=p}\bigwedge _{i\in I}(\frac{\mathrm i}2\mathrm dz_ i\wedge \mathrm d\bar z_ i)$,

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. $\square$