Dtyjlui

I already asked about this a couple of weeks ago but had introduced some rather annoying notation. I decided to reformulate the question in a more compact format. (edit: old post taken down as it is basically a duplicate now)

Suppose $psi$ is the complex differential $(1,1)$-form $$psi = i sum_{j,k=1}^n h_{j,k} ~dz_j wedge dbar{z}_k$$
and that the matrix $H$ with entries $h_{j,k}$ is hermitian (i.e. $psi$ is a real form). In a lecture of mine, it was given as an exercise that $psi$ is positive (i.e. $psi(v,Jv) > 0$ for all $v ne 0$, where $J$ is the canonical almost complex structure) if and only if $H$ is positive-definite.

This differential form induces a current by integration, $$T_{psi} ~colon ~ Omega^{n-1,n-1}_c(mathbb{C^n}) rightarrow mathbb{C}, ~ phi mapsto int_{mathbb{C^n}} psi wedge phi.$$
In Harris and Griffiths' textbook "Principles of Algebraic Geometry" (on page 386 with $p=1$), a real $(1,1)$-current is defined to be positive if for every $(n-1,0)$-form $eta$ we have that $T(eta wedge bar{eta})$ is a non-negative real number.

Take such a test form $$eta = sum_{j=1}^n phi_j ~dz_J,$$ where $dz_J$ denotes $dz_1 wedge dots wedge hat{dz_j} wedge dots wedge dz_n$. Then we have $$eta wedge bar{eta} = sum_{j,k} phi_j overline{phi_k} ~dz_J wedge dbar{z}_K$$ and
begin{align*}
T_{psi}(eta wedge bar{eta}) &= i sum_{j,k} int h_{j,k} phi_j overline{phi_k} ~dz_j wedge dbar{z}_k wedge dz_J wedge dbar{z}_K \
&= i sum_{j,k} int h_{j,k} phi_j overline{phi_k} ~sigma(j,k) ~dz_1 wedge dbar{z}_1 wedge dots wedge dz_n wedge dbar{z}_n,
end{align*}
where $sigma(j,k)$ denotes the sign coming from $$dz_j wedge dbar{z}_k wedge dz_J wedge dbar{z}_K = sigma(j,k) ~dz_1 wedge dbar{z_1} wedge dots wedge dz_n wedge dbar{z}_n.$$
Thus, the current $T_{psi}$ is positive not if $H$ is positive definite but if the matrix with entries $sigma(j,k) h_{j,k}$ is positive definite.

Is it true that positivity of currents and positivity of differential forms do not give rise to the same notion? Can this be fixed?

Revisiting my question after a month, I finally realized that the answer is frustratingly simple...

The sign $sigma(j,k)$ can be written explicitly as
$$ sigma(j,k) = (-1)^{n-1 + frac{n(n-1)}{2}} (-1)^{j-1} (-1)^{k-1}. $$
The crucial (yet simple) observation is that the matrix with entries $(-1)^{j+k}h_{j,k}$ is positive definite if and only if the matrix with entries $h_{j,k}$ is.
Thus, the two notions of positivity agree up to a sign that depends only on $n$.

In the following I would like to propose a correction to the definition in Harris and Griffiths' book.
(Remark: to some extend, their definition already seems to be flawed as the exponent of the $i$ does not depend on $n$.)
Let us say that a real $(p,p)$-current $T$ is positive if for any compactly supported $(n-p,0)$-test form $eta$ we have
$$ (-1)^{frac{(n-p)(n-p-1)}{2}} i^{n-p} T(eta wedge bar{eta}) geq 0. $$
For $T=T_{psi}$ as in the question we get ($p=1$)
$$ (-1)^{frac{(n-1)(n-2)}{2}} i^{n-1} T_{psi}(eta wedge bar{eta}) =\
= i^n int_{mathbb{C}^n} left( sum_{j,k=1}^n (-1)^{j+k} phi_j h_{j,k} bar{phi}_k right) dz_1 wedge dbar{z}_1 wedge dots wedge dz_n wedge dbar{z}_n \
= 2^n int_{mathbb{C}^n} left( sum_{j,k=1}^n (-1)^{j+k} phi_j h_{j,k} bar{phi}_k right) dx_1 wedge dy_1 wedge dots wedge dx_n wedge dy_n, $$
where the sign from the definition of positivity cancels with the sign in $sigma(j,k)$.
Thus, $T_{psi}$ is positive if and only if this integral is always positive if and only if $((-1)^{j+k}h_{j,k})$ is positive definite if and only if $(h_{j,k})$ is positive definite if and only if $psi$ is positive, as desired.

To further motivate the particular choice of sign in the definition (for now, we only verified it for $p=1$), let us consider the current defined by integration over $mathbb{C}^{n-p} times {0} subset mathbb{C}^n$, which ought to be a positive current.
Indeed,
$$ int_{mathbb{C}^{n-p} times {0}} phi bar{phi} ~dz_1 wedge dots wedge dz_{n-p} wedge dbar{z}_1 wedge dots wedge dbar{z}_{n-p} \
= (-1)^{frac{(n-p)(n-p-1)}{2}} int_{mathbb{C}^{n-p} times {0}} |phi|^2 ~dz_1 wedge dbar{z}_1 wedge dots wedge dz_{n-p} wedge dbar{z}_{n-p} \
= (-1)^{frac{(n-p)(n-p-1)}{2}} (-2i)^{n-p} int_{mathbb{C}^{n-p} times {0}} |phi|^2 ~dx_1 wedge dy_1 wedge dots wedge dx_{n-p} wedge dy_{n-p}. $$