# Currents as differential forms and positivity

Currents can be regarded as "differential forms with distribution coefficients". My understanding is the following: Given a distribution $$T$$ we define a current $$Tdx_I$$ by $$Tdx_I(\phi dx_J) = T(\phi) dx_I \wedge dx_J,$$ where $$\phi dx_J$$ is a differential form, and in order to get a complex number (which the output of a current should be), we identify $$dx_I \wedge dx_J$$ with $$1$$ if it equals $$dx_1 \wedge \dots \wedge dx_n$$, with $$-1$$ if it equals $$-dx_1 \wedge \dots \wedge dx_n$$ and with $$0$$ otherwise.

Thus, a $$p$$-current can be regarded as $$T = \sum_{|I|=p} T_I dx_I,~~~ T_I(\phi) := T(\phi \star dx_I),$$ where $$\star$$ is the Hodge star operator. Now suppose we are given a $$(p,p)$$-current on a complex manifold, $$T = \sum_{|I|=p=|J|} T_{I,J} dz_I \wedge d\bar{z}_J.$$ Evaluating $$T$$ on a $$(n-p,n-p)$$-form $$\omega = \sum_{|I|=p=|J|} \phi_{I,J} dz_{\star I} \wedge d\bar{z}_{\star J},$$ where I write $$\star I$$ for the multiindex containing all entries that $$I$$ does not contian, yields $$T(\omega) = (-1)^{p(n-p)} \sum_{|I|=p=|J|} sign(I,J) T_{I,J}(\phi_{I,J}),$$ where a sign appears because $$dz_I \wedge dz_{\star I}$$ might be $$-dz_1 \wedge \dots \wedge dz_n$$. Any actual form $$\psi$$ induces a current $$T_{\psi}$$ by integration. Consider the case $$p=1$$ and $$\psi = i \sum_{j,k} h_{j,k} dz_j \wedge d\bar{z}_k.$$ Then $$T_{j,k}(\phi) = \int_{\mathbb{C}} h_{j,k} \phi ~dz_1 \wedge \dots \wedge d\bar{z}_n.$$ A current is said to be positive if $$i^{p(p-1)/2} T(\eta \wedge \overline\eta)$$ is a real positive number for any $$(n-p,0)$$-form $$\eta$$. This should yield the same notion of positivity for forms when the current is induced by a form. However, if I plug in such $$\omega = \eta \wedge \overline{\eta}$$, then the coefficient $$\phi_{j,k}$$ in $$\omega$$ is of the form $$\phi_j \overline{\phi_k}$$ and I get $$T_{\psi}(\eta \wedge \overline{\eta}) = (-1)^{p(n-p)} \int_{\mathbb{C}} \left( \sum_{j,k} sign(j,k) \phi_j h_{j,k} \overline{\phi_k} \right)~dz_1 \wedge \dots \wedge d\bar{z}_n.$$ If the signs weren't there, then positivity would be equivalent to $$(h_{j,k})$$ being a hermitian matrix, which is what we want. However, due to the signs, forms and currents associated to forms do not seem to be exactly the same.

How can this be resolved? Is my understanding of the identification of currents as forms flawed?