I'm studing currents from Demailly's Complex Geometry, and the author defines the direct image of a current by a $C^{\infty}$ map and also for the case of a submersion. My question is about the compatibility of the direct image of the wedge product of smooth forms, in particular if $F:X\to Y$ is a submersion and $\omega,\tau$ are smooth $(p,p)$ forms, is it true that $F_{*}(\omega\wedge\tau)=F_{*}(\omega)\wedge F_{*}(\tau)$? This is true on the locus where $F$ is not singular, but it could exist a set where $F$ could not be defined, I think for example to a birational map, for example a blow-up or something like this.
Edit: I'll try to explain better. (I'm sorry but this argument is new to me and very difficult too.) If $\omega$ and $\tau$ are smooth forms of bidegree $p$, I can always define the wedge product as smooth forms, but I can see it also as a wedge product of currents in particular (as I understand it from Demailly). So, according to Demailly's book, I can define the direct image of the current $\omega\wedge\tau$; is this correct? Now, I have no idea when the wedge product of current is defined, but let's suppose that $F_{*}(\omega)\wedge F_{*}(\tau)$ is well defined. Now considering $F_{*}(\omega)$ and $F_{*}(\tau)$ as currents, is it true that $F_{*}(\omega)\wedge F_{*}(\tau)=F_{*}(\omega\wedge\tau)$?