Proof of holomorphic Lefschetz fixed point formula using currents in Griffiths and Harris I am trying to understand the proof of the Holomorphic Lefschetz fixed point formula on page 426 in Griffiths and Harris.  However, I find their use of currents extremely confusing.  They seem to go back and forth between currents and forms without a thought, which is confusing to me as I don't know much about currents.
Definitions: given a manifold $M$ and a smooth $p$-form $\phi$, the current induced by $\phi$ is the linear functional $T_\phi:\Omega^{n-p}_c(M)\to\mathbb{R}$ given by $T_\phi(\omega)=\int_M\phi\wedge\omega$.  Here $\Omega^{n-p}_c(M)$ denotes compactly supported smooth $(n-p)$-forms.  A current $T$ restricts to an open subset $U$ by extending a compactly supported form $\omega$ on $U$ by $0$ and then applying $T$.  In my case I am interested in complex currents, which are defined analagously with respect to Dolbeault cohomology.
As best as I can tell, the situation is that we have we have a compact manifold $M$ of dimension $n$, a compact submanifold $A$ of dimension $k$, and smooth currents $T_\phi$ and $T_\psi$ where $\phi$ and $\psi$ are closed smooth $k$-forms on $M$.  It seems they are implicitly using that if $T_\phi$ restricted to a coordinate neighborhood $U$ of $M$ is zero, then $\int_A\phi=\int_{A-U}\phi$, and further if $T_\phi=T_\psi$ on an open set $W$ such that $A - U \subset W$ then $\int_{A-U}\phi=\int_{A-U}\psi$.  
Do the statements above make sense?  Are they true?  
Or do Griffiths and Harris mean something else?  (Note I have used somewhat different notation than they used to not clutter this question.  )
BOUNTY:  I'm setting a bounty for someone who can give a satisfactory explanation as to why, in the notation of the book,
$$\int_{\Gamma_f}\phi=-\int_{\Gamma_f-\cup B_\epsilon(p_\alpha,p_\alpha)} \bar\partial k$$
This could mean proving or giving a reference to the two statements I stated above, of if those statements are not correct, then proving the above equality using the information given on page 426.
UPDATE:  It seems that Poincare duality for noncompact manifolds implies by the hypotheses that on $U$, $\phi=0$ up to an exact form, and on $W$, $\phi=\psi$ up to an exact form.  However, $A-U$ will have boundary.  So I still haven't solved the problem.
 A: First, I will answer your question about the equality you quote from G&H; in the meanwhile, it could happen that I also answer your two general questions about currents.
The setting is the following: we are considering currents in the product space $M\times M$, where $M$ is a $n$-dimensional complex manifold; $\Delta$ is the diagonal, i.e.
$$\Delta=\{(x,x)\ :\ x\in M\}$$
and it is obviously isomorphic to $M$ (useless here, however). 
$T_\Delta$ is the current of integration along the diagonal, so, for every test form $u\in\mathcal{D}^{2n}(M\times M)$ (smooth compactly supported $2n$-forms), 
$$T_\Delta(u)=\int_\Delta u\;.$$
By employing the decomposition in bidegrees, we can define $T^0_\Delta$ as the component of $T_\Delta$ which acts only on forms of bidegree $(n,n-*)-(0,*)$, i.e. the component of bidimension (bitype in the terminology of G&H) $(0,*)-(n,n-*)$.
Now, $T_\Delta^0$ is a current, but is not represented by integration against a smooth form, as the ones you defined (it is represented by integration against a form with measure coefficients, but let's not enter such details).
By means of the Bochner-Martinelli formula, we can locally define (around fixed points $\{p_\alpha\}$ for $\alpha$ in some set of indexes) currents of compact support (contained in given balls of radius $2\epsilon$ around those points) $k_\alpha=\rho_\alpha k_{BM}$ such that $\overline{\partial}k_\alpha=T_\Delta^0$ on a ball of center $(p_\alpha,p_\alpha)$ and radius $\epsilon$.
This means that, for every $u\in\mathcal{D}^{2n}(B_\epsilon(p_\alpha,p_\alpha))$,
$$T_{\Delta}^0(u)=\overline{\partial}k(u)$$
(meaning that we take the $\overline{\partial}$ of $k$ as a current and we apply it - which is again a current - to the form $u$).
Now, defining $k=\sum_\alpha k_\alpha$, you obtain a globally defined current on $M\times M$.
By inspection, one notices that $k$ is given by integration against a form which is smooth on $M\times M\setminus\Delta$, i.e. there exists $\omega_{BM}\in\Omega^{2n+1}(M\times M\setminus \Delta)$ such that
$$k(u)=\int_{M\times M}\omega_{BM}\wedge u$$
(this implies that $\omega_{BM}$ has locally integrable coefficients on $M\times M$, which is true and can be verified by looking at the order of "pole" along $\Delta$).
Therefore, outside $\Delta$, also $\overline{\partial} k$ is represented  by integration against some form $\eta\in\Omega^{2n}(M\times M\setminus \Delta)$. The words "outside $\Delta$" are quite important here: we cannot say that there is such an $\eta$ so that
$$\overline{\partial}k(v)=\int_{M\times M}\eta\wedge v$$
for every $v\in\mathcal{D}^{2n}(M\times M)$!! We don't know what happens along $\Delta$.
We can say, tho
$$\overline{\partial}k(v)=\int_{M\times M}\eta\wedge v$$
for every $v\in\mathcal{D}^{2n}(M\times M\setminus\Delta)$, i.e. for forms with compact support outside $\Delta$.
However, a closer inspection of $k_{BM}$, reveals that its $\overline{\partial}$ vanishes outside the diagonal, so the only part which survives is given by $\overline{\partial}\rho_\alpha\wedge k_{BM}$, which is again integrable all over $M\times M$, the "poles" along the diagonal being the same. Therefore, 
$$v\mapsto\int_{M\times M}\eta\wedge v$$
is a well defined current $T_\eta$ for $v\in\mathcal{D}^{2n}(M\times M)$. But, again, it does not coincide with $\overline{\partial} k$! We have
$$\overline{\partial} k= T_\eta+R$$
where $R$ is a current, of which we only know that it is of locally finite mass and supported on the diagonal.
So, summing up, we define the current
$$\phi=T_{\Delta}^0-\overline{\partial}k\;.$$
Now, let $u$ be a $2n$-form with compact support in $B_\epsilon(p_\alpha,p_\alpha)$; on such a ball, we arranged things so that $T_{\Delta}^0=\overline{\partial} k=\overline{\partial} k_\alpha$ (the last equality holding only on that ball, not on the whole space). Then we have
$$\phi(u)=T_\Delta^0(u)-\overline{\partial}k (u)=0$$
as $u$ has support in $B_\epsilon(p_\alpha,p_\alpha)$.
In terms of $\eta$ and $R$, we see that
$$0=T_{\Delta}^0(u)-T_\eta(u)-R(u)$$
The first and last current are supported on $\Delta$, so the contributions outside $\Delta$ all come from $T_\eta$; it means that the support of $T_\eta$ does not intersect $B_\epsilon$. That is
$$T_\eta(u)=0$$
for every $u$ a $2n$-form with compact support in $B_\epsilon(p_\alpha,p_\alpha)$. I.e.
$$\int_{B_\epsilon}\eta\wedge u=0$$
for every $u$, which easily implies that $\eta\vert_{B_\epsilon}=0$: take $\tau_n$ a function with values in $[0,1]$, which is $1$ inside a ball of radius $\epsilon-\epsilon/n$ and supported in $B_\epsilon$. By dominated convergence
$$0=\int_{B_\epsilon}\eta\wedge (\tau_n \eta^*)\to\int_{B_\epsilon}\eta\wedge \eta^*=\|\eta\|_{L^2(B_\epsilon)}^2$$
which then has to be zero, so $\eta\vert_{B_\epsilon}=0$.
Now, $\phi$ is represented by a smooth form (as G&H says) at least on 
$$U=(M\times M\setminus\Delta)\cup\bigcup_{\alpha} B_{\epsilon}(p_\alpha,p_\alpha)$$
and such a smooth form is our $\eta$: given $u\in\mathcal{D}^{2n}(U)$, we can write
$$u=\sum_{\alpha}\sigma_\alpha u + \left(u-\sum_\alpha \sigma_\alpha u\right)=\sum u_\alpha + u'$$
for suitable $\sigma_\alpha$ so that $u_\alpha$ is supported in $B_\epsilon$ and $u'$ is $0$ on (a neighbourhood of) $\Delta$. So
$$\phi(u)=T_\Delta^0(u)-T_\eta(u)-R(u)=T_\Delta^0(u')-T_\eta(u')-R(u')=-T_\eta(u')=-T_\eta(u)$$
because $u'$ has support which doesn't meet $\Delta$, where $T_\Delta^0$ and $R$ live.
Again, this implies that the form that represents $\phi$ (which we will denote by $\psi$, so that $\phi=T_\psi$) and the form that represents $-T_\eta$ (i.e. $-\eta$) coincide on $U$.
Now, as $\Gamma_f$ is a $2n$-dimensional submanifold of $U$, we have
$$\int_{\Gamma_f}\psi=-\int_{\Gamma_f}\eta$$
but as $\eta$ vanishes on all $B_\epsilon$, this is the same as
$$-\int_{\Gamma_f\setminus\bigcup B_\epsilon}\eta\;.$$

Sorry if I changed notation a bit, but I wanted to keep currents and forms as much separated as possible, because you said that it was their interplay that confused you.
