On pages 708 in the proof of the following theorem:
Theorem If $Z$ is nonempty then $H^{p,q}(M)=0$ for $p\ne q$; where $M$ is a compact Kahler manifold and $v$ a holomorphic vector field having a set $Z$ of isolated zeros.
Now in the proof of this theorem on page 709 they write:
Step One in the Proof. We denote by $\iota (v)$ the operation of contraction of a differential form with the vector field $v$. This operator was already encountered in the proof of the Bott residue theorem. If locally: $v = \sum_i v_i(z) \frac{\partial }{\partial z_i}$ and $\varphi = \frac{1}{p!q!} \sum_{I,J} \varphi_{IJ} dz_I \wedge d\bar{z_J}$ is a $(p,q)$ form, then $\iota(v)\varphi = \frac{1}{(p-1)!q!}\sum_{I,J} \bigg(\sum_{i\in I} \pm v_i \varphi_{IJ} dz_{I-\{i\}} \wedge d\bar{z_J}\bigg)$. From this the formal rules $$\iota(v): A^{p,q}(M)\to A^{p-1,q}(M)$$ $$\iota(v)^2=0$$ $$\iota(v)\bar{\partial}+\bar{\partial} \iota(v) =0 $$ $$\iota(v)(\varphi \wedge \psi) = \iota(v)\varphi \wedge \psi +(-1)^{\deg \varphi} \varphi \wedge \iota(v) \psi ,$$ are easily verified.
Now, I would like to show the last three identities, but I am not sure how exacly?
For example the first identity we have: $$\iota(v) \iota(v) \varphi = \iota(v) \bigg( \frac{1}{(p-1)!q!}\sum_{I,J} \bigg(\sum_{i\in I} \pm v_i \varphi_{IJ} dz_{I-\{i\}} \wedge d\bar{z_J}\bigg)\bigg)$$
How to continue? I would like some help also with the other identities.
Thanks.