Reconstructing the metric on $CP^2$ with special one forms

Question

I know that $(z_1,z_2)$ are the affine\inhomogeneous coordinates on the complex projective space $CP^2$. Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$. I want to rewrite the Fubini Study metric on $CP^2$ in terms of the forms $(Y_1,Y_2, Y_3, Y_4)$ . The expressions for the one forms in terms of $(z1,z2)$ are given below \begin{split} &&Y1=\frac{|z_1|^2+|z_2|^2\sqrt{1+|z_1|^2+|z_2|^2}}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz1-\frac{z1\bar{z}_2(-1+\sqrt{1+|z_1|^2+|z_2|^2})}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz2\\ &&Y2=\frac{|z_2|^2+|z_1|^2\sqrt{1+|z_1|^2+|z_2|^2}}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz2-\frac{z2\bar{z}_1(-1+\sqrt{1+|z_1|^2+|z_2|^2})}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz1\\ &&Y_3=-\bar{Y}_1\\ &&Y_4=-\bar{Y}_2 \end{split}

I have been able to workout the following relations $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ Is there a way to get the relation using the above condition so that I can rewrite the Fubini-Study metric in terms of these forms? Special Case: To make my question clear let us consider the case $z2=0$. Then we see the relationship reduces to the following $$ \begin{split} \frac{ dz_1}{1+|z_1|^2} &=Y_1\\ \frac{d\bar{z}_1}{1+|z_1|^2}&=-Y_3 \end{split} $$ Taking the product of the above equation gives $$ \begin{split} ds^2=\frac{ dz_1d\bar{z}_1}{(1+|z_1|^2)^2}=-Y_1Y_3=Y_1\bar{Y}_1 \end{split} $$ This is the Fubini Study metric on $CP^1$ with inhomogeneous/affine coordinates $(z_1,z_2)$. I am expecting to rewrite using the forms the Fubini-Study metric on $CP^2$.

Answer 1

My calculation has the exterior derivative of the left hand side being $$\frac{(1+|z_2|^2)d\bar{z}_1\wedge dz_1 + (1+|z_1|^2)d\bar{z}_2\wedge dz_2 - z_1\bar{z}_2 d\bar{z}_1 \wedge dz_2 -z_2\bar{z}_1 d\bar{z}_2\wedge dz_1}{(1+|z_1|^2+|z_2|^2)^2}.$$ The right hand side is closed. So the two sides cannot be equal on any open set of any 4-manifold on which $z_1,z_2$ are local complex coordinates and $y_1,y_2$ functions of them.