I am now reading these lectures by Stefan Vandoren on complex geometry. Everything is fine in general, hiwever I am confused with how he defines 1-form on a Calabi-Yau 1-fold (or 2-form on CY$_2$).
In particular, if we look at his (5.12), where the 1-form $\Omega$ on CY$_1$ is defined, this can be related to Weierstrass functions by the usual embedding $z\mapsto [1:\wp(z):\wp'(z)]$ of a 2-torus into projective space $CP^2$. Then \begin{equation} \Omega=dz=\frac{d\wp}{\wp'(z)}=\frac{dx}{y}, \end{equation} that is more or less the same as in the paper with $x=z_1/z_0$ and $y=z_2/z_0$.
Then he turns to Excercise 5.1, where global properties of the form are invesitgated. Here is my first question: why he defines a 1-form $\Omega_1$ separately instead of just taking transformation of $\Omega$? We go from the patch with $z_0\neq 0$ to $z_1\neq 0$ with $\tilde{x}=z_0/z_1$ and $\tilde{y}=z_2/z_1$. Then we just write \begin{equation} \Omega=\frac{dx}{dy}=-\frac{\tilde{x}^{-2}d\tilde{x}}{\tilde{x}^{-1}\tilde{y}}=-\frac{d\tilde{x}}{\tilde{x}\tilde{y}}. \end{equation} The last term above is precisely his $\Omega_1$. However he starts from another definition for it and then discovers the above identities.
The second question is actually the same but in other notations. In Exercise 5.2 Vandoren consider a curve in $CP^2$ defined by $$F(z_0,z_1,z_2)=z_0^n+z_1^n+z_2^n=0$$ with some integer $n$. The question is, why he defines the form to be $$\Omega=\frac{dx}{n dy^{n-1}}?$$ As I understand this should be somehow related to the following identities \begin{equation} 0=df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy=n(x^{n-1}dx+ny^{n-1}dy), \end{equation} however I have no idea, what the relation is. Here $F=z_0^3 f(x,y)$ and $x=z_1/z_0,y=z_2/z_0$.
Finally, he again defines the form $\Omega_1$ on the patch $z_1\neq 0$ as $$\Omega_1=-\frac{d\tilde{x}}{n \tilde{y}^{n-1}},$$ and then derives $n=3$ from the demand that $\Omega_0=\Omega_1$ on intersection. The question is again the same: why couldn't I use instead of his $\Omega_1$, just a coordinate transform of $\Omega_0$ to the patch $z_1\neq 0$? Like this $$\Omega_0=-\frac{\tilde{x}^{n-3} d\tilde{x}}{n \tilde{y}^{n-1}}.$$
I am a physicist, so it could be that I just do not understand some basic and trivial concept about globally defined differential forms. I would be happy if you could answer my questions. After some search on MathSE I couldn't find any similar questions.
I've found the answer. The point is that the n-form on the n-dimensional hypersurface is said to be a Lerya residue of a (n+1)-form, that is singular on the hypersurface.
If we say that on each patch of the projective plane the n-form should be given by the Lerya residue, then the globality condition gives the power of the polynomial, that defines the surface.