Consider the following algebraic curve:
$$C=\left\{ (x,y) \in \mathbb{C}^2 | x^3+y^3+3\lambda xy + 1=0 \right\} $$
…where $\lambda^3 \neq -1$. I think this is called the Hesse pencil, and it’s a Riemann surface. Then:
- I’m told that the differential form
$$\frac{dx}{y^2+\lambda x}=-\frac{dy}{x^2+\lambda y}$$
is a holomorphic differential form on $C$. How do I actually check that? I know from Miranda’s book that a holomorphic $1$-form on a Riemann surface $X$ is a collection of holomorphic $1$-forms (in the complex-analytic sense) on each chart such that they correctly transform to each other when two charts have overlapping domains. I also know that a holomorphic $1$-form on an open set $V \subset \mathbb{C}$ is an expression of the form $\omega = f(z)dz$.
However, I’m not sure how to apply this to this case. Are the expressions $\frac{dx}{y^2+\lambda x}$ and $\frac{dy}{x^2+\lambda y}$ locally defined (in one chart given by, I guess, the coordinates $(x,y))$? In that case, if I understand correctly, I should check that the functions $\frac{1}{y^2+\lambda x}$ and $-\frac{1}{x^2+\lambda y}$ are holomorphic over $C$, but I don’t really know how to do that since they’re defined on $\mathbb{C}^2$ and hence applying the Cauchy-Riemann equations doesn’t seem to be the way to go here (I usually check that $\partial f/\partial \bar{z}=0$ to see that a function $f(z)$ is holomorphic on $\mathbb{C}$… should I do this for $x$ and $y$ in the case of $f(x,y)$?).
- I’m also told that this holomorphic differential form extends to a holomorphic differential form on the compactification of $C$, which I think is the locus in $\mathbb{PC}^2$ of the homogenization of $x^3+y^3+3\lambda xy + 1$, that is
$$\hat{C}=\left\{ (x,y,z) \in \mathbb{PC}^2 | x^3+y^3+z^3+3\lambda xyz=0 \right\} $$
Again, I’m not sure how to do that. I know that in theory, I should check that the differential form transforms to a holomorphic form in some coordinates including the points at infinity, but I haven’t seen any example of this kind of computation (I’m kind of new to Riemann surfaces). Any help or references to a similar example would be appreciated.