Holomorphic 1-forms on a smooth affine plane curve I'm looking at Miranda's Algebraic Curves and Riemann Surfaces. In particular, Chapter IV.I, question C : Let $X$ be a smooth affine plane curve defined by $f(u,v) =0$. Show that $du$ and $dv$ define holomorphic 1-forms on $X$, as do $p(u,v)du$ and $p(u,v)dv$ for any polynomial $p(u,v)$. Show that if $r(u,v)$ is any rational function, then $r(u,v)du$ and $r(u,v)dv$ are meromorphic 1-forms on $X$. Show that $(\partial f/\partial u)du = -(\partial f/\partial v)dv$ as holomorphic 1-forms on $X$. 
I understand what a holomorphic 1-form is, but I'm not sure how I'm supposed to show that these particular examples are 1-forms on $X$. The reason for this mainly is that I'm unsure of when something fails to be a holomorphic 1-form. 

To answer your questions you should know the following: $X$ has a holomorphic atlas (F. Kirwan, "Complex Algebraic Curves", Cambridge University Press 1992, Proposition 5.27), and definitions of holomorphic/meromorphic differential forms (Chapter 6, Definitions 6.4, 6.5 from the book by Kirwan).
Fix a point $p$ of $X$. Let $t$, $t(p) = 0$ be a local coordinate (a chart) defined in an open neighborhood of $p$ in $X$. Then there exist two holomorphic (near $0$) functions $u(t)$, $v(t)$ which define the inverse chart. Therefore, we have the following.


*

*$du$, $dv$ are holomorphic near $p$ on $X$ since $u'(t)dt$, $v'(t)dt$ are holomorphic near $0$.

*$r(u, v)du$ and $r(u, v)dv$ are meromorphic since $r(u(t), v(t))u'(t)dt$ and $r(u(t), v(t))v'(t)dt$ are meromorphic near $0$.

*You will get the desired equality by differentiating the identity $f(u(t), v(t)) = 0$.


The book "Plane Algebraic Curves" by E. Brieskorn and H. Knorrer (pp. 628-631) may be also helpful.
