Definition of meromorphic form. I stumbled upon the term meromorphic 2-form on a complex manifold. How is that defined? I know what a holomorphic 2-form is, but what does it mean to be meromorphic? A meromorphic function is one that can be $\infty$ at isolated points, but I can't imagine how a form is meromorphic.
 A: If you've taken calculus, you have more than likely already met some meromorphic differential forms: for example, we often see the differential form $$\frac{xdy-ydx}{x^2+y^2}$$ and get to know its properties when discussing line integrals. This is exactly an example of a meromorphic differential: you can tell because the coefficients of the $dx$ and $dy$ terms are meromorphic functions. Every meromorphic differential looks like this in local coordinates: since you're asking about a 2-form, yours will look like $\sum f dx_idx_j$ for $f$ a meromorphic function and the $x$s as local coordinates.
If you're interested in a sheafier description of this, you can think about rational sections in algebraic geometry which are exactly the analogue of meromorphic sections in complex analysis. You can either think about tensoring your bundle/sheaf with the sheaf of meromorphic/rational functions and taking a global section, or think about taking a regular section on some dense open set. I leave it to you to verify that these are really the same things.
