# Understanding meromorphic/holomorphic forms on Riemann surface

I refer to Rick Miranda - Algebraic curves and Riemann surfaces chapter IV.1 (p. 105, p. 106, p. 107). I think I understand the regular Euclidean $$\mathbb C$$ case:

• the idea of meromorphic/holomorphic $$1$$-form on open set $$V_1$$ of $$\mathbb C$$: $$\omega_1 = f(z)dz$$, for $$f$$ mero/holo function on $$V$$ and

• the idea of the transformation rule: for $$\omega_2 = g(w)dw$$ on open set $$V_2$$ of $$\mathbb C$$ with $$g$$ mero/holo on $$V$$, we say that $$\omega_1$$ transforms to $$\omega_2$$ under $$T$$ if $$g(w)=f(T(w))T'(w)$$ for some holo $$T: V_2 \to V_1$$

Where it gets fuzzy for me is the case of Riemann surfaces. I wish Miranda would have 1st defined for charts on Riemann surface, but Miranda instead goes straight to Riemann surfaces. Apparently $$\omega$$, a mero/holo $$1$$-form on Riemann surface $$X$$ (in this book, all Riemann surfaces are connected), is a 'collection' (see (A1)) of mero/holo $$\{\omega_{\phi} | \phi: U \to V \ \text{is a chart in, I think, the max atlas of X}\} \tag{see (A2)}$$ such that for all charts $$\phi_1: U_1 \to V_1$$, $$\phi_2: U_2 \to V_2$$, with overlapping domains, we have that $$\omega_{\phi_1}$$ transforms to $$\omega_{\phi_2}$$ under $$T=\phi_1 \circ \phi_2^{-1}$$. I guess this is $$T: \phi_2(U_1 \cap U_2) \to \phi_1 (U_1 \cap U_2)$$.

Ostensibly, we have that for, say, $$\omega_{\phi_1}$$, the expression for $$\omega_{\phi_1}$$ is like '$$\omega_{\phi_1} = f_1(z) dz$$', for coordinate $$z = \phi_1(x)$$ and some mero/holo $$f_1=f_1(z)$$ on open subset $$V_1$$ of $$\mathbb C$$. But what I expected was something an expression involving some mero/holo $$h_1=h_1(x)$$ on the chart $$U_1$$ of $$X$$, like

1. $$\omega$$ is some map $$\omega: X \to \{\text{probably some bundle thing in complex geometry that I didn't learn yet}\},$$ where

2. the restriction $$\omega|_{U_1}$$ is a well-defined (because of the transformation rule for overlapping domains) mero/holo $$1$$-form on the chart domain $$U_1$$, given as $$\omega|_{U_1} = h_1(x) dx$$, where the '$$|_{U_1}$$', is just omitted. And then

3. we can map this from $$X$$ to $$\mathbb C$$ like maybe there's some correspondence to the mero/holo $$1$$-form '$$\omega|_{V_1}$$' on the chart image $$V_1$$, given as something like $$\omega|_{V_1} = (h_1 \circ \phi_1^{-1})(z) dz$$ or even like $$(h_1 \circ \phi_1^{-1})(z) d(\phi_1^{-1}(z))$$. This way $$f_1 = h_1 \circ \phi_1^{-1}: \phi_1(U_1) =V_1 \to U_1 \to \mathbb C$$.

Question 1: Are $$\omega$$'s indeed locally like $$\omega|_U = h(x) dx$$ and then converted from $$X$$'s local coordinate $$x$$ on $$U$$ into $$\mathbb C$$'s local coordinate $$z$$ on $$V$$ into '$$\omega|_{V}$$' = $$(h \circ \phi^{-1})(z) dz$$?

Question 2: Later on, there's a definition for order. How should I understand the definition for order in terms of the above? In particular, is my definition as follows correct?

1. The definition is given as '$$ord_p(\omega) := ord_0(f)$$', for '$$\omega = f(z) dz$$', where $$z=\phi(x)$$, for chart $$\phi: (U,p) \to (V,0)$$, centred at $$p \in U$$. I understand this as $$ord_p(\omega)$$ $$:= ord_{\{\phi(p)=0\}}(f \circ \phi^{-1}(z))$$, for $$\omega|_V = (f \circ \phi_1^{-1})(z) dz$$, which in turn is from $$\omega|_U = f(x) dx$$.

2. Therefore, I can make this kind of definition chain: $$ord_p(\omega) := ord_p(\omega|_U)$$ and then $$ord_p(\omega|_U) := ord_p(f)$$ (and then finally $$ord_p(f) := ord_{\{\phi(p)=0\}} (f \circ \phi^{-1})$$).

3. In particular, this is why I was hoping we would 1st have a definition for $$1$$-forms on charts: like if a Riemann surface $$X$$ is covered by a single chart $$\phi: U = X \to V$$ then we can do for its 1-forms $$\omega$$ like $$ord_p(\omega|_U) := ord_p(f)$$ (where $$\omega$$ = $$\omega|_U$$ since $$U=X$$).

• Question 2.1: Btw, for the original definition of '$$ord_p(\omega) := ord_0(f)$$', for '$$\omega = f(z) dz$$', can I just instead of any chart, that's not necessarily centred at $$p$$? This way, I would define $$ord_p(\omega) := ord_{\phi(p)}(f)$$, whether or not the chart $$\phi: U \to V$$, that gives us the local coordinate $$z=\phi(x)$$, is centred at $$p$$. Of course, it's more convenient to have Laurent series about 0, but just wondering if there's anything particular about the number 0.

Edit: Btw, there's also this thing in the text (but this is on 2-forms now) i noticed that goes like $$\int \int_{T} \eta = \int \int_{\phi(T)} f(z, \overline z) dz \wedge d \overline z,$$ where '$$\eta = f(z, \overline z) dz \wedge d \overline z$$'. I mean, if '$$\eta = f(z, \overline z) dz \wedge d \overline z$$', then one might think you wouldn't have to change the region of integration when replacing $$\eta$$ with $$f(z, \overline z) dz \wedge d \overline z$$. If this were 1-form, like $$\eta = f(z) dz$$, I'd think '$$f(z)$$' is actually like $$f \circ \phi^{-1}(z)$$

(A1): I guess similar to how a holo function on a non-connected open set is a 'collection' of holo functions on connected open sets.

(A2): I think initially mero/holo $$1$$-form is defined in Def IV.1.7/3 for every chart in max atlas and then later it's defined for every chart in an atlas in Lemma IV.1.8/4.

• A one-form is something which is well defined to integrate in a chart independent way. If it integrates to $F(p)$ then the one-form is $dF(p)$, in a chart it will integrate to $F(\phi(z))$ and the one-form is $dF(\phi(z))= \phi'(z) F'(\phi(z))dz$. A meromorphic one-form is when the $\phi'(z) F'(\phi(z))$ are meromorphic, and $F$ is only well-defined locally, it may have some branch points and $dF$ may not integrate to $0$ on closed-curves (try with $F(s)=\log s,s\in\Bbb{C}$ and $F(u)=u,u\in \Bbb{C/(Z+iZ)}$). The order of the poles and zeros of $\phi'(z) F'(\phi(z))$ don't depend on $\phi$. Nov 4, 2020 at 9:48
• @reuns confused by your comment. the one i thing i picked up on was integration. actually in miranda's text i notice there's something that goes like $\int \int_{T} \eta = \int \int_{\phi(T)} f(z, \overline z) dz \wedge d \overline z$, where '$\eta = f(z, \overline z) dz \wedge d \overline z$'. I mean, if '$\eta = f(z, \overline z) dz \wedge d \overline z$', then one might think you wouldn't have to change the region of integration when replacing $\eta$ with $f(z, \overline z) dz \wedge d \overline z$. I'll add this to the post.
– BCLC
Nov 6, 2020 at 13:56

Here is an attempt to answer your questions. Recall that a Riemann surface is a manifold $$M$$ of dimension $$2$$ such that there exists a complex atlas, that is a collection of charts $$(U,\phi_U)$$, with $$\phi_U : U \to \phi(U)\subset \mathbb{C}$$, with transition functions that are holomorphic.

Question 1 The definition of a holomorphic / meromorphic $$1$$-form on a Riemann surface is the following. Let $$\omega$$ be a $$1$$-form on $$M$$. It is holomorphic / meromorphic if for any $$p\in M$$, there exists a complex chart $$(\phi,U)$$ with $$p\in U$$, such that the push-forward $$1$$-form $$\phi_*\omega$$ on $$\phi(U) \subset \mathbb{C}$$ is a holomorphic / meromorphic one form. As a meromorphic $$1$$-form is defined to be of the form $$f(z)\mathrm{d}z$$ on open subsets of $$\mathbb{C}$$, $$\omega$$ is a meromorphic $$1$$-form if for any $$p\in M$$, there exists a complex chart $$(\phi,U)$$ with $$p\in U$$, such that there exists a meromorhic function $$f$$ on $$\phi(U)\subset \mathbb{C}$$ with $$\phi_*\omega = f\mathrm{d}z$$, that is $$\omega = \phi^*\left(f(z)\mathrm{d}z\right)$$. One can show that in an holomorphic atlas, the meromorphic functions $$f$$ behave very-well under change of charts.

Question 2 In a manifold $$M^n$$, the definition of a chart centered at $$p\in M$$ is a chart $$(U,\phi)$$ with $$\phi : U \to \mathbb{R}^n$$, with $$\phi(p) = 0$$. This is so that anything at $$p$$ can be read (in the charts centered at $$p$$) at the origin. Thus, for the definition of order, we define the order of a meromorphic $$1$$-form $$\omega$$ at $$p$$ to be the order of the meromorphic $$1$$-form $$\phi_* \omega$$ on $$\phi(U)\subset \mathbb{C}$$ at $$0$$, for $$(U,\phi)$$ centered charts at $$p$$. If one requires the chart not to be centered, one could define it to be the order at $$\phi(p)$$ of the meromorphic $$1$$-form $$\phi_*\omega$$. Note that any chart can be translated to a chart centered at $$p$$, so this is not a restrictive definition. The fact that transition functions are holomorphic shows that the order of a meromorphic $$1$$-form is a well-defined notion and does not depend on the chart, and is an intrinsic definition.

As order is a pointwise notion that depends on the local behavior, you are right when saying that the order of $$\omega$$ at $$p$$ is the same as the order of $$\omega|_U$$ at $$p$$.

If $$M$$ is covered by a single chart, then $$M$$ is an open subset of $$\mathbb{C}$$! Therefore, any local holomorphic / meromorphic function can be written as a global holomorphic / meromorphic function (this is a complex analysis result), and thus any holomorphic / meromorphic $$1$$-form is globally of the form $$f(z)\mathrm{d}z$$.

For the integration part of your question. A complex manifold is canonically oriented by its complex structure. In the case of a Riemann surface, there is a canonical volume form, defined on open subset $$U$$ by $$\phi^* \left(\frac{i}{2}\mathrm{d}z\wedge \mathrm{d}\overline{z} \right)$$. This is because if $$z = x+iy$$, $$\mathrm{d}z\wedge \mathrm{d}\overline{z} = -2i\mathrm{d}x\wedge\mathrm{d}y$$. One can choose the volume form to be $$\mathrm{d}z\wedge \mathrm{d}\overline{z}$$, it does not really matter (it seems it is what Miranda has chosen). Thus, if $$\eta$$ is a holomorphic / meromorphic $$2$$-form on $$M$$, in a chart $$(U,\phi)$$, there exists a holomorphic / meromorphic function $$f$$ such that $$\eta = \phi^* \left( f(z) i\mathrm{d}z\wedge \mathrm{d}\overline{z} \right)$$. This is because $$\Lambda^2(\phi(U))$$ is a rank $$1$$ trivial bundle with global section $$\mathrm{d}z\wedge \mathrm{d}\overline{z}$$ nowhere vanishing, so every $$2$$-form on $$\phi(U)$$ can be written $$f \times \mathrm{d}z\wedge \mathrm{d}\overline{z}$$. By the theory of integration of $$n$$-form over an oriented $$n$$ dimensional manifold $$M$$, the definition of $$\int_U\eta$$ is \begin{align} \int_{U}\eta = \int_{\phi(U)} \phi_*\eta = \int_{\phi(U)}f\mathrm{d}z\wedge\mathrm{d}\overline{z} \end{align} the $$f(z,\bar z)$$ part is just a notation in complex geometry. The function $$f$$ just depends on $$z$$ as a complex coordinate (and therefore on two real variables $$x$$ and $$y$$) but for theoretical purpose, considering $$f$$ as a function of $$z$$ and $$\bar z$$ is convenient. For example, a smooth function $$f$$ is holomorphic of and only if $$\dfrac{\partial}{\partial \bar z}f = 0$$ in this notations.