Understanding meromorphic/holomorphic forms on Riemann surface

Question

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?

4. 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 $\omega|_U = f(x) dx$.

5. 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 $ord_p(f) := ord_{\{\phi(p)=0\}} (f \circ \phi^{-1})$).

6. 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.

Answer 1

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.