# Comparing the proofs of Riemann Roch & Serre Duality in Forster's and Miranda's book

I am learning about Riemann surfaces, using the books of Forster and Miranda. When stating and proving Riemann-Roch and Serre Duality in both books a space appears which seem somehow related, but this relation is not immediately clear (to me).

These spaces are the following:
Let $X$ be a compact Riemann surface.

In Miranda's book $$H^1(D)= \tau [D](X)/\operatorname {Im}(\alpha_D),$$ where $\tau[D](X)$ is the group of Laurent tail divisors on $X$ corresponding to the Divisor $D$ and $\alpha_D$ is the truncation map which removes all terms of order $-D(p)$ or higher from the Laurent series of a meromorphic function. I.e. $H^1(D)$ measures the failure of being able to solve Mittag-Leffler problems on $X$ of finding a meromorphic function with a given tail.

In Forsters book the space of interest is the first cohomology group $H^1(X,\mathcal O_D)$ with $\mathcal O_D$ the sheaf of meromorphic functions $f$ with $\operatorname{div} f \ge -D$.

With $\Omega^1_D$ the sheaf of meromorphic 1-forms $\omega$ satisfying $\operatorname{div} \omega \ge -D$.
Serre duality then reads as $$H^1(D)= \Omega^1_{-D}(X),~~~ \text{ resp } ~~~H^1(X,\mathcal O_D)= \Omega^1_{-D}(X).$$

In both cases the duality map is given by the residue map. And the proofs are somehow similar, as the strategy is to shaw that the suitable defined resiude map is an isomorphism.

In order to calculate the Residue of a cohomology class, Forster introduces so called Mittag-Leffler distributions, which for a given open covering $\mathcal U$ are cochains $\mu \in C^0(\mathcal U, \mathcal M^1)$ with $\delta \mu \in Z^1(\mathcal U, \Omega^1)$, i.e. $\mu_j- \mu_i$ is holomorphic on $U_i \cap U_j$ and defines $\operatorname{Res} (\mu)= \sum_{a \in X} \operatorname{Res}_a (\mu)$.
He then proves $\operatorname{Res}([\delta \mu])= \operatorname{Res}(\mu)$.

I got the feeling, that both authors do basically the same, but use different words to describe it. Using both versions of Serre duality, the spaces are isomorphic, but there seems to be a more direct connection.
So what is it? In particular, how are Mittag-Leffler Distributions related to Mittag-Leffler problems? So far I could only observe that for a Mittag-Leffler distribution the Laurent series of the $\mu_i$ have the same principal part.

The bounty ended and unfortunately the question was not answered satisfactorily, even a year later. Hopefully somebody can step in.

• If around $p$, $f(z)=\sum_{n=-k}^\infty c_n z^n$, then $(\alpha_D)f(z)=\sum_{n=-k}^{-D(p)-1} c_n z^n$. So iff $f$ is holomorphic, then $\alpha_D f \equiv 0$, i.e. $L(D)=\operatorname{ker} \alpha_D$, whereas $H^1(D)=\operatorname{coker} \alpha_D$. – klirk Aug 8 '17 at 17:34
• So you are considering the injective linear map $\phi(f) = -\sum_{p \in X,n \ge 1} \frac{Res(f(z) z^{n-1},p)}{(s-p)^n}$ where $\frac{1}{(z-p)^n}$ is purely formal, to indicate a pole at $p \in X$, and $\alpha_D(f) = -\sum_{p \in X,n \ge 1} \frac{Res(f(z) z^{n-1},p)}{(s-p)^n} 1_{n > D(p)}$ – reuns Aug 8 '17 at 18:24
• Honestly, I don't know. Miranda fixes around each point in $X$ a chart and then works with the Laurent series in these charts, so we don't really need to introduce residues in the formal sum. – klirk Aug 8 '17 at 18:48
• Another important difference: For Miranda, a Riemann surface is a smooth complex-algebraic curve, in particular, it always can be realized as a ramified cover over $CP^1$. For Forster, a Riemann surface is a complex 1-dimensional manifold. In the end, under compactness assumption, this amounts to the same thing, but a priori, you do not know that a compact connected Riemann surface has a nonconstant holomorphic function to $CP^1$. Thus, Forster's methods are analytic in nature. – Moishe Kohan Aug 10 '17 at 23:13

$\def\CC{\mathbb{C}}\def\cO{\mathcal{O}}\def\cK{\mathcal{K}}$There are three definitions of coherent sheaf cohomology I'll want to compare here:

(1) Miranda's definition (which is pretty close to the adelic definition of Weil).

(2) Sheaf cohomology of algebraic functions in the Zariski topology.

(3) Sheaf cohomology of holomorphic functions in the analytic topology (Forster's definition).

The equality of (2) and (3) is Serre's GAGA, and I'm not sure I can say anything insightful about it. This answer will focus on the equivalence of (1) and (2).

Let $D$ be a divisor supported on a finite subset $S$ of $X$. Let $U$ be the open set $X \setminus S$. Let $\cO_U$ be the sheaf on $X$ obtained by pushing forward the structure sheaf $\cO$ from $U$. Note that $\cO_U$ is acyclic, since it is pushed forward from the affine variety $U$. For $s \in S$, let $(\cO_s)_s$ be the skyscraper sheaf whose value at $s$ is the local ring $\cO_s$. Let $\cK$ be the field of meromorphic functions on $X$ and let $\cK_s$ be the skyscraper sheaf equal to $\cK$ at $s$. Note that, since they are skyscraper sheaves, $(\cO_s)_s$ and $\cK_s$ are acylic.

Then we have a short exact sequence of sheaves: $$0 \to \cO \to \cO_U \oplus \bigoplus_{s \in S} (\cO_s)_s \to \bigoplus_{s \in S} \cK_s \to 0.$$ Exactness can be easily checked on stalks. So $H^1(X, \cO)$ is the cokernel of $$H^0(X \setminus S, \cO) \oplus \bigoplus_{s \in S} \cO_s \longrightarrow \bigoplus_{s \in S} \cK.$$ We can quotient by the image of the second summand, and get that $H^1(X, \cO)$ is the cokernel of $$H^0(X \setminus S, \cO) \longrightarrow \bigoplus_{s \in S} \cK/\cO_s.$$

The quotient $\bigoplus_{s \in S} \cK/\cO_s$ is finite sums $\sum^{\mathrm{finite}}_{j < 0} c_j z_s^j$ where $z_s$ is a coordinate near $s$. So $$H^1(X, \cO) \cong \mathrm{CoKer} \left(H^0(X \setminus S, \cO) \longrightarrow \bigoplus_{s \in S} \left\{ \sum\nolimits^{\mathrm{finite}}_{j < 0} c_j z_s^j \right\} \right).$$

If we tensor everything with $\cO(D)$, we similarly get $$H^1(X, \cO) \cong \mathrm{CoKer} \left(H^0(X \setminus S, \cO) \longrightarrow \bigoplus_{s \in S} \left\{ \sum\nolimits^{\mathrm{finite}}_{j < -D_s} c_j z_s^j \right\} \right).$$

Now, this still isn't quite Miranda's definition. The meromorphic functions on $X$ are $\lim\limits_{S \to} H^0(X \setminus S, \cO)$ and the Laurent tails are the limit of the right hand side of $S$. So Miranda is working with $$\mathrm{CoKer} \left( \lim\limits_{S \to}H^0(X \setminus S, \cO) \longrightarrow \lim\limits_{S \to} \bigoplus_{s \in S} \left\{ \sum\nolimits^{\mathrm{finite}}_{j < -D_s} c_j z_s^j \right\} \right).$$

Our previous computation shows that the cokernel stabilizes at $H^1(X, \cO_D)$ once $S$ is large enough to support $D$, and cohomology commutes with forward limits, so Miranda's definition works.

• I appreciate that you took the time to write this down, but it does not really answer my question – klirk Aug 16 '17 at 10:09
• I verify that Miranda's definition gives $H^1$ as defined by Forster (up to GAGA issues). I thought that was what you were looking for; sorry if it isn't. – David E Speyer Aug 16 '17 at 12:21
• First, it is not clear to me how the definitions of sheaf cohomology differ in these books. Secondly, Miranda does not (explicitly) use sheaf cohomology to proof Riemann-Roch and Serre Duality. Or are you talking about $H^1$ as defined in my question? If so, still don't get much insight from your answer, but this is possibly because I don't have much background in Algebra/Algebraic Geometry. As I already have mentioned in my question, I could simply compare the different versions of Serre Duality and get that the spaces are isomorphic. – klirk Aug 16 '17 at 12:45
• I am mor interested in how the differences and paralles in the proof of Serre duality occur and how to "translate" the ideas of one author in the language of the others and vice versa. (e.g Mittag-Leffler distributions, Residue map etc) – klirk Aug 16 '17 at 12:46