The degree of a meromorphic form in a genus $g$ Riemann surface is equal to $2g-2$… is this in some way related to the Gauss-Bonnet theorem?

Question

I’m following an introductory course on Riemann surfaces. Today, the lecturer proved the fact that the degree of a meromorphic form in a genus $g$ Riemann surface is equal to $2g-2$ (we can deduce from this, for instance, that the sum of orders of zeros of a holomorphic nontrivial form is $2g-2$ as well).

He then said as a side remark that this result is related to the Gauss-Bonnet theorem from differential geometry (he called this result “flat Gauss-Bonnet”), in the sense that (in some intuitive way, I reckon) “all the curvature would be concentrated in the singularities”, where “conical singularities act like Dirac masses” (or something along that lines… I can’t precisely recall his words since I didn’t totally grasp what his point was).

I more or less could picture what he meant in a handwavy way, but I prefer trying to get a more solid understanding over things. Is there some precise sense in which the singularities of a meromorphic function on a Riemann surface are related to the (Gaussian…? Geodesic…?) curvature of that surface? Is the result about the degree of meromorphic functions actually related to the Gauss-Bonnet theorem? Would there be any other result generalizing both, or a more general framework in which this analogy could be made explicit? In general, how can I learn more about this subject?

(I have some basic knowledge of differential geometry, if that helps. But in any case references on the subject would be welcome.)

Answer 1

The formalism of distributional curvature concentrated at finitely many points and the corresponding Gauss-Bonnet formula was worked out in:

Troyanov, Marc, Les surfaces euclidiennes à singularités coniques. (Euclidean surfaces with cone singularities), Enseign. Math., II. Sér. 32, 79-94 (1986). ZBL0611.53035.

Troyanov, Marc, Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc. 324, No. 2, 793-821 (1991). ZBL0724.53023.

McOwen, Robert C., Point singularities and conformal metrics on Riemann surfaces, Proc. Am. Math. Soc. 103, No. 1, 222-224 (1988). ZBL0657.30033.

Answer 2

I don't know for certain if this is the correct approach, but it seems like the "natural" thing to do.

Suppose we are given a meromorphic $1$-form $\eta$. Then we can define a meromorphic quadratic differential $q=\eta\otimes\eta$, and we can define $g=|q|$ which is a "pseudo-metric". It may have points where the metric vanishes, and it may also have poles. In a local holomorphic coordinate $z$, we have $q=f(z)dz\otimes dz$ for some meromorphic function $f:U\to\mathbb{C}_\infty$, so the metric in local coordinates can be written as $|f(z)|(dx^2+dy^2)$. Away from the poles of $f$, this metric is flat (which is pretty much the statement that $f$ is holomorphic away from the poles). So the only contributions to the curvature come in the form of some Dirac delta functions at the poles, which (I would imagine) is related to the order of the poles of $f$.

If you were to work out the details, I think you should find that $$\text{deg}(\eta)=2g-2\iff\int_X\Theta=2-2g$$ where $\Theta$ denotes the Pfaffian of the curvature $2$-form of the Levi-Civita connection of the pseudo-metric $g$ (divided by $2\pi$). Admittedly, I have not tried to verify these details, but if something is unclear (or incorrect) please let me know.