Question

While working on a project for mathematics I came across the following lemma: [Kock]

If $X$ is a curve defined over an algebraically closed subfield $N$ of $\mathbb{C}$ and let $t:X\rightarrow \mathbb{P}_{\mathbb{C}}^1$ be a finite morphism defined over $N$, then the critical values of $t$ are $N$-rational.

I have only recently started learning about this sort of stuff and have been utlising Forster's Lectures on Riemann Surfaces. I restated the theorem as follows:

If $X$ is a compact connected Riemann surface defined over an algebraically closed subfield $N$ of $\mathbb{C}$ and let $t:X\rightarrow \mathbb{P}_{\mathbb{C}}^1$ be a non-constant holomorphic map defined over $N$, then the critical values of $t$ are $N$-rational.

I believe this is allowable as compact Riemann surfaces can be embedded in a projective space where they are the common zero of a set of polynomials over $N$. (My wording here may be a little off - feel free to correct me).

Kock in his proof uses the following fact that seems simple but has me absolutely stumped (even after days of research):

The set of critical values of $t$ are given by $t(supp(\Omega_{X/\mathbb{P}_{\mathbb{C}}^1}))$

What is the definition of $supp(\Omega_{X/\mathbb{P}_{\mathbb{C}}^1})$? And how does this fact follow?

Answer 1

This is quite standard and probably the question belongs to http://math.stackexchange.com rather than Mathoverflow, anyway let me give an answer.

The holomorphic map $t \colon X \to \mathbb{P}^1$ induces a natural map $dt$ between tangent bundles, hence a short exact sequence of coherent sheaves on $X$: $$0 \longrightarrow T_X \stackrel{dt}{\longrightarrow} t^*T_{P^1} \longrightarrow T_{X/P^1} \longrightarrow 0,$$ where the cokernel $T_{X/P^1}$ is called the relative tangent sheaf of $t$.

Notice that, by the Jacobian criterion, the critical points of $t$ are precisely the points were $dt$ has not maximal rank, so the critical values (branch points) of $t$ are given by $t(\textrm{Supp}(T_{X/P^1}))$.

Dualizing the previous sequence one obtains

$$0 \longrightarrow t^*\Omega_{P^1} \stackrel{(dt)^*}{\longrightarrow} \Omega_X \longrightarrow \Omega_{X/P^1} \longrightarrow 0,$$

where the map $(dt)^*$ is induced by the pullback of the holomorphic $1$-forms and the cokernel $\Omega_{X/P^1}= \operatorname{Ext}^1(T_{X/P^1}, \, \mathcal{O}_X)$ is called the relative cotangent sheaf.

Since clearly $\textrm{Supp}(T_{X/P^1})=\textrm{Supp}(\Omega_{X/P^1})$, the claim follows.