MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
I assume the critical values of a holomorphic map are its branch points hence the title. – Konrad Pilch Aug 11 '11 at 6:05
up vote 4 down vote accepted

This is quite standard and probably the question belongs to 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.

share|cite|improve this answer
Thank you for the answer! Just a quick question: What do you mean by $f$ in your short exact sequences? Is that supposed to be $t$? – Konrad Pilch Aug 11 '11 at 18:30
Yes, of course I meant $t$. I fixed the answer, thank you for the remark – Francesco Polizzi Aug 11 '11 at 21:21

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.