Branch points of a non-constant holomorphic map between compact riemann surfaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:12:11Z http://mathoverflow.net/feeds/question/72658 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72658/branch-points-of-a-non-constant-holomorphic-map-between-compact-riemann-surfaces Branch points of a non-constant holomorphic map between compact riemann surfaces Konrad Pilch 2011-08-11T06:02:57Z 2011-08-11T21:19:17Z <p>While working on a project for mathematics I came across the following lemma: [Kock]</p> <p>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. </p> <p>I have only recently started learning about this sort of stuff and have been utlising Forster's <i>Lectures on Riemann Surfaces</i>. I restated the theorem as follows:</p> <p>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. </p> <p>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).</p> <p>Kock in his proof uses the following fact that seems simple but has me absolutely stumped (even after days of research):</p> <p>The set of critical values of $t$ are given by $t(supp(\Omega_{X/\mathbb{P}_{\mathbb{C}}^1}))$</p> <p>What is the definition of $supp(\Omega_{X/\mathbb{P}_{\mathbb{C}}^1})$? And how does this fact follow?</p> http://mathoverflow.net/questions/72658/branch-points-of-a-non-constant-holomorphic-map-between-compact-riemann-surfaces/72667#72667 Answer by Francesco Polizzi for Branch points of a non-constant holomorphic map between compact riemann surfaces Francesco Polizzi 2011-08-11T08:37:37Z 2011-08-11T21:19:17Z <p>This is quite standard and probably the question belongs to <a href="http://math.stackexchange.com" rel="nofollow">http://math.stackexchange.com</a> rather than Mathoverflow, anyway let me give an answer.</p> <p>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 $X$: <code>$$0 \longrightarrow T_X \stackrel{dt}{\longrightarrow} t^*T_{P^1} \longrightarrow T_{X/P^1} \longrightarrow 0,$$</code> where the cokernel $T_{X/P^1}$ is called the <em>relative tangent sheaf</em> of $t$.</p> <p>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 $t(\textrm{Supp}(T_{X/P^1}))$.</p> <p>Dualizing the previous sequence one obtains</p> <p><code>$$0 \longrightarrow t^*\Omega_{P^1} \stackrel{(dt)^*}{\longrightarrow} \Omega_X \longrightarrow \Omega_{X/P^1} \longrightarrow 0,$$</code></p> <p>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 <em>relative cotangent sheaf</em>.</p> <p>Since clearly $\textrm{Supp}(T_{X/P^1})=\textrm{Supp}(\Omega_{X/P^1})$, the claim follows.</p>