Comments – Lecture I

I gave the first lecture of class today.  Thanks to all the class participants and especially to David and Jinpeng, the note takers, and to everyone who asked questions (great questions).  My class list has the names of Alok, David, Hsuan-Yi, Jinpeng, Angel, Qijun, Adam and Ryan on it.  There were a couple of other participants as well.   Please send an e-mail with your name and I will add you to the blog.

We began by asking a simple question: how many polynomials are there?  The dimension of the space \(P_n\) of polynomials of degree \(\le n\) is \(\max\{0,n+1\}\).  We interpreted this space algebro-geometrically as the space of meromorphic functions subordinated to the divisor \( n[p]\), where \(p\) is the point at infinity on the complex projective line \( {\mathbb C}{\mathbb P}^1\).   Then we thought about the analogous problem for an elliptic curve \({\mathbb C}/\Lambda\) (where one can use the classical theory of Weierstrass’ elliptic functions) and (without details) on a Riemann surface of genus 2.  These results are all compatible with Riemann’s inequality

\[ \dim {\mathscr O}(D) \ge |D| – g + 1 \]

for a divisor \(D\) on a compact Riemann surface of genus \(g\).  Later, Riemann’s student Roch refined this to the Riemann-Roch theorem

\[ \dim {\mathscr O}(D)  – \dim {\mathscr O}(K-D) =|D| – g + 1 \]

where the second quantity on the left is called the superabundance of \(D\).

In the second part of the lecture we interpreted the left hand side in the language of partial differential equations.  We saw that associated to a divisor \(D\) there is a complex line bundle \(L=L_D\) such that the left side of the Riemann-Roch theorem is the Euler characteristic of the length two chain complex

\[  C^\infty(L)  \longrightarrow  C^\infty(T^{0,1}\otimes L) \]

where the operator is the Cauchy-Riemann operator \(\partial/\partial{\bar z}\).

Several people asked me for some reading on Riemann surfaces.  There are infinitely many possibilities for this.  Here are some notes from my Oxford colleague Nigel Hitchin which provide a little background to what we used today.  (I learned most of this from Gunning’s Lectures on Riemann SurfacesThere are some nice lecture notes by Kapovich giving a classical take on the Riemann-Roch theorem.) But the Riemann surfaces appeared only for an example in today’s lecture.

On Thursday we will begin discussion of the Atiyah and Bott Lefschetz theorem papers. We will look mainly at section 1 of paper I, with some examples from sections 3 and 4 of paper II

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