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I'm trying to understand Atiyah--Singer by looking at the usual starting point of $CP^1$ and the Dirac--Dolbeault operator. If I've reduced everything down correctly, then in this case the theorem gives $$ \text{Index}(\overline{\partial} + \overline{\partial}^*) = \frac{1}{2}\int_{CP^1} \text{ch}_1(T(CP^1)), $$ where ch$_1(T(CP^1))$ is the first Chern class of the tangent bundle $CP^1$. I would like a direct explanation of why this is true.

flag	asked Jun 11 2011 at 19:03 Jean Delinez 1,527●6●13

For the RHS, see en.wikipedia.org/wiki/… – Steve Huntsman Jun 11 2011 at 20:44

I don't know helpful my cryptic comment will be, but you can compute both sides explicitly in this case. The right side is $1$ because $ch_1=c_1=2$ (times fundamental class). So is the left: use the Dolbeault resolution and the fact the holomorphic functions on $CP^1$ are constant... – Donu Arapura Jun 11 2011 at 21:21

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@ Steve, Donu: Thanks for your answers, but I'm not really looking to show that both sides are equal (as you show, this is easy) I'm looking for someone who understands the Atiyah--Singer proof to show what it reduces down to in this simple case. – Jean Delinez Jun 11 2011 at 22:02

Yes, I see. I have to admit, I've only really looked as special cases, e.g Hirzebruch-Riemann-Roch. In any case, my (limited) understanding is there are two steps. The first (and probably the deepest) is to see that there is a universal formula in Chern classes of the manifold and symbol. Then to discover the explicit form by computing enough examples. Hopefully, someone else gave give you more insight. – Donu Arapura Jun 11 2011 at 22:57

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I'm a little confused about what you want - the Atiyah-Singer index theorem for the Dolbeault operator on a Kahler manifold is precisely the Hirzebruch-Riemann-Roch theorem. So for $CP^1$ it looks like you're asking about the classical Riemann-Roch theorem. The proof of Atiyah-Singer doesn't really tell you anything new about complex geometry (in fact, if you follow the global proof then the first step is to embed $CP^1$ into Euclidean space...), and the proof that Atiyah-Singer implies Riemann-Roch just involves playing around with Clifford algebras and characteristic classes. – Paul Siegel Jun 12 2011 at 0:44

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Atiyah--Singer for the Complex Projective Line

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