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I would like to find some good references (or any insight) that would help me to understand a few articles mentioning rigidity of Hirzeburch genus. One of the consequences of this phenomenon is that whenever you have a Hamiltonian circle action (with isolated fixed points?) on a symplectic manifold $M$, you can calculate the Hirzebruch genus $\chi_y(M)$ from the Betti numbers of $M$. This is written, for example, in Remark 4.15(i) in https://arxiv.org/abs/1604.00277.

I tried to look into classical papers on rigidity, for example, Bott and Taubes 1989 https://pdfs.semanticscholar.org/e058/c27d3a9d551852d4325739559e069da33598.pdf , but I am not able to see a connection to the article above (due to my very low knowledge in the area). So I wonder if there are some intermediate references, that explain a bit what is rigidity and how it affects the data of fixed points of $S^1$-actions (including the weights) on a manifold.

Also, if you could give some general answer (intuition, reference) for the question in the title, it would be great as well.

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    $\begingroup$ Very quickly, the Hirzebruch genus is the generating function for indices $\chi^p$, each of which is the index of the elliptic operator $\bar\partial+\bar\partial^\ast$ on $\bigoplus_{q\;\text{even}}\Omega^{p,q}(M)$. Rigidity concerns the $S^1$ acting trivially in some ways on the virtual index (i.e. cancellations in kernel and cokernel of the operator). $\endgroup$
    – Chris Gerig
    Commented Feb 15, 2020 at 17:18
  • $\begingroup$ Thanks Chris. What you say works for all almost complex manifolds? Do you think there is nice readable reference for this? (if the reference speaks of connection to localisation formulas, even better). $\endgroup$
    – aglearner
    Commented Feb 15, 2020 at 18:39
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    $\begingroup$ The book "Manifolds and Modular Forms" by Hirzebruch, Berger and Jung may be helpful. See especially section 5.7. $\endgroup$
    – Gustavo Granja
    Commented Feb 19, 2020 at 9:47

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