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I am reading Lie Groupoids and Lie Algebroids in Differential Geometry by Kirill Mackenzie.

In appendix (page $291$), before discussing about Atiyah sequence associated to a Principal bundle, the author says the following:

The first advantage of the Atiyah sequence concept is that it allows the standard definitions and basic properties of infinitesimal connections and their curvature forms to be presented quickly and clearly, in an algebraically natural manner. The correspondence between the two standard definitions of a connection is seen to be a particular case of the correspondence between right- and left-split maps in an exact sequence; curvature is seen to measure precisely the extent to which a connection fails to preserve Lie brackets; associated connections, the Bianchi identities and the structural equation appear in a clear and natural algebraic manner. This approach also allows that infinitesimal connection theory should be ragarded not so much as a theory about principal bundles as about their first-order approimations- the Atiyah sequence or Lie algebroid.

The account given here is a fairly rapid rehearsal of the Atiyah sequence approach to the most basic and general concepts of infinitesimal connection theory. At each stage the correspondence of this formulation with the standard on is established. The reader may wish to continue this programme by rewriting further parts of infinitesimal connection theory in terms of Atiyah sequences.

Questions :

  1. What are the other advantages of using Atiyah sequence to study connections?
  2. What are the other possible set up of standard connection theory that can be seen in terms of Atiyah sequences (other than what is mentioned above)?
  3. What could be the motivation to think of connection as in the set up of Atiyah sequence?
  4. Is there an account of Chern-Weil theory using Atiyah sequence definition of connections?
  5. Is there an account for studying Characteristic classes of Principal bundles, using Atiyah sequences?
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  • $\begingroup$ Reason for downvote? $\endgroup$ – Praphulla Koushik Aug 6 '19 at 14:39
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I'll try to say something about question 1).

One major advantage of the Atiyah sequence is that it gives a (very elegant) sheaf-theoretic characterization of connections. Namely, if $E_{G}$ is a principal $G$-bundle over a manifold $M,$ then the Atiyah sequence

$0\rightarrow E_{G}(\mathfrak{g})\rightarrow At(E_{G})\rightarrow \Theta_{M}\rightarrow 0$

is a short exact sequence of (locally free) sheaves over $M.$ In the smooth category this is not so important. But in the complex analytic, or algebraic category, this description allows one to bring to bear the power of sheaf theory in studying various relevant problems. Moreover, this formalism allows you to study the behavior of the notion of connection in these various categories in a fairly uniform way.

As an example of the sheaf machinery, the Atiyah sheaf $At(E_{G})$ governs deformations of the pair $(E_{G}, M),$ in the sense that first order deformations of the pair $(E_{G}, M)$ are given by the sheaf cohomology group $H^{1}(M, At(E_{G})).$

In finer categories than the smooth category, the existence of a connection, i.e. a splitting of the Atiyah sequence is not guaranteed, and in particular it implies that deformations of the pair $(E_{G}, M)$ decouple, in the sense that

$H^{1}(M, At(E_{G}))\simeq H^{1}(M, \Theta_{M})\oplus H^{1}(M, E_{G}(\mathfrak{g})).$

This is only one tip of a very deep iceberg...

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  • $\begingroup$ Thanks. I do not know clearly how does a locally free sheaf gives a vector bundle (as in Algebraic geometry)... I only know something like that exists... So, you are saying use of Atiyah sequence is helpful in complex/algebriac geometry where we can use sheaf theory... Ok.. Do you have any personal choice of paper that discuss this in more detail (an introductory one would be helpful :)) $\endgroup$ – Praphulla Koushik Aug 6 '19 at 17:22
  • $\begingroup$ Unfortunately, I don't know an introductory reference, but I learned a lot by reading the paper of Chan arxiv.org/pdf/1406.6753.pdf and the original paper of Atiyah ams.org/journals/tran/1957-085-01/S0002-9947-1957-0086359-5/…. See also this thesis of Chen math.upenn.edu/grad/dissertations/ChenThesis.pdf for some useful computations. $\endgroup$ – Andy Sanders Aug 6 '19 at 18:04
  • $\begingroup$ Thank you. I will see those references $\endgroup$ – Praphulla Koushik Aug 6 '19 at 18:06

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