Reference on learning Donaldson's construction on sections

Question

I am aiming to understand the constructions on the high dimensional contact and symplectic topology. Giroux gives the sketches of the proofs on the relations between open books and contact structures and as far as I understand his construction is based on Donaldson's constructions on sections. The similar thing happens in symplectic topology as well, the existence of Lefschetz fibrations comes from Donaldson.

To have a complete understanding, I decided to focus on understanding Donaldson's work which requires a good level of knowledge on differential analysis and topology of complex manifolds. Here is path that I am planning to follow:

Step 1: Learn Sheaf Theory and Cohomology of Sheaves (referring to Bredon).

Step 2: Learn Differential Geometry of Complex Manifolds (referring to Raymond O. Wells, Jr.)

Step 3: Read Donaldson's papers.

I would be happy to be advised about if the path I have planned really looks like a good one. Are there any other sources that would work better? Or any topics that I need to cover before passing to Donaldson's papers? I appreciate your help.

Answer 1

Well this is a very hard area of geometry indeed and your three points relating to your plan of attack (as it were) will provide you with a solid grounding, but if you're anything like me, point 3 'read Donaldson's paper' will leave you vastly prematurely aged.

You are probably aware that Contact manifolds are the 'conformal' equivalent of the symplectic manifolds

For Contact geometry, I would:

1. Seek the following paper written by Ibort, Martinez-Torres & Presas, the objective was to study co-dimension $2$ contact submanifolds on a general contact manifold. This was shown to be non-acheivable and moves you nicely to
2. Donaldson's paper on the Lefschetz pencil decomposition of a Symplectic Manifold - J.Differ.Geom 53, 1999. Also seek out Donaldson's Lefschetz Fibrations in Symplectic Geometry, Doc. Math. J.(1998)

The goal in the first paper was to create a co-dimension $2$ fibre over a contact manifold by considering fibration over the sphere. Donaldson went on to prove that every Symplectic Four-manifold admits a Lefschetz pencil.

if the above is too advanced at your stage in research, then could I point you to https://mathoverflow.net/questions/152373/trying-to-understand-lefschetz-pencils for a great discussion on these pencils.

I hope I have been of use,

Your humble servant,

Bacon