I've just begun working on my bachelor thesis on the "Lefschetz Theorem on Hyperplane Sections" (see for example http://en.wikipedia.org/wiki/Lefschetz_hyperplane_theorem). The goal of the thesis is to explain Andreotti and Frankel's proof of the theorem.

My only reference so far is the book "Morse Theory" (1973) by J. Milnor of which only the first 42 pages are of interest to me. However I'm struggling a lot with the actual proof of the Lefschetz hyperplane theorem in the way it is presented there.

Can anyone please refer me to some other material on this subject?

In Milnor's book the original paper by S. Lefschetz from 1924 and the paper by A. Andreotti and T. Frankel from 1959 are also mentioned. However I doubt that these will be of much help to me. Mainly because in the first, Lefschetz's original proof is stated which is of no (significant) interest to my thesis and as the latter one is an academic paper just 3 or 4 pages in length, I assume that the presentation of the proof there is even more condensed than in Milnor's book.

I've also found some references on the Wikipedia site (http://en.wikipedia.org/wiki/Lefschetz_hyperplane_theorem#References) but I've got no idea which of these books might be suitable for me. Please bear in mind that I'm a third year mathematics student with no significant knowledge in differential and algebraic geometry nor differential topology.

Thanks in advance for any suggestions!


[Perhaps, this is a long comment rather than an answer.]

Well, there is a proof in C. Voisin's second book on Hodge theory, and it is quite similar to the Andreotti-Frankel approach, but it requires not less knowledge of the subject than the Andreotti-Frankel paper. The treatment by Milnor is the most elementary I know. Either way, you'll have to gain knowledge in geometry and topology for this. When you've got sufficient knowledge of these subjects one day, I'm sure you will appreciate the text of Andreotti and Frankel.

Addendum: Concerning differential topology I recommend Milnor, Topology from the differential viewpoint (or point of view?) as a starting point. This may even suffice for understanding Morse theory. The possible references for the algebraic geometry you need depends highly on what you already know and how much time you want to spend. For a quick first reading, you could work through Mumford, Algebraic geometry I; it requires very few algebra and gets deep enough for the basics, but it won't give you the tool of (co-)homology vanishing for Stein spaces. Concerning his I have to think about a good reference other than the early articles of Stein, Grauert and Remmert. Perhaps Voisin's books contain a readable treatment.

By the way, you are right not to have a deep look at Lefschetz's paper at the beginning, but for historical reasons it may be worth having a look at the very original source as soon you understood some proofs in modern language. Somewhere in between, there is the (rather from the topological point of view) article of Klaus Lamotke, The topology of complex projective varieties after S. Lefschetz, I really liked in the days I was more comfortable with algebraic topology than with geometry. I'm not sure if it contains a complete proof of the theorem on hyper plane sections, though. Edit: it does! See paragraph 3, in particular 3.6. For a deep understanding of this article one needs particular knowledge of algebraic topology provided by several textbooks like Dold or Spanier.


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