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What are some good references (papers/lecture notes/books) on Picard-Lefschetz theory, Morse theory and complex manifolds? I am looking for material at a level as introductory as possible, as my background is in physics and I have little knowledge of pure mathematics. I have an elementary understanding of point-set topology, real manifolds and symplectic geometry, but almost no algebraic topology. Which texts are fairly self-contained and enjoyable to read?

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Milnor's "Morse Theory" is a classical source for the Morse Theory. Complex manifolds is an enormous field. You have to be more specific. You can start with by reading Chapter 0 of Griffiths-Harris "Principles of Algebraic Geometry". As for the P-L theory, I do not think there are any textbook-level treatments, "Applied Picard–Lefschetz theory" by V.Vassiliev is probably your best option. However, Milnor's another gem "Singular points of Complex Hypersurfaces" may be a good starting point for the P-L theory. Also, you can read Chapter 4, Part 2 of Griffiths-Harris: Even though it only covers maps from surfaces to curves, it is instructive to understand this case first.

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Reference for Picard-Lefschetz theory:

  1. Claire Voisin. Hodge Theory and complex algebraic geometry II cha.2,3
  2. Klaus Lamotke. The topology of complex projective variety after S. Lefschetz
  3. Liviu I. Nicolaescu. Notes on the Topology of Complex Singularities
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