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