These lectures are centered around the following result and its various special cases, applications, and extensions:
Theorem. There is an algebraically integrable system on the moduli space of meromorphic Higgs bundles on a curve.
This was proved independently by Markman [M] and Bottacin [Bo], and is closely related to results of Mukai [Mu] and Tyurin [T]. It incorporates and generalizes earlier work of Hitchin [H] and many others. The theorem combines ideas from algebraic geometry and symplectic geometry. In keeping with the expository aim of the lectures, the bulk of these notes concerns not the theorem and its applications, but the many ingredients which go into its proof. It is my hope that students with a fairly modest background in geometry will be able to work through these notes, learning a fair amount of algebraic geometry and symplectic geometry along the way. They may also be motivated to follow some of the leads in the last section towards open problems and further development of the subject.
The symplectic geometry needed for the statement and proof of the theorem is covered in Sections 3.2, 3.3, and 3.7, while the algebraic geometry is in Sections 3.4, 3.5, 3.6. Section 3.2 introduces the basics of symplectic and Poisson manifolds, while Section 3.3 discusses integrable systems. The notions of moment map and symplectic reduction, which are used in the proof, are explained in Section 3.7.