Lecture Notes

These lecture notes were prepared by Kartik Venkatram, a student in the class, in collaboration with Prof. Auroux.

1 Review of differential forms, Lie derivative, and de Rham cohomology (PDF)
2 Cup-product and Poincaré duality in de Rham cohomology; symplectic vector spaces and linear algebra; symplectic manifolds, first examples; symplectomorphisms (PDF)
3 Symplectic form on the cotangent bundle; symplectic and Lagrangian submanifolds; conormal bundles; graphs of symplectomorphisms as Lagrangian submanifolds in products; isotopies and vector fields; Hamiltonian vector fields; classical mechanics (PDF)
4 Symplectic vector fields, flux; isotopy and deformation equivalence; Moser's theorem; Darboux's theorem (PDF)
5 Tubular neighborhoods; local version of Moser's theorem; Weinstein's neighborhood theorem (PDF)
6 Tangent space to the group of symplectomorphisms; fixed points of symplectomorphisms; Arnold's conjecture; Morse theory: Gradient trajectories, Morse complex, homology; action functional on the loop space, and the basic idea of Floer homology (PDF)
7 More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triples (PDF)
8 Almost-complex structures: Existence and contractibility; almost-complex submanifolds vs. symplectic submanifolds; Sp(2n), O(2n), GL(n,C), and U(n); connections: definition, connection 1-form (PDF)
9 Horizontal distributions; metric connections; curvature of a connection: Intrinsic definition; expression in terms of connection 1-form (PDF)
10 Twisted de Rham operator; Levi-Civita connection on (TM,g); Chern classes of complex vector bundles (via curvature and Chern-Weil); Euler class and top Chern class (PDF)
11 Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundles (PDF)
12 Chern classes of the tangent bundle; cohomological criterion for existence of almost-complex structures on a 4-manifold, examples; splitting of tangent and cotangent bundles of (M,J), types; complex manifolds, Dolbeault cohomology (PDF)
13 Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space (PDF)
14 Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theorem (PDF)
15 Hodge * operator on a Riemannian manifold; d* operator; Laplacian, harmonic forms; Hodge decomposition theorem; differential operators; symbol, ellipticity; existence of parametrix (PDF)
16 Elliptic regularity, Green's operator; Hodge * operator and complex Hodge theory on a Kähler manifold; relation between real and complex Laplacians (PDF)
17 Hodge diamond; hard Lefschetz theorem; holomorphic vector bundles; canonical connection and curvature (PDF)
18 Holomorphic sections and projective embeddings; ampleness; Donaldson's proof of the Kodaira embedding theorem: local model; concentrated approximately holomorphic sections (PDF)
19 Donaldson's proof of the Kodaira embedding theorem: Estimates; concentrated sections; approximation lemma (PDF)
20 Proof of the approximation lemma; examples of compact 4-manifolds without almost-complex structures, without symplectic structures, without complex structures; Kodaira-Thurston manifold (PDF)
21 Symplectic fibrations; Thurston's construction of symplectic forms; symplectic Lefschetz fibrations, Gompf and Donaldson theorems (PDF)
22 Symplectic sum along codimension 2 symplectic submanifolds; Gompf's construction of symplectic 4-manifolds with arbitrary pi_1 (PDF)
23 Symplectic branched covers of symplectic 4-manifolds (PDF)
24 Homeomorphism classification of simply connected 4-manifolds; intersection pairings; spin^c structures; spin^c connections; Dirac operator (PDF)
25 Seiberg-Witten equations; gauge group; moduli space; linearized equations; compactness of moduli space (PDF)
26 Seiberg-Witten invariant; properties; vanishing for manifolds of positive scalar curvature; vanishing for connected sums; Taubes non-vanishing for symplectic manifolds; examples of non-symplectic 4-manifolds, of non-diffeomorphic homeomorphic manifolds