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

LEC # | TOPICS |
---|---|

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 |