Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Course Description

This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.

Main Topics

  • Review of differential forms and de Rham cohomology
  • Symplectic manifolds; symplectomorphisms; Lagrangian submanifolds
  • Darboux and Moser theorems, Lagrangian neighborhood theorem
  • Complex vector bundles
  • Almost-complex structures, compatibility, integrability
  • Kähler manifolds, Dolbeault cohomology, Hodge theory, projective embeddings
  • Smooth 4-manifolds, intersection pairing, topological invariants
  • Smooth 4-manifolds vs. symplectic 4-manifolds vs. complex surfaces

The symplectic geometry part of the course follows the book by Ana Cannas da Silva, Lectures on Symplectic Geometry (Lecture Notes in Mathematics 1764, Springer-Verlag); the discussion of Kähler geometry mostly follows the book by R. O. Wells, Differential Analysis on Complex Manifolds (Springer GTM 65).

Prerequisites

Geometry of Manifolds (18.965) or equivalent: manifolds, vector fields, differential forms, vector bundles, homology, cohomology.

Homework

Grading for this course is based on homework. Homework assignments are due every 3 weeks or so.

Texts

There is no required text. The following references are useful:

Amazon logo Cannas da Silva, A. Lectures on Symplectic Geometry (Lecture Notes in Mathematics). New York City, NY: Springer, 2001. ISBN: 9783540421955.

Amazon logo Wells, R. O. Differential Analysis on Complex Manifolds. New York City, NY: Springer, 1980. ISBN: 9780387904191.

Amazon logo McDuff, D., and D. Salamon. Introduction to Symplectic Topology. New York City, NY: Oxford University Press, 1999. ISBN: 9780198504511.

Amazon logo Morgan, J. W. The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (Mathematical Notes 44). Princeton, NJ: Princeton University Press, 1995, ISBN: 9780691025971.

Calendar

LEC # TOPICS KEY DATES
1 Review of differential forms, Lie derivative, and de Rham cohomology.  
2 Cup-product and Poincaré duality in de Rham cohomology; symplectic vector spaces and linear algebra; symplectic manifolds, first examples; symplectomorphisms  
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  
4 Symplectic vector fields, flux; isotopy and deformation equivalence; Moser's theorem; Darboux's theorem  
5 Tubular neighborhoods; local version of Moser's theorem; Weinstein's neighborhood theorem  
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  
7 More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triples Homework 1 due
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  
9 Horizontal distributions; metric connections; curvature of a connection: Intrinsic definition; expression in terms of connection 1-form  
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  
11 Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundles  
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 Homework 2 due
13 Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space  
14 Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theorem  
15 Hodge * operator on a Riemannian manifold; d* operator; Laplacian, harmonic forms; Hodge decomposition theorem; differential operators; symbol, ellipticity; existence of parametrix  
16 Elliptic regularity, Green's operator; Hodge * operator and complex Hodge theory on a Kähler manifold; relation between real and complex Laplacians  
17 Hodge diamond; hard Lefschetz theorem; holomorphic vector bundles; canonical connection and curvature  
18 Holomorphic sections and projective embeddings; ampleness; Donaldson's proof of the Kodaira embedding theorem: local model; concentrated approximately holomorphic sections Homework 3 due
19 Donaldson's proof of the Kodaira embedding theorem: Estimates; concentrated sections; approximation lemma  
20 Proof of the approximation lemma; examples of compact 4-manifolds without almost-complex structures, without symplectic structures, without complex structures; Kodaira-Thurston manifold  
21 Symplectic fibrations; Thurston's construction of symplectic forms; symplectic Lefschetz fibrations, Gompf and Donaldson theorems  
22 Symplectic sum along codimension 2 symplectic submanifolds; Gompf's construction of symplectic 4-manifolds with arbitrary pi_1  
23 Symplectic branched covers of symplectic 4-manifolds.  
24 Homeomorphism classification of simply connected 4-manifolds; intersection pairings; spin^c structures; spin^c connections; Dirac operator  
25 Seiberg-Witten equations; gauge group; moduli space; linearized equations; compactness of moduli space  
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