18.966 - Geometry of Manifolds - Spring 2008

D. Auroux - Mondays & Wednesdays, 9:30-11, in 2-102.

Homework

• Homework 1 (handed out in lecture on Tue Feb 19, due on Wed Feb 27).
  Solutions
• Homework 2 (handed out in lecture on Mon Mar 10, due on Wed Mar 19).
  Solutions
• Homework 3 (handed out in lecture on Wed Apr 23, due on Wed Apr 30).
  Solutions

Lecture summary

• Wed Feb 6: review of differential forms, Lie derivative, and De Rham cohomology; cup-product and Poincaré duality.
• Mon Feb 11: symplectic vector spaces and linear algebra; symplectic manifolds, first examples; symplectomorphisms; symplectic form on the cotangent bundle (A. Cannas da Silva, Lectures on Symplectic Geometry, Lect. Notes in Math. 1764, Springer, pp. 3-13).
• Wed Feb 13: symplectic and Lagrangian submanifolds; graphs in T*X; graphs of symplectomorphisms as Lagrangian submanifolds in products; isotopies and vector fields; Hamiltonian vector fields; symplectic vector fields, flux; classical mechanics (Cannas pp. 15-19, 35-37, 105-107).
• Tue Feb 19: isotopy and deformation equivalence; Moser's theorem; Darboux's theorem; tubular neighborhoods (Cannas pp. 37-46).
• Mon Feb 25: local version of Moser's theorem; Weinstein's neighborhood theorem (Cannas pp. 40, 45-52).
• Wed Feb 27: tangent space to the group of symplectomorphisms; fixed points of symplectomorphisms; Arnold's conjecture (Cannas pp. 53-56); Morse theory: gradient trajectories, Morse complex, homology.
• Mon Mar 3: action functional on the loop space, Floer homology; Hamiltonian group actions, moment maps, symplectic reduction (Cannas pp. 127-136, 141-146).
• Wed Mar 5: contact geometry (Cannas pp. 57-66); almost-complex structures; compatibility with a symplectic structure; polar decomposition (Cannas pp. 67-70).
• Mon Mar 10: almost-complex structures: existence and contractibility; almost-complex submanifolds vs. symplectic submanifolds; Sp(2n), O(2n), GL(n,C), and U(n) (Cannas pp. 71-76).
• Mon Mar 17: connections: definition, connection 1-form; horizontal distributions; metric connections; curvature (R. O. Wells, Differential Analysis on Complex Manifolds, Springer GTM 65, pp. 65-72).
• Wed Mar 19: curvature (continued); twisted de Rham operator; Levi-Civita connection on (TM,g); compatibility with J (Wells pp. 72-77).
• Mon Mar 31: Chern classes of complex vector bundles (via curvature and Chern-Weil); Euler class and top Chern class; degree of a line bundle over a closed oriented surface; naturality properties of Chern classes and topological definition; equivalence between the two definitions (Wells pp. 84-97; see also Milnor and Stasheff, Characteristic classes).
• Wed Apr 2: Chern classes continued; classification of complex line bundles; Chern classes of the tangent bundle.
• Mon Apr 7: obstruction to existence of almost-complex structures on a 4-manifold, examples; splitting of tangent and cotangent bundles of (M,J), types; complex manifolds (Cannas pp. 78-80, 83-87).
• Wed Apr 9: Dolbeault cohomology; Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space. (Cannas pp. 81-82, 88-89).
• Mon Apr 14: deformations of complex structures; Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form (Cannas pp. 90-97).
• Wed Apr 16: complex projective manifolds; Hodge decomposition theorem; Hodge * operator on a Riemannian manifold; d* operator; Laplacian, harmonic forms; differential operators; symbol, ellipticity (Cannas pp. 98-99; Wells pp. 114-116).
• Wed Apr 23: elliptic operators; existence of parametrix; elliptic regularity, Green's operator; Hodge * operator on a Kähler manifold (Cannas pp. 98-99; Wells pp. 114-116, 136-141, 154-163).
• Mon Apr 28: complex Hodge theory on a Kähler manifold; relation between real and complex Laplacians; Hodge diamond (Cannas pp. 99-100; Wells pp. 191-199).
• Wed Apr 30: hard Lefschetz theorem (statement); holomorphic vector bundles; canonical connection and curvature (Wells pp. 77-80).
• Mon May 5: holomorphic sections and projective embeddings; ampleness; Donaldson's proof of the Kodaira embedding theorem: local model; concentrated sections.
• Wed May 7: examples of compact 4-manifolds without almost-complex structures, without symplectic structures, without complex structures; Kodaira-Thurston manifold; symplectic fibrations; Thurston's construction of symplectic forms (Cannas pp. 101-103; McDuff-Salamon, Introduction to Symplectic Topology pp. 197-203).
• Mon May 12: symplectic sum along codimension 2 symplectic submanifolds; Gompf's construction of symplectic 4-manifolds with arbitrary pi_1 (McDuff-Salamon pp. 253-256); homeomorphism classification of simply connected 4-manifolds (Freedman); intersection pairings; Seiberg-Witten invariants: vanishing and non-vanishing results.
• Wed May 14: spin^c structures; spin^c connections; Dirac operator; Seiberg-Witten equations; gauge group; moduli space; Seiberg-Witten invariant (J. Morgan, The Seiberg-Witten equations and applications to the topology of smooth 4-manifolds, Princeton Univ. Press)

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 a long digression into complex geometry.

The main topics of the course are:

• 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.
• More (to be determined).

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).

Miscellaneous information

Prerequisites: 18.965 or equivalent: manifolds, vector fields, differential forms, vector bundles, homology, cohomology.

Schedule: Mondays and Wednesdays, 9:30-11 in 2-102

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

References: no required text. The following references are useful:

• A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer-Verlag
• R. O. Wells, Differential Analysis on Complex Manifolds, Springer GTM 65, Springer-Verlag
• D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press
• J.W. Morgan, The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds, Mathematical Notes 44, Princeton University Press