everything.study
18.117

Topics in Several Complex Variables

3-0-9 units · Graduate

Harmonic theory on complex manifolds, Hodge decomposition theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of Stein manifolds. As time permits students also study holomorphic vector bundles on Kahler manifolds.

Spring 2005

Topics

  • Functions of one Complex Variable, Cauchy Integral Formula, Taylor Series, Analytic Continuation
  • Cauchy Integral Formula (cont.), Inhomogeneous C.R. Equation, Riemann Equation in One Variable, Functions of Several Complex Variables
  • The Inhomogeneous Cauchy-Riemann Equation in Several Variables, Hartog’s Theorem
  • Applying Hartog’s Theorem, The Dolbeault Complex, Exactness of the Dolbeault Complex on Polydisks
  • The Holomorphic Version of the Poincare Lemma
  • The Inverse Function Theorem and the Implicit Function Theorem for Holomorphic Mappings
  • Complex Manifolds: Affine and Projective Varieties
  • Complex Manifolds: Affine and Projective Varieties (cont.)
  • Sheaf Theory and Sheaf Cohomology
  • The DeRham Theorem for Acyclic Covers
  • Linear Aspects of Symplectic and Kaehler Geometry
  • The Local Geometry of Kaehler Manifolds, Strictly Pluri-subharmonic Functions and Pseudoconvexity
  • The Ricci Form and the Kaehler Einstein Equation
  • The Fubini Study Metric on CP n
  • Differential Operators on R n and Manifolds
  • Smoothing Operators, Fourier Analysis on the n-torus
  • Pseudodifferential Operators on T n and Open Subsets of T n , Elliptic Operators on Compact Manifolds
  • Systems of Elliptic Operators and Elliptic Operators on Vector Bundles
  • Elliptic Complexes and Examples
  • Hodge Theory, the *-operator
  • Computing the *-operator
  • The *-operator in Kaehler Geometry
  • The *-operator in Kaehler Geometry (cont.)
  • The Symplectic Version of the Hodge Theory
  • The Symplectic Version of the Hodge Theory (cont.)
  • The Brylinski Conjecture and the Hard Lefchetz Theorem, Hodge Theory on Riemannian Manifolds
  • Basic Facts About Representations of SL(2,R), SL(2,R) Modules of Finite H-type
  • Hodge Theory on Kaehler Manifolds
  • Hodge Theory on Kaehler Manifolds (cont.)
  • Actions of Lie Groups on Manifolds, Hamiltonian G Actions on Symplectic Manifolds
  • Symplectic Reduction
  • Kaehler Reduction and GIT Theory
  • Toric Varieties
  • The Cohomology Groups of Toric Varieties
  • Stanley’s Proof of the McMullen Conjecture

Readings

  • Hormander, L. An Introduction to Complex Analysis in Several Variables . 2nd revised ed. Vol. 7, North-Holland Mathematical Library . Amsterdam-London: North-Holland Publishing Co. ; New York, NY: American Elsevier Publishing Co., Inc., 1973. ISBN: 0444105239.
    Listed reading
  • Griffiths, P., and J. Harris. Principles of Algebraic Geometry . New York, NY: John Wiley and Sons, 1978. ISBN: 0471327921.
    Listed reading

Prerequisites & course information

Prereq: 18.112 and 18.965 Acad Year 2027-2028: G (Spring) Acad Year 2028-2029: Not offered 3-0-9 units Can be repeated for credit.

Sources