Special Topics Course

Math 581 DA: Introduction to Several Complex Variables and Almost-Complex Geometry

S. Ivashkovich

Autumn 2001, MWF 11:30-12:20

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• Complex and symplectic structures on real vector spaces, Hermitian metrics.

• Holomorphic functions of several complex variables: Cauchy-Riemann

conditions, separate analyticity, extension properties.

• Plurisubharmonic functions and domains of holomorphy. Biholomorphic mappings.

• L^2-estimates for the solutions of the ¶-equation. Applications.

• Projections and analytic sets. Bishop compactness theorem.

Complex manifolds. Riemannian metrics and complex structures on real surfaces.

• Complex and holomorphic bundles, analytic sheafs. Holomorphicity of complex bundles on Riemann surfaces. Cauchy-Riemann type operators, Chern class and Riemann-Roch Formula. Sobolev spaces, index, injectivity and surjectivity of CR-type operators.

• Almost-complex manifolds. Contractibility of the space of adapted structures. Existence of complex curves and generic non existence of holomorphic functions. Existence of plurisubharmonic functions, integrability condition of Newlander-Nirenberg.

• Kobayashi hyperbolicty of complex and almost-complex manifolds.

• Complex curves in symplectic geometry: domains in symplectic vector space, symplectomorphisms, Gromov's non-squeezing Theorem.

Textbooks

1. McDuff D., Salomon D.: J-holomorphic Curves and Quantum Cohomology. A.M.S. Univ. Lect. Ser. v. 6.

2. Hummel C.: Gromov Compactness Theorem for Pseudoholomorphic Curves. Birhäuser.

3. Hörmander L.: An Introduction to Complex Analysis in Several Variables. Van Nostrand, Princeton.

4. S. Ivashkovich, V. Shevchishin : Complex Curves in Almost Complex Manifolds and Envelopes of Meromorphy. Ruhr-University-Bochum Graduirentenkollegs Lecture Notes Series, Issue 36, pp. 1-186 (1999), see also math.CV/9912046.