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MA954 - Complex Geometry

Lecturer: Dr Weiyi Zhang

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Prerequisites: Familiarity with topics covered in MA3H5 Manifolds, MA3B8 Complex Analysis, MA3H6 Algebraic Topology.

MA475 Riemann Surfaces, MA4C0 Differential Geometry, MA4A5 Algebraic Geometry, MA4J7 Cohomology and Poincare duality would be certainly very helpful.

Content:

The primary goal of this Module is to present some fundamental techniques from several complex variables, Hermitian differential geometry (and partial differential equations, potential theory, functional analysis), to study the geometry of complex, and in particular, Kaehler manifolds. Hodge theory will be one important major topic of this course.

-Basics/definitions concerning complex manifolds, vector bundles and sheaf theory

-Some selected topics from several complex variables: the Cauchy integral, the Cauchy-Riemann equations, Hartogs’s principle, plurisubharmonic functions, domains of holomorphy, holomorphic convexity, Riemann extension theorem, Hormander’s L2 estimates …

-Hermitian differential geometry, curvature of Hermitian holomorphic vector bundles, Chern classes

-Some elliptic operator theory, Kaehler manifolds, Hodge decomposition, Kodaira embedding,

- Outlook on the topology of varieties, Morse theory, Lefschetz pencils, variation of Hodge structures, Clemens-Schmid exact sequences, etc.

References:

R.O. Wells: Differential Analysis on Complex Manifolds

C. Voisin: Hodge Theory and Complex Algebraic Geometry I/II

K. Fritzsche, H. Grauert: From Holomorphic Functions To Complex Manifolds

Topics in Transcendental Algebraic Geometry, ed. Ph. Griffiths

P. Griffiths, J. Harris: Principles of Algebraic Geometry 

V. Kulikov, P.F. Kurchanov: Complex Algebraic Varieties: Periods of Integrals and Hodge Structures, in: Algebraic Geometry III, Encyclopedia of Math. Sciences Vol. 36, Parshin/Shafarevich eds.

S.S. Chern: Complex manifolds without potential theory


D. Huybrechts: Complex geometry: An Introduction