Homepage of M392C, 59000: Complex Geometry; Fall 2004

Motto

"The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles" (The Clay Mathematics Institute's cryptic description of the Hodge conjecture, one of the seven one-million dollar millenium problems)

Essays Introductory essays written by the students in this course:

  • Craig Michoski: Braided Jets
  • Michael Ortiz: Manifolds and Holonomy
  • Phillip Schmitz: Calabi-Yau manifolds and mirror symmetry
  • Andrea Young: Gradient Kähler-Ricci Solitons

    Course

  • Instructor: Tamás Hausel, Office Hours: TTh 11-12, Office: RLM 11.168, Phone: 471-7169, Email: hausel@math.utexas.edu
  • Time: TTh 9.30am-11am, Room: RLM 10.176
  • Syllabus

    Description of course

    The course will give a rapid introduction to complex algebraic geometry from a differential geometry point of view. We quickly introduce complex and Kähler manifolds. The main theme of the course will be Hodge theory on Kähler manifolds. We will cover the so-called Kähler package. I.e. we will study harmonic forms and the Hodge theorem, the Hodge decomposition and the Hard Lefschetz theorem and discuss the Hodge conjecture. If time permits we may introduce Deligne's mixed Hodge structures on any algebraic variety or study deformation theory and mirror symmetry or characteristic p methods or some combination of these.

    Prerequisites

    The basic prerequisite is the two prelim courses in topology, in other words the notion of smooth manifolds, vector fields and differential forms on such and their cohomology. Familiarity with complex analysis in many variables is an advantage. If you are uncertain if this course is for you, you can send me an e-mail.

    Text books

    These books are available reserved in the library, and can be purchased in the University Coop. The Book we will follow in this course is:

  • Voisin: Hodge theory and complex algebraic geometry. I. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge, 2002. x+322 pp. ISBN 0-521-80260-1

    A more encyclopedic approach is in the classical:

  • Griffiths and Harris: Principles of Algebraic Geometry,Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. xii+813 pp. ISBN 0-471-32792-1

    For complex analysis:

  • Hörmander: An introduction to complex analysis in several variables. Third edition. North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, 1990. xii+254 pp. ISBN 0-444-88446-7

    For differentiable manifolds:

  • F. Warner: Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.

    Links

  • The Newlander-Nirenberg paper on the integrability of almost complex structures
  • Calabi's paper on the octanionic construction of a large number of almost complex structures on compact $6$-manifolds which are non-integrable.

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