M392C
SYLLABUS

COURSE INFO

COURSE: COMPLEX GEOMETRY

UNIQUE NO: 59000

INSTRUCTOR INFO

INSTRUCTOR: TAMAS HAUSEL

PHONE: 471-7169

OFFICE: RLM 11.168

OFFICE HOURS: TTh 11-12; also by appt.

EMAIL: hausel@math.utexas.edu

COURSE WEB SITE http://www.math.utexas.edu/~hausel/complex/

GRADES

Grade (A) will be given for writing and reviewing a survey paper on a subject related to Complex geometry and chosen by the student. Every survey paper will be reviewed by the instructor and another student before it will get revised by the author. DEADLINES: First version of the survey paper is due by November 18. Comments by the reviewers are due by November 23rd. Final version of the survey papers are due by December 3.

BOOKS

During this course "Book" will stand for

  • 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 differentiable manifolds and Hodge theory on 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.

    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

    SYLLABUS

    We will roughly follow the following timeline (Chapter N means, Chapter N in the Book):

    WEEK 1: Holomorphic functions of many variables, Chapter I.1

    WEEK 2-3: Complex manifolds, Chapter I.2

    WEEK 4-5: Kähler manifolds, Chapter I.3

    WEEK 6-7: Sheaves and Cohomology, Chapter I.4

    WEEK 7-11: Harmonic forms and cohomology, Chapter II.1

    WEEK 12-13: The case of Kähler manifolds

    WEEK 14-15: One or more of the five applications: 1. Hodge conjecture 2. Isoperimetric inequalities 3. Mirror symmetry 4. Face vectors of simple polytopes 5. Relation to Arithmetic: Weil Conjectures

    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)