The course aims to give an introduction to cohomology and its applications to geometry. In particular we want to develop the basics of cohomology theory of sheaves, and we want to make the connection (de Rham, Dolbeault, Hodge) with singular cohomology as treated in Algebraic Topology.
Topics we plan to discuss (subject to changes):
1. basic notions from category theory
2. sheaves
3. derived functors; cohomology of sheaves
4. complex manifolds and some differential geometry
5. de Rham cohomology of complex manifolds
6. the Hodge theorem
7. applications and examples
Here is the final assignment:
Some exercises (more to be added during the semester):
Brief note on hermitian metrics:
There is no lecture on Oct. 2. Here are some suggestions for study:
. Note that most relevant books are all available in the library of the KdV Institute; on the ground floor, immediately to the left of the entrance, there are shelves with books reserved for study.
On October 23 we have an autumn break.
Some literature:
M.F. Atiyah, I.G. MacDonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.
R. Bott and L.W. Tu, Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982.
G.E. Bredon, Sheaf theory. McGraw-Hill Book Co., New York-Toronto, Ont.-London 1967 xi+272 pp. [Second edition: Graduate Texts in Mathematics, 170. Springer-Verlag, New York, 1997. xii+502 pp.]
H. Cartan and S. Eilenberg, Homological algebra. Princeton University Press, Princeton, N. J., 1956. xv+390 pp.
S.S. Chern, Complex manifolds without potential theory. Second edition. Universitext. Springer-Verlag, New York-Heidelberg, 1979.
R. Godement, Topologie algébrique et théorie des faisceaux. Actualites Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13 Hermann, Paris 1958 viii+283 pp.
Ph. Griffiths and J. Harris, Principles of Algebraic Geometry Pure and Applied Mathematics. Wiley-Interscience, New York, 1978.
A. Grothendieck, Sur quelques points d'algèbre homologique. Tôhoku Math. J. (2) 9 1957 119--221.
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer. Cited as HAG.
P. Hilton and U. Stammbach, A course in homological algebra. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1971.
B. Iversen, Cohomology of sheaves. Universitext. Springer-Verlag, Berlin, 1986. xii+464 pp.
M. Kashiwara and P. Schapira, Categories and sheaves. Grundlehren der Mathematischen Wissenschaften 332. Springer-Verlag, Berlin, 2006. x+497 pp.
S. Lang, Algebra. Second edition: Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. Revised third edition: Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 2002.
S. MacLane, Categories for the working mathematician. Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York-Berlin, 1971.
F. Oort, Categorieën. Syllabus, Math. Instituut UvA, 1966.
J. Strooker, Introduction to categories, homological algebra and sheaf cohomology. Cambridge University Press, Cambridge-New York-Melbourne, 1978. ix+246 pp.
R.O. Wells, Differential analysis on complex manifolds. Prentice-Hall Series in Modern Analysis. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. x+252 pp. [2nd and 3rd ed: Graduate Texts in Mathematics, 65. Springer]
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