Toric Varieties (Spring 2024)
What is this course about?
Toric varieties provide a sort of elementary way to see many examples and phenomena in algebraic geometry, partly due to its close connections with simplicial geometry. As noted by Fulton: “toric varieties have provided a remarkably fertile testing ground for general theories”, the concreteness of toric varieties offers an excellent context for someone encountering the powerful techniques of modern algebraic geometry for the first time.
In this course, I will give a basic introduction to the properties of toric varieties, with many examples, and then concentrate on the cohomology of sheaves and intersection theory on toric varieties.
Prerequisite
Basic knowledge of algebraic varieties
Literature
[1] D. Cox, J. Little, H. Schenck, Toric varieties, Grad. Stud. Math. 124, American Mathematical Society, Providence, RI, 2011. xxiv+841 pp.
[2] W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, Princeton University Press, Princeton, NJ, 1993. xii+157 pp.
[3] T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Translated from the Japanese. Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin, 1988. viii+212 pp.
Place and time
Mondays, 13:30-15:05; Venue: A3-1a-204
Wednesdays, 09:50-11:25; Venue: A3-1a-205
ZOOM: 787 662 9899, PW: BIMSA
Starting from Mar. 4
WeChat Group
We have created a WeChat Group for communication. You could write to me if you would like to join.
Material covered
Mar. 4: Introduction with some examples.
Mar. 6: Introduction with some examples (continued).
Mar. 11: Convex polyhedral cones.
Mar. 13: Convex polyhedral cones (continued).
Mar. 18: Affine toric varieties.
Mar. 20: Affine toric varieties (continued).
Mar. 25: Fans and toric varieties.
Mar. 27: Fans and toric varieties (continued).
Apr. 1: One-parameter subgroups.
Apr. 3: Orbits.
Apr. 8: Orbits (continued).
Apr. 10: Toric varieties from polytopes.
Apr. 15: Smoothness and normality.
Apr. 17: Compactness and properness.
Apr. 22: Fundamental groups and Euler characteristics.
Apr. 24: Divisors.
Apr. 29: Line bundles.
May 1: Holiday, no class.
May 6: Line bundles (continued).
May 8: Line bundles (continued).
May 13: Line bundles (continued).
May 15: Cohomology of line bundles.
May 20: Moment maps.
May 22: Chow groups.
May 27: The cohomology ring.
PS
There is also a webpage at BIMSA dedicated to this course where you could find all the videos and notes for this course.