Course in Algebraic Cobordism

A course in algebraic cobordism

Course description

The goal of this course is to give an introduction to the theory of oriented cohomology, a construction of the universal such theory, algebraic cobordism, and a description of its basic properties. If time permits, we will also present a number of applications, such as degree formulas and relations to Donaldson-Thomas theory, as well as a discussion of variants of the theory, such as the algebraic cobordism of varieties with vector bundle, due to Lee-Pandharipande. The original definition of algebraic cobordism is a bit complicated, so we will try to give a new approach to the theory, developing algebraic cobordism directly from the presentation given by double point cobordisms (we'll see how successful this attempt will be!).

Basic knowledge of algebraic geometry will be assumed, although very little from sheaf theory or cohomology of sheaves will be needed. We will make extensive use of results from resolution of singularities. Also, many ideas will be taken from Fulton's Intersection Theory.

Schedule

The course will meet Mondays and Tuesdays, 12-14h (c.t.) in 4N (4th floor, North side seminar room). First meeting: Tues., Oct. 16.

References

M. Levine, F. Morel, Algebraic cobordism. Monographs in Mathematics, 246 pp., Springer, Berlin 2007.

M. Levine, R. Pandharipande, Algebraic cobordism revisited Invent. Math. 176 (2009), no. 1, 63--130.

Y.-P. Lee, R. Pandharipande, Algebraic cobordism of bundles on varieties. Preprint 2010, arXiv:1002.1500

Y.-J. Tzeng, Universal formulas for counting nodal curves on surfaces. Current developments in mathematics, 2010, 95--115, Int. Press, Somerville, MA, 2011.

W. Fulton, Intersection theory. Second edition. Springer-Verlag, Berlin, 1998.