Abstract: In these talks I will introduce various notions of positivity (e.g. ampleness) for vector bundles on projective varieties and explain through examples how the birational geometry of the variety is governed by positivity of the cotangent bundle.
Abstract: In this series of lectures we will cover the following topics:
(1) Review on stability of vector bundles on curves.
(2) Stability conditions on derived categories.
(3) Moduli spaces of stable objects and variation of stability.
(4) The case of surfaces.
(5) The higher dimensional case and open problems.
Abstract: Holomorphic curves have many fascinating applications to algebraic geometry, symplectic geometry, and string theory from physics, but they can be difficult to find and visualise. I will explain how in many situations, we can study holomorphic curves by studying piecewise linear graphs called tropical curves.
Abstract: The course is centered on the Chow group of algebraic cycles on an algebraic variety. Some introductions to this material are Hartshorne's section II.6 on Divisors and Appendix A on Intersection Theory. We will mostly use what we need from Fulton's Intersection Theory without proof.
One main theme of the course is the notion of "decomposition of the diagonal", which describes the geometric consequences that hold for a variety with "small" Chow groups. The basic argument by Bloch and Srinivas is very simple, once you know the formal properties of Chow groups.
The course will end with applications of these ideas to birational geometry. In particular, we will use Chow groups to prove a striking recent result: many Fano hypersurfaces X in projective space are not stably rational. (That is, no product of X with projective space is birational to projective space.)
|Monday||August 15||Hanna Neumann Seminar Room (G058|
|9-9:30||Welcome and registration||in lobby of John Dedman|
|Tuesday||August 16||Hanna Neumann Seminar Room (G058)|
|Wednesday||August 17||Hanna Neumann Seminar Room (G058)|
|Thursday||August 18||Hanna Neumann Seminar Room (G058)|
|Friday||August 19||John Dedman Building, room G35|