Bayer:



This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

  <1> Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories. [1, 4, 5]
  <2> Wall-crossing and moduli spaces. I will continue the introduction of stability conditions, and proceed to explain the concept of wall-crossing, both in theory, and in examples. [0, 1, 3, 5]
  <3> Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction. [6]
<4,5> Applications: Moduli spaces of sheaves on K3 surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on K3 surfaces. [0, 7]

References:


[0] D. Arcara, A. Bertram, I. Coskun, J. Huizenga, "The minimal model program for the Hilbert scheme of points on P^2 and Bridgeland stability", Adv. Math., 235:580-626, 2013. (arXiv:math/1203.0316);
[1] T. Bridgeland, "Stability condition on triangulated categories", Annals of Math. 166 no. 2, 317-345 (2007) (arXiv:math/0212237);
[2] T. Bridgeland, "Spaces of stability conditions", Algebraic geometry-Seattle 2005. Part 1, 1-21, Proc. Sympos. Pure Math., 80, AMS. (arXiv:math/0611510);
[3] T. Bridgeland, "Stability conditions on K3 surfaces", Duke Math. J. 141, no. 2, 241-291 (2008) (arXiv:math/0307164);
[4] A. Caldararu, "Derived categories of sheaves: a skimming", In Snowbird lectures in algebraic geometry, volume 388 of Contemp. Math., pages 43-75. AMS (arXiv:math/0501094);
[5] A. Bayer, "A tour to stability conditions on derived categories", Informal notes available on my homepage;
[6] A. Bayer, E. Macri, "Projectivity and birational geometry of Bridgeland moduli spaces", (arXiv:1203.4613);
[7] A. Bayer, E. Macri, "MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations", (arXiv:1301.6968);