Overview
We plan to investigate the relationship between the geometry of a projective variety and its derived category of coherent sheaves. We intend to prove some of the landmarks along the way while gently introducing a geometrically motivated approach to derived categories of coherent sheaves. In the second half, we will see applications such as the Derived and Twisted Derived Global Torelli theorem for K3 surfaces and interesting connections between derived categories and moduli spaces of sheaves. The main reference will be the classic "Fourier-Mukai Transforms in Algebraic Geometry" by Huybrechts.This reading seminar will be held during the Summer Semester 2025 at the Institute of Mathematics of the University of Bonn, and is co-organised with Samir Geiger.
Schedule
Following talks have been programmed.
- Talk 1: Sprint through derived categories. Give a short overview of the basics concerning the derived category of coherent sheaves on a projective variety, as well as some of the important derived functors. Talk about derived functors in algebraic geometry (Ch. 3.3), state Serre-Duality (in the language of Serre functors), and Grothendieck-Verdier duality. (Dave Bowman)
- Talk 2: Why Calabi-Yau's are a problem? We will see why Calabi-Yau varieties play a special role in the study of the interactions between the geometry of varieties and and their derived category of coherent sheaves. Explain the ideas behind Proposition 4.11.
- Talk 3: Fourier-Mukai transforms. The main result of this talk is the Bondal-Orlov Theorem (Proposition 4.11), which states that, under some conditions, from an equivalence of the derived categories of coherent sheaves of two varieties, we can conclude an isomorphism of the varieties.
- Talk 4: Derived equivalence vs birationality. State the D-equivalence conjecture and explain the proof of Proposition 6.19. Illustrate the intimate relationship between birationality and D-equivalence. Try to focus on the ideas and provide as much detail as needed within the given timeframe.
- Talk 5: Equivalence criteria. The idea of this talk is to prove criteria to conclude the equivalence between two derived categories of coherent sheaves, i.e., Proposition 7.6 and Proposition 7.11. Prove also Proposition 7.1, which gives us criteria to use Proposition 7.6. (Javier Fernández)
- Talk 6: Descending to cohomology. In this talk, we will see how Fourier-Mukai functors give maps on cohomology. Blackbox the Grothendieck-Riemann-Roch formula and show Corollary 5.29. Show that derived equivalences give rise to Hodge isometries on the Mukai lattices. The cornerstones for the proof are Proposition 5.33, Proposition 5.39, and finally Proposition 5.44.
- Talk 7: D-equivalence for abelian varieties. Define the Albanese variety, the dual Abelian variety, and the Poincar\'e bundle. The main goal of this talk is to prove the result by Mukai (Proposition 9.19) about the equivalence between the derived categories of sheaves of an Abelian variety and its dual. (Fabian Schnelle)
- Talk 8: Derived Global Torelli theorem for K3 surfaces. We will see a derived version of the celebrated Global Torelli theorem for K3 surfaces. State and explain the classical Global Torelli (Theorem 10.4). Prove 10.7 and the Derived Global Torelli (Theorem 10.10). Prove 10.25 using the characterisation of Fourier-Mukai functors being equivalences. (Brais Gerpe Vilas)
- Talk 9: Mukai reflections. This talk is based on the article "Some examples of Mukai reflections on K3s" by Yoshioka. The main result of this talk is to prove that, under some conditions, moduli spaces of stable sheaves are deformation equivalent to Hilbert schemes of points by means of reflections in the Mukai lattice, which are particular examples of Fourier-Mukai transforms. (Alejandro Ovalle)
- Talk 10: Twisted derived equivalences and Twisted Derived Global Torelli. Consider how to improve result 10.25 in the case of non-fine moduli spaces of sheaves. Introduce twisted sheaves and twisted universal families. Prove the Twisted Derived Global Torelli Theorem from the acticle "Equivalences of twisted K3s" by Huybrechts and Stellari. (Samir Geiger)
Contact
If you have any questions or would like to participate, please contact me at alejandro.ovalle@uni-bonn.de or contact Samir at s42sgeig@uni-bonn.de.