Course Description:
This reading course is ideal for students in Algebraic Geometry II or those studying algebraic geometry independently. The course primarily covers two topics:
I. Derived Categories. Introduced by Grothendieck and Verdier, these are crucial invariants of algebraic varieties.
II. Geometric Invariant Theory (GIT). This theory is essential for studying quotients in algebraic geometry and constructing moduli spaces.
Additionally, it studies the link between these two topics:
III. Derived Categories of GIT quotients. Focusing on Halpern-Leistner and Ballard, Favero, & Katzarkov’s categorification of Kirwan surjectivity theorem.
The course will operate in a seminar format, with each participant contributing one or two sessions, each spanning 2 to 3 hours.
Meeting Time and Venue: Weekly on Tuesdays, 15:00 -17:50 at Room 5510 (Lift 25-26).
Course Outline: Here (updated constantly).
Course Schedule:
Lecture 1 (Feb 4): Overview. (Lecture by organizer, Qingyuan Jiang)
Part I: Derived Categories
Lecture 2 (Feb 11): Foundations of Derived Categories and Fourier-Mukai Transform. (Lecture by Shuai Zong)
Lecture 3 (Feb 18): Semiorthogonal Decompositions and Exceptional Collections. (Lecture by Shuai Zong)
Lecture 4 (Feb 25): Quotient Stacks. Derived Categories and Birational Geometry. (Lecture by Shuai Zong)
Part II: Geometric Invariant Theory (GIT)
Lecture 5 (Mar 4): Introduction to GIT. (Lecture by Linfang Jiang)
Lecture 6 (Mar 11): Mumford’s (Quasi-)Projective GIT. (Lecture by Huey Tai)
Lecture 7 (Mar 18): Hilbert–Mumford Criteria and Kemf–Ness Stratifications. (Lecture by Yang Hu)
Lecture 8 (Mar 25): Applications of GIT: Vector Bundles on Curves (Lecture by Tairun Chen)
Part III: Derived Categories of GIT Quotients
Lecture 9 (Apr 1): Homological Structures on Unstable Strata and Quantization Theorem. (Lecture by Wenwei Liu)
Lecture 10 (Apr 8): Semiorthogonal Decompositions and Derived Kirwan Subjectivity. (Lecture by Lei You)
Lecture 11 (Apr 29): Variant of GIT Quotients and Derived Categories. (Lecture by Tairun Chen)
Lecture 12 (May 6): Applications: Moduli Spaces of Vector Bundles on Curves. (Lecture by organizer, Qingyuan Jiang)
Main references:
[HL] Halpern-Leistner, Daniel. The derived category of a GIT quotient. Journal of the American Mathematical Society 28, no. 3 (2015): 871-912.
[BFK] Ballard, Matthew, David Favero, and Ludmil Katzarkov. Variation of geometric invariant theory quotients and derived categories. Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 746 (2019): 235-303.
[Seg] Segal, Ed. Equivalences between GIT quotients of Landau-Ginzburg B-models. Communications in mathematical physics 304 (2011): 411-432.
References on Derived Categories:
[Cal] Caldararu, Andrei. Derived categories of sheaves: a skimming. arXiv: math/0501094 (2005).
[Huy] Huybrechts, Daniel. Fourier-Mukai transforms in algebraic geometry. Clarendon Press, 2006.
[Bei] Beilinson, A. Coherent sheaves on 𝑃𝑛 and problems of linear algebra. Funktsional’nyi Analiz i ego Prilozheniya 12, no. 3 (1978): 68-69.
[BO]Bondal, Alexei, and Dmitri Orlov. Semiorthogonal decomposition for algebraic varieties. arXiv preprint alg-geom/9506012 (1995).
[Kuz] Kuznetsov, Alexander. Semiorthogonal decompositions in algebraic geometry. ICCMtalk.arXiv:1404.3143 (2014).
[Kap] Kapranov, M. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math., 92 (1988), 479-508.
References on GIT:
[Dol] Dolgachev, I. Lectures on Invariant Theory. London Mathematical Society Lecture Note Series/Cambridge University Press 296 (2003).
[Hos] Hoskins, Victoria. Moduli spaces and geometric invariant theory. Lecture notes. (2016).
[New] Newstead, Peter. Geometric invariant theory. 3rdcycle.Guanajuato(Mexique),2006,pp.17.cel-00392098.
[MFK] Mumford, David, John Fogarty, and Frances Kirwan. Geometric invariant theory. Vol. 34. Springer Science & Business Media, 1994.
Additional references on GIT
[Ber] Bertram, A. Notes on GIT. Math 7800-Spring 2022.
[Muk] Mukai, Shigeru. An introduction to invariants and moduli. Vol. 81. Cambridge University Press, 2003.
[Kir] Kirwan, Frances. Cohomology of quotients in symplectic and algebraic geometry. Vol.31. Princeton University Press, 1984.
[DH] Dolgachev, Igor V., and Yi Hu. Variation of geometric invariant theory quotients. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 87 (1998): 5-51.
[Tha] Thaddeus, Michael. Geometric invariant theory and flips. Journal of the American Mathematical Society 9,
no. 3 (1996): 691-723.
[Tho] Thomas, Richard. Notes on GIT and symplectic reduction for bundles and varieties. arXiv:0512411 (2005).