The seminar in the winter term 2012/13
In the winter term 2012/13, the topic of the seminar is Singularity Categories which takes place at the Leibniz Universität Hannover. The aim of this seminar is to understand (graded) singularity categories in interesting special cases. Here is a detailed overview:
1st session on November 3, 2012
- 10:00 – 11:15
Wolfgang Ebeling: Singularities - 11:30 – 12:45
David Ploog: Singularity categories - 14:15 – 15:30
Alexey Basalaev: Simple (ADE) singularities - 15:45 – 17:00
Nathan Broomhead: Ueda's Proof for HMFgr(ADE)
2nd session on December 1, 2012
- 10:00 – 11:15
David Ploog: Knörrer periodicity - 11:30 – 12:45
Mark Blume: Explicit matrix calculation for HMFgr(ADE) - 14:15 – 15:30
Greg Stevenson: Analytical invariance of singularity categories - 15:45 – 17:00
Andreas Hochenegger: Hyperplane sections and singularity categories
3rd session on January 12, 2012
- 10:00 – 11:15
Zain Shaikh: Fuchsian singularities - 11:30 – 12:45
Andreas Hochenegger: Exceptional unimodal singularities I - 14:15 – 15:30
David Ploog: Exceptional unimodal singularities II - 15:45 – 17:00
Martin Kalck: Cluster categories and MCM(ADE)
Description of the topics
The time slot of a talk will be 75 minutes but deviations in both directions are possible. The idea behind this 75 minutes is that an average talk lasts about 60 – 70 minutes, therefore leaving some time for discussions.
- Singularities
Examples, some notions (isolated singularity, hypersurface, complete intersection, Gorenstein), Milnor fibre and Milnor lattice, modality, Arnold's classification scheme. (Simple singularities should be mentioned, but will be treated in more detail in another talk.) - Singularity categories
They come in various guises: D/Dhf for abstract triangulated categories [O1]; Dsg(X) for schemes [O0]; stable module categories (Buchweitz' paper from last session, but Orlov also gives the comparison); stable matrix factorisations (Eisenbud, but Orlov compares as well, "Landau-Ginzburg models"). Properties: always triangulated; Hom finite is the singular locus is compact; have Serre duality. Mention Orlov's theorem for the graded case [O2]. (Perhaps give quick proof.) - Simple (ADE) singularities
An expository talk to make simple singularities as concrete as possible. Mention some equivalent notions: zero modality; rational double point (say what a rational singularity is). See [Du] for a long list (of this, A1-A5' and B1 might be interesting for our audience.) Resolution is the same as smoothing. May mention McKay correspondence. - Ueda's proof for HMFgr(ADE)
The appendix of [KSTU] is independent of the rest of the article and very short. Geigle-Lenzing weighted projective lines are used, which should be explained (they are more familiar in representation theory than in algebraic geometry). Ueda goes further in [U1] which could also be covered. - Knörrer periodicity
An aside on a classical, important fact about stable categories. Yoshino [Y, Section 12] treats this using MCM modules. Orlov [O0] gives a more geometrically flavoured proof. - Explicit matrix calculation for HMFgr(ADE)
Just treat one or two cases from [KSTU] in detail - with explicit computation of the matrices. - Analytical invariance of singularity categories
An aside: Orlov observes in [O3] that Dsg(X) is (i) in general not idempotent complete and (ii) not an invariant of the analytical type of singularity, and that these two defects are related. - Hyperplane sections and singularity categories
Another aside in spirit of the Lefschetz hyperplane section theorem, by Ueda [U2]. - Fuchsian singularities
Fuchsian groups acting on the hyperbolic plane, triangle singularities, the 14 exceptional unimodal singularities of Arnold's strange duality as special cases. The setup of [Do] is a modern way to get there, [M] a classical one. The first two pages of [L] may be helpful as well (Looijenga seems to have coined the term "Fuchsian singularity", by the way.) - Exceptional unimodal singularities I [KST2]
- Exceptional unimodal singularities II [KMU]
- Cluster categories and MCM(ADE)
The most algebraic talk in this series. Amiot et al describe in [AIR] the (ungraded!) stable category for the simple singularities, using cluster categories.
Literature
[AIR] Stable Categories of Cohen-Macaulay Modules and Cluster Categories by C. Amiot, O. Iyama and I. Reiten.[BH] A Generalized Construction of Mirror Manifolds by P. Berglund and T. Hübsch.
[Do] McKay's Correspondence for Cocompact Discrete Subgroups of SU(1,1) by I. Dolgachev.
[Du] Fifteen Characterisations of Rational Double Points and Simple Critical Points by A. Durfee.
[KMU] A Note on Exceptional Unimodal Singularities and K3 Surfaces by M. Kobayashi, M. Mase and K. Ueda.
[KST2] Triangulated Categories of Matrix Factorizations for Regular Systems of Weights with ε=-1 by H. Kajiura, K. Saito and A. Takahashi.
[KSTU] Matrix Factorizations and Representations of Quivers II: type ADE case by H. Kajiura, K. Saito and A. Takahashi; with an appendix by K. Ueda.
[L] The Smoothing Components of a Triangle Singularity II by E. Looijenga.
[M] On the Three-Dimensional Brieskorn Manifolds M(p,q,r) by J. Milnor.
[O0] Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models by D. Orlov. (No gradings. Knörrer periodicity in Section 2.)
[O1] Triangulated Categories of Singularities and Equivalences Between Landau-Ginzburg Models by D. Orlov. (This contains "the" theorem of Orlov on graded singularity categories. Also starts with an extremely general definition of Dsg.)
[O2] Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities by D. Orlov. (Relation between graded singularities for homogenous W=0 and graded matrix factorisations of W.)
[O3] Formal Completions and Idempotent Completions of Triangulated Categories of Singularities by D. Orlov. (Analytically/formally equivalent singularities, e.g. two double points, can have the same singularity category. The idempotent closure of Dsg is an invariant of the analytic equivalence class.)
[U1] On Graded Stable Derived Categories of Isolated Gorenstein Quotient Singularities by K. Ueda.
[U2] Hyperplane Sections and Stable Derived Categories by K. Ueda.
[Y] Cohen-Macaulay Modules over Cohen-Macaulay Rings by Y. Yoshino.