EPSRC Warwick Symposium 2014-15:
Derived Categories and Applications

Sep 2014 - Aug 2015, University of Warwick

Main Venue Program Participants Registration
Workshops Concentration Seminars

  1. Symposium Seminar Series
  2. Seminars of the “Moduli Spaces and Derived Categories” concentration period
  3. Seminars of the “Geometry from stability conditions” concentration period
  4. Symposium minicourse “Introduction to spherical and P^n twists”
  5. Symposium minicourse “Geometric representation theory and categorical braid group actions”

Symposium Seminar Series

The Symposium Seminar meets on most Monday afternoons throughout the Symposium. The subjects of the talks cover all things derived and their applications to classical algebraic geometry. The talks are sometimes followed by a social dinner for the participants.

Next Meeting:

Future Meetings:

Past Meetings:

Symposium mini-course “Introduction to spherical and P-twists”

Timothy Logvinenko (Cardiff) and Ciaran Meachan (Edinburgh)
Tuesdays starting 3rd Feb, 16:00 - 18:00, B3.01, Warwick Math Institute

The notion of a spherical twist originated as a mirror-symmetric analogue of a Dehn twist around a Lagrangian sphere on a symplectic manifold. The corresponding notion is an autoequivalence of the derived category D(X) of an algebraic variety X constructed out of a "spherical" object. These were the first examples of genuinely derived autoequivalences and they quickly became indispensible in the study of Aut D(X) and its links with birational geometry of X. It was also observed that spherical twists can give rise to natural categorical group actions on D(X) and thus they became a fundamental tool in geometrical representation theory. A parallel minicourse by Sergey Arkhipov will explore this connection further.

In this introductory lecture course, aimed at graduate students and early postdocs, we will present a cohesive and thorough account of our current understanding of this theory. We will start with the symplectic foundations and connections to mirror symmetry, define the original Seidel-Thomas objects, present their various generalisations such as spherical functors, P-objects and P-functors, and then proceed to some current work-in-progress. Each lecture will be followed by an examples class where we will work together with the students through many of the explicit examples and applications which appear in the literature.

Lecture plan:

L1 (Tue 3rd Feb, 5pm start): Generalised Dehn twists and Homological Mirror Symmetry

In this lecture, we will unpack the physics terminology and try to explain what mathematicians should think of when they hear a physicist say "the category of branes in a topological twist of the sigma model" as well as other exotic phrases. We will also try to develop some intuition for what the Homological Mirror Symmetry conjecture actually says and then loosely describe the concept of generalised Dehn twists and their link with monodromy maps in the “stringy Kahler moduli space”.

Lecture Notes #1 (.PDF, 10 Mb)

Exercise Sheet #1 (.PDF, 0.169 Mb)

L2 (Tue 10th Feb, 5pm start): Spherical objects

In this lecture, we introduce a notion of a spherical object in the derived category D(X) of a smooth projective variety. We illustrate this notion with a number of geometrical examples on elliptic curves, K3 surfaces and CY3 threefolds. We then define the notion of a twist of D(X) around an arbitrary object E in it. We conclude by giving a sketch of the proof that the twist around a spherical object is an autoequivalence of D(X).

Lecture Notes #2 (.PDF,6 Mb)

L3 (Tue 17th Feb, 5pm start): Examples of spherical twists

In this lecture we compute explicitly several examples of well-known spherical twists. We first show that on an elliptic curve the twist around a point sheaf is simply the functor of tensoring by the corresponding line bundle, while the twist around the structure sheaf is a non-trivial derived autoequivalence which can be identified with the original Fourier-Mukai transform.

We then consider two smooth rational curves intersecting in precisely one point on a K3 surface and show that the corresponding spherical twists satisfy braid relations. This is an example of a more general phenomena

Lecture Notes #3 (.PDF,5.454 Mb)

L4 (Tue 24th Feb): P-objects

In this lecture, we will introduce the notion of a P-object and explain why it gives rise to an autoequivalence of the derived category. We will then analyse the relationship between P-objects and spherical objects when we have a one parameter deformation of the underlying variety. Throughout the talk, I will try to highlight the key steps which will need to be generalised when we come to talk about P-functors.

Lecture Notes #4 (.PDF,8.103 Mb)

L5 (Tue 3rd Mar): Spherical functors
L6 (Tue 17th Mar): P-functors
L7 (Tue 31st Mar): Spherical pairs, perverse shobers and further generalisations

Symposium mini-course “Geometric representation theory and categorical braid group actions”

Sergey Arkhipov(Aarhus)
Fridays starting 6th Feb, 10:00 - 11:00, D1.07, Warwick Math Institute

Broadly speaking, geometric representation theory is a framework in which symmetries of geometric objects act on invariants of these objects such as cohomology theories and, more generally, derived categories associated to them. Often the representation theoretic results obtained in this way are substantial and beyond the reach of purely algebraic methods.

More specifically, in an algebro-geometric setting we can consider an algebraic group G with a subgroup H. Тhe geometry of the space H\G/H produces a number of interesting algebras and their representations, both classical and categorical. These, in turn, give rise naturally to braid group actions. In this introductory lecture course I will present several examples of this, both well-known and work-in-progress.

Lecture plan:

L1 (Fri 6rd Feb): Springer correspondence

It is well known that both conjugacy classes of nilpotent matrices of size n and irreducible representations of the symmetric group in n letters are enumerated by Young diagrams of size n. Springer correspondence both explains and generalises this coincidence. Let G be a reductive algebraic group. We first recall the standard geometric objects related to G: the flag variety Fl_G, the Springer variety T^* Fl_G, the Springer desingularization of the nilpotent cone in the Lie algebra of G, and the Steinberg variety for G. Then we outline Kazhdan-Lusztig-Ginzburg construction of the group algebra for the Weyl group of G via Borel-Moore homology of the Steinberg variety. We conclude the lecture by classification of the irreducible representations of the Weyl group via Borel-Moore homology of Springer fibers.

Lecture Notes #1 (.PDF, 0.6 Mb)

Exercise Sheet #1 (.PDF, 2.021 Mb)

L2 (Fri 13th Feb): Kazhdan-Lusztig-Ginzburg construction and the finite Hecke category

We recall the classical theory of algebraic D-modules on Fl_G for a reductive algebraic group G due to Kashiwara, Beilinson and Bernstein. Its culmination is the famous Beilinson-Bernstein localization theorem for representations of the Lie algebra of G. The derived category of B-equivariant D-modules on Fl_G equipped with a convolution becomes a monoidal category whose Grotehndieck group equals the group algebra of the Weyl group. We call this the finite Hecke category. Characteristic cycle of a holonomic D-module on a variety X is a Lagrangian cycle in T^* X. This relates the finite Hecke category to the Kazhdan-Lusztig-Ginzburg approach to Springer correspondence.

L3 (Fri 27th Feb): Bezrukavnikov-Riche construction

We begin by reviewing the Springer variety T^* Fl_G, the Grothendieck variety and the Steinberg variety for a reductive group G. It is discussed in lecture 2 that Steinberg variety is a natural home for the characteristic cycles of D-modules in the finite Hecke category. A version of characteristic cycle construction of a D-module gives rise to an equivariant coherent sheaf on the Steinberg variety. We then take the category of equivariant coherent sheaves on the Steinberg variety and define a convolution monoidal structure on it. The result is the affine Hecke category. Following Bezrukavnikov-Riche we outline the construction of canonical objects in the affine Hecke category whose convolution satisfies the relations in the affine braid group. We conclude the lecture by constructing various categorical braid group actions from monoidal actions of the affine Hecke category on certain categories of algebro-geometric nature.

L4 (Mon 2nd Mar, Room D1.07, 4pm): Monoidal action of the affine Hecke category on equivariant matrix factorizations

Let G be a reductive algebraic group with the standard Borel subgroup B and let X be a G-variety. Examples show that while the affine braid group acts naturally on the B-equivariant K-group of X, the affine Hecke category does not act in general on Coh^B(X). Nor does it act on the category of Coh^B(T^*X). Instead it acts on a certain category of G-equivariant matrix factorizations on the product of T^* X with the Grothendieck variety of G.

In the lecture, we introduce the derived category of equivariant matrix factorizations on a G-variety Z with a global G-invariant function called a potential. We recall the formalism of inverse and direct images for matrix factorizations. We define a natural monoidal action of the affine Hecke category on certain derived categories of equivariant matrix factorizations, with the potential given by the moment map. It follows that the latter category possesses a categorical action of the affine braid group.

@ Warwick Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Funded by the Engineering and Physical Sciences Research Council (EPSRC)