The GiC (IPA: /ɡiːk/) seminar is an algebraic geometry seminar for those in and around Cardiff area and, more generally, in the southwest. All are welcome. The seminar meets on Wednesday afternoons 16:00 - 18:00 in M/2.06. The subjects of the talks reflect the interests of its current organisers, ranging from the classical algebraic geometry to all things derived, categorical and homotopical. The talks are sometimes followed by a social dinner for the participants.
Please address any questions or queries to one of the seminar organisers:
Date: | Wed 22nd May, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Timothy Logvinenko (Cardiff) |
“P^n-functors” | |
I will explain the new notion of (non-split) P^n-functors introduced recently in the eponymous preprint (arXiv:1905.05740). I will explain the background, define a P^n-structure on an enhanced functor, construct the induced P-twist, and give an outline of the proof that this P-twist is an autoequivalence of the target category. Then I will explain the strong monad condition. Finally, I will give some examples of non-split P^n-functors. |
No further meetings have been scheduled at this time.
Date: | Wed 15th May, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Chris Seaman (Cardiff) |
“Perverse schobers” | |
The group of autoequivalences of a given derived category of coherent sheaves on a variety X, D^b(X), has been the subject of much study. In this talk I will present some well-known results about Aut(D^b(X)), as well as more recent technology in the form of perverse schobers. These should be thought of as categorifications of perverse sheaves, a way of thinking which can be made more precise through results of Bondal, Kapranov and Schechtman. |
Date: | Wed 8th May, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Daria Poliakova (Copenhagen) |
“Homotopy limits in DG categories and applications to derived algebraic geometry” | |
I will explain a construction of Reedy-fibrant replacements for simplicial DG-categories with respect to Dwyer-Kan model structure. This allows for explicit formulas for homotopy limits (due to Block-Holstein-Wei and Arkhipov-Ørsted). As an application, I will demonstrate a computation of homotopy limit of a cosimplicial DG-algebra associated to a commutative Hopf DG-algebra and compare it to the classical cobar construction. |
Date: | Wed 10th Apr, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Lorenzo De Biase (Cardiff) |
“Generalised braid actions” | |
In this talk, after giving some background on autoequivalences of derived categories of smooth projective varieties, I will define the generalised braid category and describe its action on the derived categories of (the cotangent bundles of) full and partial flag varieties. Generalised braids are the braids whose strands are allowed to touch in a certain way. The basic building blocks of their action on flag varieties are spherical and non-split P- functors together with the twist equivalences they induce. I will describe our present progress and future expectations. This is a joint project with Rina Anno and Timothy Logvinenko. |
Date: | Wed 27th Mar, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Lorenzo De Biase (Cardiff) |
“A quick tour of deformation theory II” | |
Following the discussion of last'week, we will go through the definition of complex space and through the completeness, versality and universality properties of deformations. |
Date: | Wed 20th Mar, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Lorenzo De Biase (Cardiff) |
“A quick tour of deformation theory” | |
After recalling the basic definitions of deformation families, we will discuss the Kodaira-Spencer criterion for completeness and the Kuranishi Theorem on the existence of a versal deformation for every compact complex manifold. |
Date: | Wed 13th Mar, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Lorenzo De Biase (Cardiff) |
“Bondal-Orlov criteria for derived equivalences V” | |
We will give a brief introduction to deformation families and Kodaira-Spencer maps. |
Date: | Wed 6th Mar, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Chris Seaman (Cardiff) |
“Bondal-Orlov criteria for derived equivalences IV” | |
We shall look at a lemma by Bridgeland which states that if we have a flat map S → X and an object E ∈ D(S) such that for every closed point x ∈ X the object ι_x^* E ∈ D(S_x) is a sheaf, then E itself is a sheaf on S flat over X. |
Date: | Wed 27th Feb, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Chris Seaman (Cardiff) |
“Bondal-Orlov criteria for derived equivalences III” | |
Following on from the overview given two weeks ago, we shall present the spectral sequence argument which ensures that the Fourier-Mukai kernel of LF ∈ D(X × X) is a sheaf supported on Δ ⊂ X × X and flat over the projections to X. |
Date: | Wed 20th Feb, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Chris Seaman (Cardiff) |
“Bondal-Orlov criteria for derived equivalences II” | |
Following on from the overview given two weeks ago, we shall begin the process of fleshing out the full details of the proof of the Bondal-Orlov criterion for a Fourier-Mukai transform to be fully faithful. |
Date: | Fri 15th Feb, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
13:10-14:10 | Enrico Fatighenti (Loughborough) |
“On technical aspects of constructing Fano varieties of K3 type and IHS manifolds” | |
Subvarieties of Grassmannians (and especially Fano varieties) obtained from section of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type, for their deep links with hyperk\"ahler geometry. This talk will be mainly devoted to the construction of some new examples of such varieties. This is a work in progress with Giovanni Mongardi. |
Date: | Wed 6th Feb, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:10-18:00 | Timothy Logvinenko (Cardiff) |
“Bondal-Orlov criteria for derived equivalences I” | |
I will give a bird's eye overview of the well-known criterion by Bondal and Orlov for the Fourier-Mukai transform defined by a kernel F ∈ D(X × Y) to be fully faithful. We will look at the kernel composition LF ∈ D(X × X) and the successive steps through which it is established to be the structure sheaf of the diagonal Δ ⊂ X × X. |
Date: | Wed 30th Jan, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:10-18:00 | Lorenzo de Biase (Cardiff) |
“Derived Equivalence vs Birationality III” | |
We will continue the proof of Kawamata's result on the implication of K-equivalence from D-equivalence under the hypothesis of maximality of the (anti-canonical) Kodaira dimension. |
Date: | Wed 23rd January, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:10-18:00 | Christopher Seaman (Cardiff) |
“ Derived Equivalence vs Birationality II” | |
Carrying on from our digression last week, we shall remind ourselves of the statement of Kawamata's result and continue the proof. Time permitting, we shall address some corollaries of the result. |
Date: | Wed 16th January, 2019 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:10-18:00 | Christopher Seaman (Cardiff) |
“Derived Equivalence vs Birationality I” | |
We shall state and prove the result by Kawamata that if two smooth projective varieties are derived equivalent and the Kodaira dimension of the (anti-)canonical divisor is maximal, then the varieties are birational. |
Date: | Wed 12th December, 2018 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:10-18:00 | Timothy Logvinenko (Cardiff) |
“Nefness and birationality under derived equivalence” | |
We will state and prove the famous results of Kawamata on the preservation of the nefness of the canonical bundle and the preservation of the birational class under derived equivalences. |
Date: | Wed 28th November, 2018 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Lorenzo de Biase (Cardiff) |
“Geometrical aspects of the Fourier-Mukai kernel” | |
What happens if a Fourier-Mukai kernel $K \in D^b(XxY)$ has support not the whole product XxY but just a subvariety Z? In today's seminar we will prove some results about Supp(K): beyond their intrinsic importance, they will be the technical background for proving Kawamata's theorems on the preservation of the nefness of the canonical bundle and of the numerical Kodaira dimension under derived equivalences. |
Date: | Wed 21th November, 2018 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Christopher Seaman (Cardiff) |
“Hochschild (co)homology and geometrical aspects of the Fourier-Mukai kernel” | |
After reminding ourselves of the definitions, we will study isomorphisms of Hochschild (co)homologies induced by equivalences of derived categories. Time permitting, we will explore some results on the support of the Fourier-Mukai kernel of an equivalence. |
Date: | Wed 31st October, 2018 |
Place: | Room M/2.06 @ School of Mathematics, Cardiff University |
16:00-18:00 | Lorenzo de Biase (Cardiff) |
“Kodaira dimension under derived equivalence” | |
We will prove Orlov's result on preservation of Kodaira dimension and canonical ring under derived equivalences for smooth projective varieties. Then, we will extend the result to Hochschild cohomologies and we will recover the Hochschild-Kostant-Rosemberg isomorphism. |