Schedule for: 22w5108 - Derived Categories, Arithmetic, and Reconstruction in Algebraic Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will discuss various general types of reconstruction or characterization results, ranging from basic algebraic geometry to new results obtained in joint work with Kollár, Olsson, and Sawin, and with Alper and de Jong. In each case, one attempts to extract a complete algebraic invariant -- for example, a ring, a group, a Hodge structure, an abelian category, a tensor triangulated category, a motive, a topos, an abstract projective structure, a poset -- from a geometric object. Each type of invariant has successes and failures.

Let X be a smooth variety over a number field k. I will review the general heuristic underlying our expectation that the set of k-rational points in the degeneracy locus of a p-adic local system whose geometric monodromy is semisimple (perfect?) are not Zariski-dense and recall some of the motivations for this question. This heuristic relies on a geometric conjecture and (a weak form of) the Bombieri-Lang conjecture. In the second part of the talk, I will give an hint of the proofs of the geometric conjecture when X is a curve (joint with A. Tamagawa) and a product of 2 curves.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Let X and Y be compact hyper-Kahler manifolds deformation equivalence to the Hilbert scheme of length n subschemes of a K3 surface. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subalgebra generated by Chern classes of coherent analytic sheaves. Let f be a Hodge isometry of the second rational cohomologies of X and Y with respect to the Beauville-Bogomolov-Fujiki pairings. We prove that f is induced by an analytic correspondence. We furthermore lift f to an analytic correspondence F between their total rational cohomologies, which is a Hodge isometry with respect to the Mukai pairings, and which preserves the gradings up to sign. When X and Y are projective the correspondences f and F are algebraic.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This will be a survey talk about anabelian geometry.

I am going to discuss a progress in an ongoing project (since approx 2000) which aims to formalise the notion of an analytic covering space of a complex algebraic variety in such a way that the formal cover is unique up to abstract isomorphisms (categorically axiomatised ). It turned out deeply dependent on and related to both arithmetic geometry and transcendental number theory. Model-theoretic geometry presents aspects of both in an explicit and predictive format.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The middle cohomology of hyperkahler fourfolds of Kummer type was studied by Hassett and Tschinkel, who showed that a large portion is generated by cycle classes of fixed-point loci of symplectic involutions. In recent joint work with Katrina Honigs, we study symplectic fourfolds over arbitrary fields which are constructed as fibers of the Albanese map on moduli spaces of stable sheaves on an abelian surface. We have extended the results of Hassett and Tschinkel and characterized the Galois action on the cohomology. We do this by giving an explicit description of the symplectic involutions on the fourfolds. This has natural consequences for derived equivalences between Kummer fourfolds.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The talk is based on joint work with Sofia Tirabassi. I will be discussing tame twisted K3 surface over an algebraically closed field of positive characteristic and counting its untwisted FM partners. On the way to the counting results, we will also discuss that every tame twisted Fourier Mukai partner of a K3 surface of finite height is a moduli space of twisted sheaves over it.

In the framework of non-archimedean Berkovich geometry, questions of anabelian type are best answered using the so-called tempered fundamental group, introduced by Yves André. It is now known that in many cases, the tempered group of a curve determines its skeleton as a graph. This graph also has a natural metric. Does the tempered fundamental group determine the length of each edge? If the answer is positive in some cases for algebraic curves due to some work of Lepage, it remains unknown otherwise, even for the most simple curves : annuli. I will present a partial result in this direction, using some interesting methods of splitting radius of torsors and resolution of non-singularities.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This is a survey of the relationship between derived equivalence, the existence of rational points, and other arithmetic properties. Given smooth varieties X and Y over a field k, assume to be derived equivalent over k, how are the k-rational points of X and Y related? We summarize what is currently known for K3 surfaces as well as some important recent results of Addington-Antieau-Honigs-Frei in higher dimensions. (joint with Tschinkel)

In this talk, I will define a basic class of algebras, “homotopy path algebras”, and explain the relation between a homotopy path algebra and entrance/exit paths on an appropriately stratified classifying space that naturally gives a cellular resolution of the diagonal bimodule. An earlier result of mirror symmetry due to Bondal-Ruan and certain Berglund-Hu ̈bsch-Krawitz mirrors can be recovered as an application. I will also discuss some results on minimal cellular resolutions of diagonal bimodules. This is based on joint work with David Favero.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The classical theorem of Neukirch and Uchida says that number fields are completely determined by their absolute Galois groups. In this talk, we’ll explain joint work with Clark Barwick and Saul Glasman generalizing this reconstruction result to schemes. Given a scheme S we construct a category Gal(S) that records the Galois groups of all of the residue fields of S (with their profinite topologies) together with ramification data relating them. We’ll explain why the construction S ↦ Gal(S) is a complete invariant of normal schemes over a number field. The category Gal(S) also plays some other roles. For example, just like how there is a monodromy equivalence between representations of étale fundamental group and local systems, there is an equivalence between representations of the category Gal(S) and constructible sheaves. This invariant also gives rise to a new definition of the étale homotopy type.

A double feature talk. During the first half, I’ll discuss how, when, and where does the Frobenius pushforward generate the derived category. This is joint with Pat Lank. In the second half, I’ll introduce a general construction which extracts integral kernels form flips and show how it gives an equivalence for stratified Mukai flops. This is joint with Nitin Chidambaram and David Favero.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Conic bundles are a geometrically rich class of varieties. In the 70's, Beauville showed that over an algebraically closed field, the group of algebraically trivial curve classes on a conic bundle threefold is isomorphic to the Prym variety of a double cover naturally associated with it. In joint work with Sarah Frei, Lena Ji, Bianca Viray and Isabel Vogt, we study curve classes on (geometrically standard and geometrically ordinary) conic bundle threefolds over arbitrary fields of odd characteristic. We then use the description of these classes to study the rationality of such varieties. Indeed, Hassett-Tschinkel and Benoist-Wittenberg introduced an obstruction to rationality, namely the intermediate Jacobian torsor obstruction, closely related to the structure of the group of curve classes on threefolds. We show that this obstruction is insufficient to characterize rationality.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Roughly speaking, this means that the moduli space of curves fails to be a fine moduli space in every possible way. This is joint work with Max Lieblich.