Schedule for: 19w5164 - Interactions between Brauer Groups, Derived Categories and Birational Geometry of Projective Varieties

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the 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.

In this talk we propose a new approach to nonrationality questions. Examples will be given.

The talk will start by recalling essential dimension and looking at some elementary examples. For certain kinds of Deligne-Mumford stacks, P. Brosnan Z. Reichstein and A. Vistoli have proved a very powerful genericity theorem which reduces the calculation of essential dimension to that of gerbes. For Gm-gerbes there is a conjectural formula of J.L Colliot-Thelene, N. Karpenko and A. Merkurjev for the essential dimension of a gerbe that relates the essential dimension to Brauer invariants such as the index. In joint work with I. Biswas and N. Hoffmann, modulo this conjecture, the essential dimension of the moduli stack of vector bundles on a smooth projective curve was computed. Recently these results were extended to orbifold curves with D. Valluri.

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

I will present joint work with Martin Olsson on a conjectural Torelli-type statement for derived categories of arbitrary proper smooth varieties. The key is a replacement for the Hodge structure by “filtrations” that naturally arise from geometry.

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!

Topological Hochschild homology gives rise to a number of intrinsic derived invariants of a variety in positive characteristic. We will explain how to compute these quantities in terms of more classical invariants of X, such as de Rham and crystalline cohomology. As a consequence, we obtain new restrictions on the Hodge numbers of derived equivalent varieties in positive characteristic. We will also present an example of two derived equivalent 3-folds in characteristic 3 with different Hodge numbers. This is joint work with Benjamin Antieau and Nick Addington.

It is natural to ask which properties of a smooth projective variety are recovered by its derived category. In this talk, I will consider the question: is the existence of a rational point preserved under derived equivalence? In recent joint work with Nicolas Addington, Benjamin Antieau, and Katrina Honigs, we show that over Q, the answer is no. We give two examples: an abelian variety and a torsor over it, and a pair of hyperkaehler fourfolds. The latter is independently interesting as a new example of a transcendental Brauer-Manin obstruction to the Hasse principle.

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

Let X be a Calabi–Yau manifold that admits a Tyurin degeneration to a union of two quasi-Fano varieties X1 and X2 intersecting along a smooth anticanonical divisor D. The “DHT mirror symmetry conjecture” implies that the Landau–Ginzburg mirrors of (X1,D) and (X2,D) can be glued to obtain the mirror of X. Initial motivation came from considering the bounded derived categories of X, X1, and X2 and symplectomorphisms on the Landau-Ginzburg models mirror to (X1,D) and (X2,D). In this talk, flipping the roles of the two categories, I will explain how periods on the Landau-Ginzburg mirrors of (X1,D) and (X2,D) are related to periods on the mirror of X. The relation among periods relates different Gromov-Witten invariants via their respective mirror maps. This is joint work with Fenglong You and Jordan Kostiuk.

The failure of the integral Hodge conjecture has been known since the famous counterexamples of Atiyah and Hirzebruch. Currently, several methods exist for producing other counterexamples. For instance, a seminal result of Colliot-Thélène and Voisin relates the failure of the integral Hodge conjecture in degree 4 to degree 3 unramified cohomology. After giving a quick overview of this method, I will discuss how it can be used to obtain new counterexamples to the integral Hodge conjecture.

The problem of determining conditions under which a rational map can exist between a pair of twisted flag varieties plays an important role in the study of algebraic groups and their torsors over general fields. A non-trivial special case arising in the theory of quadratic forms amounts to studying the extent to which an anisotropic quadratic form can become isotropic after extension to the function field of a quadric. To this end, let $p$ and $q$ be anisotropic quadratic forms over an arbitrary field, and let $k$ be the dimension of the anisotropic part of $q$ over the function field of the quadric $p=0$. We then conjecture that the dimension of $q$ lies within $k$ of an integer multiple of $2^{s+1}$, where $2^{s+1}$ is the least power of 2 bounding the dimension of $p$ from above. This generalizes the so-called ``separation theorem'' of D. Hoffmann, bridging the gap between it and some other classical results on Witt kernels of function fields of quadrics. The statement holds trivially if $k \geq 2^s - 1$. In this talk, I will discuss recent work that confirms its validity in the case where $k \leq 2^{s-1} + 2^{s-2}$ (among other cases).

Toric varieties defined over the complex numbers provide an important testing ground for computing various algebro-geometric invariants (e.g., the coherent derived category associated to a variety), as many computations of interest may be phrased entirely in terms of combinatorial data such as fans, cones, polytopes. Over general fields, we consider twisted forms of such objects called "arithmetic toric varieties", whose analysis is naturally Galois-theoretic. In this talk, we will present results on the structure of derived categories of arithmetic toric varieties via exceptional collections. In particular, we will focus on some highly symmetric classes of such objects, including centrally symmetric toric Fano varieties and toric varieties associated to root systems of type A. The latter class yields examples of varieties which are arithmetically interesting but whose derived categories are well understood. A conjecture of Orlov posits that the structure of the derived category influences the rationality type of a variety. We will discuss how this plays out in the setting of toric varieties. This is joint work with Matthew Ballard, Alexander Duncan, and Alicia Lamarche.

A healthy body of evidence says that birational geometry and derived categories are intimately bound. Even so, many basic questions are still open. One of the most central questions is the conjecture of Bondal and Orlov (later extended by Kawamata) that says two smooth projective varieties related by a flop are actually derived equivalent. The first step in resolving this question is understanding how to produce functors from rational maps. In work with Diemer and Favero, we provided a method to construct an integral kernel associated to any D-flip of normal varieties with Q-Cartier D. Conjecturally, this can be used to answer Bondal and Orlov's question. In this talk, we will discuss the construction and natural extensions of it. In particular, we will highlight work with Chidambaram, Favero, McFaddin, and Vandermolen relating to the what has been termed a Grassmann flop.

The Clifford algebras provide a connection between the study of quadratic forms and the Brauer group. We explain how a bound for the u-invariant of a field can be given purely in terms of certain period-index bounds for the Brauer group. We survey some progress in finding such bounds for function fields of p-adic curves. There are open questions in this direction for function fields of curves over number fields.

The Iitaka dimension of a line bundle D on a projective variety X is the dimension of the image of the rational map given by |mD| for large and divisible m. The Iitaka dimension is not a numerical invariant of D, and there are several approaches to constructing a "numerical dimension", which should be an analogous invariant depending only on the numerical class of D. I will discuss some divisors of dynamical origin that have unexpected behavior with respect to these invariants.

The theory of flasque resolutions gives a complete solution to the problem of determining whether an algebraic torus is stably- or retract-rational. More generally, flasque resolutions of more general connected linear algebraic groups are a powerful tool for studying R-equivalence, weak approximation, and the Hasse principle of such groups. There is a dual notion of coflasque for tori, which can also be extended to more general linear algebraic groups. ​ A common technique for classifying objects over non-closed fields is to associate a set of simple algebras to each object. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. However, it may be that there is no way to distinguish non-isomorphic objects through any such association if it is functorial in the base field. Using coflasque resolutions of general algebraic groups, we can precisely describe the set of such indistinguishable objects. ​ I will recall flasque and coflasque resolutions, discuss how they can be used to describe these indistinguishable sets, and comment on the implications for detecting arithmetic properties using Brauer groups, K-theory, and derived categories. This is based on joint work with Matthew Ballard, Alicia Lamarche, and Patrick McFaddin.

Let X be a projective algebraic surface. We recall the K-group K_{1,\ind}^{(2)}(X) of indecomposables and provide evidence that membrane integrals are sufficient to detect these indecomposable classes.

Donaldson-Thomas invariants are virtual counts of stable sheaves with fixed numerical invariants, on a Calabi-Yau threefold. I will survey old and new results with emphasis on the peculiarities arising when substituting the CY with an abelian threefold.

The concept of birational divisor arises in different contexts; for example in work of V. V. Shokurov and, independently, P. Vojta. The aim of this talk is to survey related recent developments. Some emphasis will be placed on results that deal with Brauer groups of function fields of algebraic varieties (e.g., joint work with C. Ingalls). Time permitting, I hope also (to at least briefly) mention results which deal with complexity and distribution of rational points (e.g., extensions to the recent work of Ru and Vojta).

By an observation of Reid, any birational morphism between algebraic varieties can be realized through variation of geometric invariant theory quotients (VGIT). As a consequence, one way of comparing the derived categories of birational algebraic varieties is through VGIT. It turns out that this can be done even in singular cases by, in some sense, "deriving" the VGIT procedure. I will discuss how this can be done very concretely by studying VGIT for commutative differential graded algebras. This talk is about joint work with Chidambaram motivated by work with Ballard and Diemer.

In this talk we study the behavior of special classes of fibrations onto normal projective varieties that admit a finite morphism to an abelian variety under derived equivalence of smooth projective complex varieties. Our first result is that any derived equivalence of such varieties induces a base preserving correspondence between their sets of isomorphism classes of fibrations onto smooth projective curves of genus greater or equal to two. The proof of this result involves earlier results, obtained in collaboration with M. Popa, regarding the derived invariance of the non-vanishing loci attached to the canonical bundle, and generic vanishing theory. Concerning fibrations onto higher-dimensional bases, we show how the problem of the derived invariance of fibrations is related to the conjectural derived invariance of Hodge numbers. I will report on some progress in this direction.

Cohomological invariants play an important role in the classification of G-torsors, where G is an algebraic group over a field. However these invariants are hard to compute. In case of a Weyl group G they have been recently computed (with some restrictions on the base field). A crucial role in this computation is played by a splitting principle, which roughly says that an invariant of a Weyl group is determined by its restriction to elementary abelian 2-subgroups generated by reflections. In the talk I will discuss the generalization of this principle to orthogonal reflection groups. (joint work with Christian Hirsch)

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.