Abstracts

Torsion universal 0-cycles on fibrations

Asher Auel (Yale University, New Haven)

Abstract: In the last few years, a breakthrough in the (stable) rationality problem has been due to the degeneration method for the universal triviality of the Chow group of 0-cycles, initiated by Voisin and developed by Colliot-Thélène and Pirutka. The power of this method lies in its mix of inputs from algebraic cycles, Hodge theory, algebraic K-theory, birational geometry, and singularity theory. I will survey some of these developments and show how a mix of techniques can lead to (stable) rationality obstructions for certain classes of varieties with the structure of a fibration over a rational base.

Categorical representability and applications to birational geometry

Marcello Bernardara (IMT, Toulouse )

Abstract: The aim of this talk is to define the notion of categorical representability for a (smooth) projective variety and state related questions in birational geometry. In particular, using Bondal-Larsen-Lunts motivic measure and weak factorization, it is possible to construct both a motivic obstruction to rationality and a filtration of the group of birational autoequivalences of a given variety. These give motivic evidences to categorical questions. We finally motivate by some example the categorical approach; in particular constructing a birational invariant for del Pezzo surfaces (a joint result with A.Auel), and reinterpreting the genus filtration of the Cremona group of a threefold.

Hochschild (co)homology for Azumaya orbifolds

Andrei Caldararu (University of Wisconsin, Madison)

Abstract: Tabuada and Van den Bergh have argued in a recent paper that Hochschild (co)homology of a smooth variety is not sensitive to the presence of a twisting by a Brauer class. I will argue that the situation is fundamentally different for orbifolds. Specifically, I will describe an HKR decomoposition of the Hochschild (co)homology of an orbifold endowed with a Brauer class, and I will explain how the invariants change when the Brauer class is turned on. This gives mathematical justification for certain calculations in mirror symmetry of Vafa and Witten. (Joint work with Dima Arinkin.)

Derived category of moduli of pointed stable rational curves

Anne-Maria Castravet  (Northeastern University, Boston)

Abstract: I will report on joint work with Jenia Tevelev on Kuznetsov's conjecture on the derived category of moduli of pointed stable rational curves and related spaces.

Generalized Borcea-Voisin mirror duality in any dimension

Alessandro Chiodo (IMJ-PRG,Paris)

Abstract: The ordinary Borcea-Voisin duality pairs two mirror K3 surfaces with anti-symplectic involutions and produces two mirror three-folds via a product with a fixed elliptic curve and quotient by the natural diagonal involution on the two sides. We extend Borcea-Voisin mirror duality beyond dimension three. This uses the Landau-Ginzburg model and aspects of the Landau-Ginzburg/Calabi-Yau correspondence which allow improvements to ordinary mirror symmetry theorems. This is work in collaboration with Elana Kalashnikov and Davide Cesare Veniani.

The cohomological integrality theorem and positivity in algebraic combinatorics

Ben Davison (EPFL Lausanne)

Abstract: In this talk I will discuss interactions between Donaldson--Thomas theory and algebraic combinatorics. In particular, I will explain a recent result with Sven Meinhardt, which provides a cohomological lift of the integrality conjecture (one of the central conjectures in DT theory), and explain a representative application in algebraic combinatorics - a new proof of the result of Hausel, Letellier and Villegas, on the positivity of the coefficients of the Kac polynomial a_Q(q) counting absolutely indecomposable representations of a quiver Q over a field of order q.

Toric tilings and limit linear series

Eduardo Esteves (IMPA Rio de Janeiro)

Abstract: This is a report on joint in-progress work with Omid Amini (ENS Paris). The goal is to construct a new compactification of the Jacobian over reducible nodal curves, by using what we may call toric tilings. These are unions of toric varieties described by combinatorial data which serve as bases for families of line bundles and their degenerations over reducible nodal curves. While the approach taken to date considers as objects of the moduli problem line bundles, our objects are rather these families over toric tilings. I will contend in this talk that this new compactification is suitable for the study of limit linear series on all reducible nodal curves.

Symplectic reductions and polar representations

Manfred Lehn  (Johannes Gutenberg-Universität, Mainz)

Abstract: The concept of a polar representation, due to Dadok and Kac, of a reductive group $G$ on a vector space $V$ generalises certain aspects of the geometry of adjoint representations. In particular, there is a subvector space $c$ in $V$, stable under the action of a finite subquotient $W$ of $G$ such that the natural morphism $c/W\to V//G$ is an isomorphism. In joint work with Bulois, C. Lehn and Terpereau we prove a symplectic version of this Chevalley isomorphism, namely an isomorphism of Poisson varieties $(c\oplus c^*)/W \to (V\oplus V^*)///G$, where the right hand side denotes the Marsden-Weinstein reduction, for a number of cases.

Stability on the derived category of the projective space

Emanuele Macri  (Northeastern University, Boston)

Abstract: In this talk I will present two notions of stability in the derived category of the projective space, tilt stability and Bridgeland stability. I will then show how to apply this to study space curves and their Hilbert schemes. This is joint work (in progress) with Benjamin Schmidt.

Remarks on Voevodsky's nilpotence conjecture and on Grothendieck's standard conjecture of type D

Gonçalo Tabuada  (MIT, Cambridge)

Abstract: I will explain how the recent theory of noncommutative motives enables (alternative) proofs of these conjectures for some particular families of varieties.

A trichotomy for the autoequivalence groups of derived categories on surfaces

Hokuto Uehara Tokyo Metropolitan University, Tokyo)

Abstract: We study the group of autoequivalences of the derived categories on smooth projective surfaces, and show a trichotomy of types of the groups, that is, of K3 type, of elliptic surface type and of general type. We also pose a conjecture on the description of each type of the groups, and prove it in some special cases.

Semi-orthogonal decompositions for equivariant derived categories for some reflection groups

Michel van den Bergh   (Universiteit Hasselt, Diepenbeek)

Abstract: We will discuss a number of instances of semi-orthogonal decompositions of equivariant derived categories for reflection groups. We will discuss in particular the case of the symmetric group acting on its standard representation. To prove our results we use the Springer correspondence. This is joint work with Alexander Polishchuk.

Classification of torsors in motivic homotopy theory

Matthias Wendt    (University of Freiburg, Freiburg)

Abstract: In complex geometry, the well-known Oka-Grauert theorem states that the holomorphic and continuous classification of principal bundles over Stein manifolds agree. In the talk, I will present an analogous statement in algebraic geometry, established in recent joint work with Aravind Asok and Marc Hoyois. The main result is that torsors under isotropic reductive groups over smooth affine varieties can be classified by means of motivic homotopy theory. I will explain some of the motivic homotopy theory in the background, but the focus of the talk is going to be on how to apply the general theorem to prove explicit result about classification of projective modules (as done by Asok and Fasel), octonion algebras or stably hyperbolic quadratic forms.