Title: On the motivic hyperkähler resolution conjecture
Abstract:
This is a joint work with Zhiyu Tian. Ruan's conjecture on hyperkähler resolution says that the orbifold cohomology ring of an smooth projective orbifold is isomorphic to the cohomology ring of its hyperkähler resolution of singularities. I would like to formulate the motivic analogue of this conjecture on the Chow ring (or more generally the Chow motive as an algebra in the category of Chow motives) of hyperkähler varieties. The particular interesting cases that I want to discuss are the Hilbert-Chow morphisms for the symmetric product of K3 surfaces and the similar generalized Kummer varieties. This work could be seen as a continuation of the recent work of C. Vial.
Title: The cycle classes of divisorial Maroni loci
Abstract:
To a general curve of genus g with a linear system of degree d one can associate a (d-1) tuple of integers that describe the type of scroll spanned by the fibres of the linear system on the canonically embedded curve. This gives rise to the so-called Maroni stratification of the Hurwitz space H(d,g). We determine the cycle classes of Maroni divisors in the compactified Hurwitz spaces. This is joint work with Alexis Kouvidakis.
Title: Euler characteristics and epsilon constants of curves over finite fields - some wild stuff
Abstract:
Given a smooth projective curve over a finite field equipped with an action of a finite group, we first introduce the corresponding Artin L-function and a certain equivariant Euler characteristic. The main result is a precise relation between the epsilon constants appearing in the functional equations of Artin L-functions and that Euler characteristic. This generalises a theorem of Chinburg from the tamely to the weakly ramified case.
Title: On the rationality problem of cubic fourfolds
Abstract:
The rationality problem of cubic fourfolds is one of the open problems in algebraic geometry. We have many examples of rational cubic fourfolds. It is widely believed that a very general cubic fourfold is irrational but so far we have no proof of this. People has been attacking this problem from different points of view, such as Hodge structures, derived categories, Chow groups and Grothendieck ring of varieties. In this talk, I will survey our current knowledge about this problem and discuss some recent progresses.
Abstract:
Let F:A\to B be a left-exact functor. Under suitable
finiteness conditions, the
alternating sum \sum (-1)^i [ R^iF(X) ] in the Grothendieck group
K_0(B) only depends on the class [X] in K_0(A) and one obtains a
homomorphism K_0(A) \to K_0(B). In his 2014 master's thesis Niels uit
de Bos has constructed for a large class of not necessarily additive
functors F an induced function K_0(A) \to K_0(B). In general this
induced map is not a homomorphism. Interesting examples include
`multilinear' functors such as exterior and symmetric powers. In this
lecture I will explain the results of Niels, and show how simplicial
techniques play a major role.
Abstract:
I will talk about joint work with C. Breuil and B. Schraen in which we study families of Galois representations that occur on eigenvarieties. Using a patching module as in recent work of Caraiani, Emerton, Gee, Geraghty, Paskunas and Shin one can define a local analogue of an eigenvariety and compare this to a space of local Galois representations defined in terms of (phi,Gamma)-modules. I will discuss this construction and the relation with modularity lifting theorems.
Abstract:
We prove a relation in the Grothendieck ring of varieties between
a cubic hypersurface of arbitrary dimension and its Fano variety
of lines. Using this relation we compute the Hodge structure
of the Fano variety. Finally assuming the Cancellation conjecture
we deduce implications on irrationality of a very general cubic fourfold.
Joint work with Sergey Galkin.
Abstract:
While the p-curvature conjecture of Grothendieck and Katz, which lives in mixed characteristic, remains still widely unsolved, André stated and proved its analogous for the equicharacteristic zero situation. In a similar spirit, Esnault and Langer proved a weak form of the conjecture in positive characteristic. In this talk we will explain how the strong form of the conjecture holds over uncountable fields of positive characteristic and its complete failure if the field is countable.
Abstract:
By a classical theorem of Serre and Tate, extending previous results of Néron, Ogg, and Shafarevich, an Abelian variety over a p-adic field has good reduction if and only if the Galois action on its first l-adic cohomology is unramified. In this talk, we show that if the Galois action on second l-adic cohomology of a K3 surface over a p-adic field is unramified, then the surface admits an ``RDP model'' over the that field, and good reduction (that is, a smooth model) after a finite and unramified extension. (Standing assumption: potential semi-stable reduction for K3's.) Moreover, we give examples where such an unramified extension is really needed. On our way, we establish existence and termination of certain semistable flops, and study group actions of models of varieties. This is joint work with Yuya Matsumoto.