Title: Calculations on the formal groups of Calabi-Yau threefolds of Delsarte type
Abstract: Calabi-Yau threefolds are associated with formal groups of dimension one. When they are defined over an algebraically closed field of positive characteristic, their formal groups are classified by the height. In this talk, we compute the height of the formal groups of weighted Delsarte threefolds and discuss it in the framework of the Berglund-H\"ubsch-Krawitz mirror symmetry.
Title: A new approach to the de Rham-Witt theory, after Bhatt, Lurie and Mathew
Abstract: Bhatt, Lurie, and Mathew have recently constructed de Rham-Witt-like complexes for schemes over a perfect field of positive characteristic, which coincide with the classical one in the smooth case and are of interest in certain singular cases. I will explain their approach, which exploits the (fashionable) Deligne-Ogus décalage $\eta_p$ functor, and uses only elementary homological algebra (in particular, avoids the laborious calculations involved in the so-called canonical bases for Deligne complexes of integral forms).
Title: A motivic formalism in representation theory.
Abstract: I will speak about work joint with Jens Niklas Eberhardt. A fundamental problem in representation theory is to determine the characters of all simple rational SLn(Fp)-modules. Unlike in the characteristic zero case, this is still wide open and the subject of ongoing research. Fifteen years ago, Soergel proposed a strategy using geometric methods: He translates the problem—at least for some of the simple modules—into a question about certain sheaves on a flag variety. In this talk we present a way to enhance his statements by replacing sheaves with motives and applying work of Ayoub, Cisinski-Déglise, and Geisser-Levine.
Title: CM liftings of K3 surfaces and the Tate conjecture
Abstract: I will explain that any K3 surface of finite height over a finite field admits a CM lifting (after extending the base field). In fact, one can control the CM action on the lifting. This has an application to the Tate conjecture for the square of the K3 surface. Our work is largely influenced by the works of Nygaard-Ogus and Madapusi Pera on the Tate conjecture for the K3 surface itself, and the Kuga-Satake construction plays a key role. Joint work with Kazuhiro Ito and Tetsushi Ito.
Title: On Gysin triangles of motifs with modulus
Abstract: Kahn-Saito-Yamazaki construct triangulated categories of motifs with modulus \underline{M}DM and MDM. These categories are constructed out of modulus pairs, and are related to Chow groups with modulus. In this talk we will show that \underline{M}DM admits a generalisation of the Gysin triangle from DM, as well as another interesting triangle (which we call the boundary triangle). These give motivic analogies of results on Chow groups with modulus by Miyazaki.
Title: Artin-Mazur’s height and Yobuko’s height
Abstract: A few years ago Yobuko has introduced the notion of a new delicate invariant for a proper smooth scheme over a perfect field $k$ of finite characteristic. (We call this invariant Yobuko height.) This generalizes the notion of the F-splitness due to Mehta-Ramanathan. In this talk we give relations between Artin-Mazur heights and Yobuko heights. We also give several properties of a proper log smooth scheme of Cartier type with finite Yobuko height over the log point of $k$. This includes generalizations of Joshi’s results for an F-split proper smooth scheme over $k$.
Title: Fano varieties in positive characteristic and their F-splittings
Abstract: Fano varieties is one of the fundamental classes of varieties in algebraic geometry. In the first part of the talk, we review classification of smooth Fano threefolds in positive characteristic, mainly focusing on differences from the characteristic 0 case. In the second part, we discuss Frobenius splittings of Fano varieties. We give some characterizations of non-F-split del Pezzo surfaces in positive characteristic.
Title: Duality for cohomology of local fields and curves with coefficients in abelian varieties
Abstract: I will explain a duality for cohomology of local fields and curves over perfect base (or residue) fields of positive characteristic with coefficients in abelian varieties. The cohomology mentioned here is equipped with a structure of a sheaf over a Grothendieck site called the "rational etale site" of the base field, and we consider a sheaf-theoretic relative duality. With this duality, we are able solve Grothendieck's duality conjecture in SGA 7 on special fibers of abelian varieties. Also, Tate-Shafarevich groups, as sheaves, are represented by unipotent algebraic groups. Cassels-Tate pairings are generalized with these geometric structures. I will then try to express my naive hope that, in this duality, abelian varieties and unipotent groups should be generalized to motives and curves should be generalized to morphisms between varieties. The big picture here is that there should be a relative duality and six operations formalism for mixed etale motives with p-torsion, a program that has yet to be developed.
Title: On variation of Newton polygons of $F$-isocrystals on a variety
Abstract: In a variety of characteristic $p > 0$, the existence and non-existence of F-isocrystals with non constant Newton polygons is mysterious. It is related to arithmetic and geometric properties of varieties. We discuss this problem in the case of elliptic surfaces.