COHOMOLOGY OF ALGEBRAIC VARIETIESCIRM, LUMINY, 15 - 19 OCTOBER, 2018

Abstracts

Joseph Ayoub
New realisations for motives over positive characteristic fields
Abstract: Let K be an algebraically closed field of characteristic zero endowed with a rank one
valuation having residue field k of characteristic p > 0. We construct a realisation functor
R : DM(k) → D(A)from the category of Voevodsky motives to the derived category of A-modules having the following properties. (1) The ring A is a K-algebra which has an explicit description reminiscent tothe description of algebras of abstract periods. (2) The functor R is monoidal and takes a geometric motive to a perfect complex of A-modules. (3) There are comparison maps relating R withthe classical realisations. In particular, there is a comparison isomorphism to rigid cohomology
which on the algebra A is given by an explicit integration map.
If time permits, we also state some conjectures about the ring A and the realisation functor R.

Patrick Brosnan
Fourier transform and sheaves associated to families of Hessenberg varieties
Abstract: Hessenberg varieties are certain closed subvarieties of flag varieties defined by Lietheoretic conditions and depending on a Lie algebra element. By varying the Lie algebra element
we get a projective family of varieties which is flat over the locus of regular elements of the Liealgebra. In particular, we get a monodromy action on the cohomology of the smooth fibers.
When the Lie algebra is gln , this monodromy action has a combinatorial interpretation. In fact,
there is a combinatorial formula for it. Moreover, there are long-standing conjectures due toStanley and Stembridge concerning a certain type of positivity in the formula. I will explain joint
work with Tim Chow which explains how to use Fourier transform to compute the monodromy
action (in all types), and also proves that the total direct image of the constant sheaf is a sum of
IC sheaves of local systems supported in codimension 0.

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Hélène Esnault
Cohomological dimension in pro-`-towers
Abstract: We give a proof without use of perfectoid geometry of Scholze’s vanishing theorem forétale cohomology with F` -coefficients beyond the dimension of projective varieties in a specificpro-` tower in characteristic not equal to `.

Javier FresánSymmetric power moments of Kloosterman sums, I
Abstract: We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to thecomplex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn ou to be classical, the strategy consists in first realizing themas exponential motives and computing their Hodge numbers by means of the irregular Hodge
filtration. We show that all Hodge numbers are either zero or one, which implies potential automorphicity thanks to recent results of Patrikis and Taylor. The first talk will be concerned withthe arithmetic aspects and in the second one we will present the Hodge theoretic computations.
Joint work with Claude Sabbah and Jeng-Daw Yu.

Ofer Gabber
Remarks on vanishing cycles and comparison of oriented products and rigid toposes
Abstract: to be announced

Annette Huber-Klawitter
Periods of 1-motives
Abstract: (joint work with G. Wüstholz) Roughly, 1-dimensional periods are the complex numbersobtained by integrating a differential form on an algebraic curve over Q̄ over a suitable domainof integration. One of the alternative characterisations is as periods of Deligne 1-motives.
We clear up the linear relations between these numbers, proving Kontsevich’s version of theperiod conjecture for 1-dimensional periods.
In particular,R a 1-dimensional period is shown to beR
algebraic if and only if it is of the form γ (φ + df) with γ φ = 0. We also get formulas for thespaces of periods of a given 1-motive, generalising Baker’s theorem on logarithms of algebraic
numbers.
The proof is based on a version of Wüstholz’s analytic subgroup theorem for 1-motives.

2June Huh
Kazhdan-Lusztig theory for matroids
Abstract: There is a remarkable parallel between the theory of Coxeter groups (think of thesymmetric group or the dihedral group) and matroids (think of your favorite graph or point
configuration), based on a combinatorial cohomology theory. I will give an overview of thesimilarity, and report on my joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, andBotong Wang on a cohomological approach to classical conjectures in combinatorics.

Peter Jossen
Cohomology of varieties with potential
Abstract: There are several interesting cohomology theories attached to pairs (X, f), consisting ofan algebraic variety X and a regular function f (the potential) on X. I will present one of thesecohomology theories, which takes values in a tannakian category Perv0 , whose objects are certain
perverse sheaves on the affine line. My plan is then to explain how to compute tannakian Galois
groups in this category Perv0 , and to explain how these groups are related to motivic Galois
groups of motives attached to pairs (X, f).Joint work with J. Fresán.

Emanuele Macrı̀
Derived categories of cubic fourfolds and non-commutative K3 surfaces
Abstract: The derived category of coherent sheaves on a cubic fourfold has a subcategory which
can be thought as the derived category of a non-commutative K3 surface. This subcategory wasstudied recently in the work of Kuznetsov and Addington-Thomas, among others. In this talk, Iwill present joint work in progress with Bayer, Lahoz, Nuer, Perry, Stellari, on how to construct
Bridgeland stability conditions on this subcategory. This proves a conjecture by Huybrechts, andit allows to start developing the moduli theory of semistable objects in these categories, in ananalogue way as for the classical Mukai theory for (commutative) K3 surfaces. I will also discussa few applications of these results.

Alexander Merkurjev
Versal torsors and retracts
Abstract: Let G be an algebraic group over a field and p a prime integer. We introduce the notionof a p-retract rational variety and prove that if Y → X is a p-versal G-torsor, then the classifying
space BG is a stable p-retract of X. It follows that BG is p-retract rational if and only if there isa p-versal G-torsor Y → X with X a rational variety, that is all G-torsors over infinite fields arerationally parameterized. In particular, for such groups G the unramified Galois cohomology
group is trivial.

Matthew Morrow
Relative integral p-adic Hodge theory
Abstract: Given a smooth scheme X over the ring of integers of a p-adic field, we introduce thenotion of a relative Breuil-Kisin-Fargues module M on X. Each such M simultaneously encodes
the data of a lisse étale sheaf, a module with flat connection, and a crystal, whose cohomologies
are then intertwined by a relative form of the Ainf cohomology introduced in ”Integral p-adic
Hodge theory” by Bhatt-M-Scholze. They are moreover closely related to other work in relative
p-adic Hodge theory, notably Faltings small generalised representations and his relative Fontaine
Lafaille theory.
Joint with Takeshi Tsuji.
3Andrei Negut
Hilbert schemes of K3 surfacesAbstract: We give a geometric representation theory proof of a mild version of the BeauvilleVoisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map
restricted to the subring of Chow generated by tautological classes. Although other geometricproofs of this result are known, our approach involves lifting formulas of Lehn and Li-QinWang from cohomology to Chow, and using them to quickly solve the problem by invoking theirreducibility criteria of Virasoro algebra modules, due to Feigin-Fuchs.
Joint work with Davesh Maulik.

Wiesława Nizioł
Comparison theorems for rigid analytic varieties
Abstract: I will discuss p-adic comparison theorems for rigid analytic varieties.
This is joint work with Pierre Colmez.

Stefan Patrikis
Deformations of Galois representations: examples and applications
Abstract: The deformation theory of Galois representations, and in particular the question of how
we can lift mod p representations of the Galois group of a number field to p-adic representations,is one of the main tools for understanding the conjectured modularity and motivic origin of padic representations. I will survey some illustrative examples in this subject and describe someapplications to the construction of motives with exceptional Galois groups.
Some of this is joint work with N. Fakhruddin and C. Khare, and some is joint with G. Boxer, F.
Calegari, M. Emerton, B. Levin, and K. Madapusi Pera.

Claude Sabbah
Symmetric power moments of Kloosterman sums, IIAbstract: We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to thecomplex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be classical, the strategy consists in first realizingthem as exponential motives and computing their Hodge numbers by means of the irregular
Hodge filtration. We show that all Hodge numbers are either zero or one, which implies potential automorphicity thanks to recent results of Patrikis and Taylor. The first talk will be concernedwith the arithmetic aspects and in the second one we will present the Hodge theoretic computations.
Joint work with Javier Fresan and Jeng-Daw Yu.

Takeshi Saito
Characteristic cycle of constructible sheaves and restriction to curves
Abstract: The characteristic cycle is defined for a constructible sheaf on a smooth scheme overa perfect field as a cycle on the cotangent bundle. After briefly recalling its basic properties, I
explain that it is determined by its rank and the conductor of the restrictions to curves.
4Christian Schnell
Extension theorems for holomorphic forms on complex spaces
Abstract: Suppose we have an algebraic (or holomorphic) differential form, defined on the smooth
locus of an algebraic variety (or analytic space). Under what conditions does it extend to analgebraic (or holomorphic) differential form on a resolution of singularities? In 2011, Greb,Kebekus, Kovacs, and Peternell proved that such an extension always exists on algebraic varieties
with klt singularities. I will explain how to generalize their result to a much larger class ofsingular spaces, with the help of Hodge modules and the decomposition theorem.
This is joint work with Kebekus.

Qizheng Yin
Special subvarieties in holomorphic symplectic varieties
Abstract: The classical Bogomolov-Mumford theorem states that every algebraic K3 surface admits a rational curve. In my talk I shall discuss generalizations of this theorem to holomorphic
symplectic varieties, namely the search of special subvarieties swept out by rational/constantcycle subvarieties. Various results and open questions will be presented, and the relation to theBeauville-Voisin conjectures will be mentioned.
Based on joint work with G. Oberdieck, J. Shen, and X. Zhao.

Weizhe Zheng
Compatible systems along the boundary
Abstract: A theorem of Deligne says that compatible systems of `-adic sheaves on a smooth curve
over a finite field are compatible along the boundary. I will present an extension of Deligne’s
theorem to schemes of finite type over the ring of integers of a local field. This has applications
to the equicharacteristic case of some conjectures on `-independence. I will also discuss therelationship with compatible wild ramification.
This is joint work with Qing Lu.

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