Seminar on Motives, Standard Conjectures and Weil Cohomology Theories, Spring 2022

Description of the seminar

The primary goal of the seminar would be to get acquainted with the language of motives and the questions surrounding them.

The first few talks aim at defining the category of correspondences and the symmetrical monoidal category of Chow Motives M(k), where k is an algebraically closed field. After that we will discuss classical Weil cohomology theories H*, tying them in with motives due to the property of factoring through the category M(k) and their realization functors. (We will follow the chapter of stacks project [0FFG]).

Later we will study the interaction between standard conjectures and motives: in particular we will focus on semisimplicity (and how it doesn't need standard conjectures in order to be proved [J]), on the existence of graded decomposition on the motive h(X), on the property of being Tannakian and the Motivic Galois Group. (We will use Milne's paper [M1] in order to navigate the topics, and as references André's book [A] and papers [J] & [S]).

Depending on the audience's preferences (as well as how much time we have left), we can either dig in some concrete computations from Scholl's paper [S] (graded decomposition for motives h(A) of abelian varieties, and computation of motives h1(X) and h2d−1(X), with X variety of dimension d [S]) or discuss Motives over finite fields (following [M2]).

Logistics Info

The seminar will be held on Monday afternoons from 1:00pm to 2:20pm in Room 528 in the Math Department at Columbia.

Please email me at mp3947 at columbia dot edu if you are interested in giving a talk and/or you want to be added to the mailing list. Below there is a tentative list of topics: but please feel free to make suggestions about any related topic which you may be interested in.

References

Intersection Theory , Fulton's book [F]
Monoidal categories [0FFJ] and additive monoidal categories [0FN9]
Weil Cohomology Theories , Stacks Project, Tag [0FFG].
Introduction to étale cohomology Lombardo & Maffei's CourseNote [LM]
Survey on Motives: Motives -- Grothendieck’s Dream Milne's note [M1]
Motives, numerical equivalence, and semi-simplicity Jannsen's paper [J]
Classical Motives School's paper [S]
Another survey's paper, Realization functors Riou's paper [R]
Algebraic Cycles and the Weil Conjectures Kleiman's paper [K]
Une introduction aux Motifs André's book [A]
Motives over finite fields Milne's paper [M2]
Le Groupe Fondamental de la Droite Projective Moins Trois Points Deligne's paper [D]
Survey on Motivic Galois Group: Bruno Kahn's Slides [K]

List of Talks

Feb 21
Speaker: Morena Porzio
Title: Motivations to Motives and Seminar Roadmap
Abstract: I will talk about why we could be interested in the category of motives, giving some motivations and the roadmap of the seminar.
We will also review the bare minimum needed to talk about intersection theory and adequate equivalence relations.
Feb 28
Speaker: Morena Porzio
Title: Category of correspondences and the one of rational motives
Abstract: The program for today is to see the construction of the category of correspondences and the one of rational motives M_rat(k).
Then we will discuss some of the properties of M_rat(k), in particular its symmetric monoidal structure and additive structure. The existence of (left) duals is postponed to next week.
Mar 07
Speaker: Morena Porzio
Title: Classical Weil cohomology theories and their factorization through the category of Chow motives
Abstract: We will resume the proof that Mrat(k) is Karoubian and has left duals. Then we will focus on Classical Weil cohomology theories, in particular on their factorization through
the category of rational motives Mrat(k). In order to do so, some formal properties of Weil cohomology theories will be recalled (but without proof).
Mar 21
Speaker: Caleb Ji
Title: The Standard Conjectures on Algebraic Cycles
Abstract: In his work on motives and the Weil conjectures, Grothendieck formulated a set of conjectures on algebraic cycles which he called the standard conjectures.
We explain what they are and the known implications between them. Then we will explain their application to the Weil conjectures, and in particular to the proof of the Riemann hypothesis over function fields.
Caleb Ji's Notes: Standard Conjectures
Mar 28
Speaker: Morena Porzio
Title: Manin's identity principle and abelianity and semisimplicity of M_~(k)
Abstract: Today we will compute the motives of some concrete examples using Manin's identity principle, and looking at the relation between the motive of a curve and its Jacobian. Then we will see why
the category of motives fails to be abelian and why the category of motives M_~(k) is semisimple iff ~ is the numerical equivalence. At the end, we will discuss which direction to focus on
in the upcoming meetings (if you have preferences, feel free to share them!).
Apr 04
Speaker: Avi Zeff
Title: The Tannakian formalism and the motivic Galois group
Abstract: We'll discuss rigid symmetric monoidal categories, their properties, and some examples, and then see how this structure gives rise to the Tannakian formalism. We'll then try to apply
this formalism to the category of motives, using the standard conjectures, and try to describe the motivic Galos group.
Apr 11
Speaker: Morena Porzio
Title: The Tannakian formalism and the motivic Galois group (continuation)
Abstract: We'll resume discussing Tannakian categories from where we left: in particular we'll see the general notion of Tannakian category via fibre functors, how to associate an affine gerbe to any Tannakian
category C, and then how to realize C as a category of representations. After that we will focus on particular fibre functors, namely the realization functors associated with specific weil cohomology theories.
We'll see an overview of what is known/folklore about the motivic galois groups for Hodge structures of weight k and for l-adic cohomology theory.
Apr 18
Speaker: Morena Porzio
Title: Numerical Motives over finite fields
Abstract: Following Milne's Motives over Finite Fields, we will focus on the category of numerical motives over finite fields F_q assuming the truth of Tate's conjecture. In particular we will study
the fiber functors w_l for l-adic cohomology theories, with the aim of describing the motivic galois group GMot_{num}(F_q).
Apr 25
Speaker: Morena Porzio
Title: Motivic Galois Group of M_num(F_q) w.r.t. l-adic cohomology and over the algebraic closure of Q
Abstract: We will finish the discussion about the essential image of the realization functor w_l : M_num(F_q) \otimes Q_l ---> V_l(F_q) to then pass to the Motivic Galois
group of it, giving a description of it as a proreductive affine group scheme. Then we will see that for any fiber functor w defined over the algebraic closure of Q the automorphisms group
Aut^{\otimes}(w) is a group scheme P(q) whose characters coincide with the Weil q-numbers.

Tentative list of Topics

Crash Lecture on Intersection Product on CH^*(X), with X a smooth projective scheme over k ([F])
Correspondences and Chow motives (sections 3 and 4 of [0FFG])
Classical Weil cohomology theories, discussion of the axioms (de Rham cohomology - étale cohomology) (section 7 of [0FFG], section 1 and 3 of [LM], [0FWC])
Factorization through the category of Chow Motives (section 7 of [0FFG])
Relation between the motive of a curve and its Jacobian variety([S])
Standard Conjectures and their consequences on Mnum(k) ([M1])
Semisemplicity ([J])
M'num(k) is Tannakian and Motivic Galois Group [M1] & [A])
Study of realization functors for different cohomology theories ([R])
Graded decomposition for the motives of abelian varieties ([S])
The Picard and Albanese motives h1(X) and h2d−1(X), with X variety of dimension d ([S])
Motives over finite fields ([M2])