I will be running a weekly learning seminar on algebraic cycles and motives in Spring 2025.

If you are interested, let me know, fill out the Schej to help us decide a meeting time, and add yourself to the mailing list for announcements.

Content

Cohomology provides a way to translate problems from the foreboding world of algebraic geometry to the more accessible landscape of linear algebra. The theory of motives generalizes and unifies the various cohomology theories associated to an algebraic variety. Indeed, a (pure of weight $k$, irreducible) motive (over $\Q$ with coefficients in $\Q$) is a direct summand of the $k^{\text{th}}$ cohomology of a smooth projective algebraic variety.

An algebraic cycle is a formal linear combination of subvarieties of an algebraic variety. Grothendieck’s standard conjectures assert the existence of enough algebraic cycles to construct an interesting category of motives. (Algebraic cycles are also at the heart of the famously lucrative Hodge conjecture.)

We will probably start by covering portions of the following:

• Chapter 11 of Complex Geometry and Hodge Theory Vol. 1, by Claire Voisin
• Hodge Cycles on Algebraic Varieties, by Pierre Deligne (notes by James Milne)
• Algebraic Cycles and the Weil Conjectures, by Daniel Kleiman
• The Standard Conjectures, by Daniel Kleiman

Because the standard conjectures remain open, in practice people often replace the category of algebraic cycles with either the category of motivated cycles (due to Yves André) or the category of absolute Hodge cycles (due to Deligne). For example, we can’t prove that all Hodge classes on an abelian variety are algebraic (the Hodge conjecture for abelian varieties) but we do know that they are motivated. If we have time, I would like to discuss André’s category of motivated cycles.

Prerequisites

Depends on the participants! Currently, I don’t expect that much will be strictly necessary beyond the first-year courses. We will make occasional use of sheaves and complex manifolds (e.g. to the level of 2.1, 2.3, 4.1, and 4.3 of Voisin). I encourage participants to familiarize themselves with the Hodge and Lefschetz decompositions of complex algebraic geometry, as it may be difficult to appreciate some of the general theory without that motivation.

Schedule

TBD! Fill out the Schej to help decide :)

Because I have some upcoming travel, we may not have our first meeting until Week 3.

I will speak by default. If you would like to speak then please reach out and we can discuss scheduling, but there’s no pressure at all to do so.

Notes

Notes will be posted here on the day of each lecture.