Motives Today

Hotel Villa Madruzzo, Cognola (Trento), Italy

June, 2019


Archivio APT Trento - foto Hufton&Crow (4) - Muse Museo delle Scienze

MUSE - Archivio APT Trento – Foto Hufton & Crow


Scientific Organizers

  • Enrico Arbarello (Roma)
  • Gilberto Bini (Milano)
  • Claudio Fontanari (Trento)

Format and speakers

This school consists of three courses (of about 6 hours each, to be spread on five days) delivered by:

Program: titles and abstracts of the courses

 Click here to see the program.

Giuseppe Ancona: Motives over finite fields and over the complex numbers

We will discuss examples where questions on algebraic cycles are easier over the complex numbers and others which are easier over finite fields. We will try to give intuitions on the differences between these two situations. A preliminary list of results we will treat is the following: the Kunneth conjecture for varieties over finite fields (Katz-Messing 1973), the Bloch Beilinson conjecture for elliptic curves over finite fields (Kahn 2002 and Jannsen 2007), standard conjectures for complex abelian varieties (Kleiman 1968 and Lieberman 1968), partial results on standard conjectures for abelian varieties over finite fields (Clozel 1999 and Ancona 2018).


Robert Laterveer: Motives and non-rationality

In the first half, I intend to give a gentle introduction to algebraic cycles, Chow groups and the various categories of pure motives. In the second half, I will give an overview of results of the last few years concerning non-rationality of certain families of algebraic varieties (Voisin, Totaro, Colliot-Thelene-Pirutka, …). These results rely on Chow-theoretic techniques, in particular the notion of “integral decomposition of the diagonal”.


Charles Vial: Weight decompositions for Chow motives

Some 25 years ago, Jacob Murre introduced the notion of Chow-Kuenneth decomposition, or weight decomposition, for Chow motives and conjectured that the motive of every smooth projective variety admits such a decomposition, and also conjectured how such decompositions should behave. I will start by explaining the link between Murre’s conjectures, the existence of a conjectural Bloch-Beilinson filtration on the Chow ring of smooth projective varieties, Kimura’s finite-dimensionality conjecture, and the conservativity conjecture. After reviewing examples of varieties for which a weight decomposition does exist, I will more specifically focus on the case of abelian varieties, where weight decompositions with further structures exist (compatibility with the intersection product). Finally, I will suggest that the motives of hyperKaehler varieties admit similar weight decompositions as that of abelian varieties.

Confirmed Participants

Click here to see the list.


 For registering in the school please complete and submit our registration form to the Secretary of CIRM. Young researchers are encouraged to participate. It is possible to support the full board and lodging in a shared double bedroom at the hotel to some of them upon request. Unfortunately, it is not possible to reimbourse travel expenses. People interested in having such a support should send a short CV to the Secretary of CIRM together with the registration form.

No registration fee will be charged.


Deadline for registration: April 15, 2019.

 Logistic Information

The school will be held in the conference hall of the Hotel Villa Madruzzo in Cognola (Trento). Participants will be lodged in the same hotel. Arrival day is Sunday, June 2nd  in the afternoon/evening. Lectures will start in the morning of Monday, June 3rd  and end at lunchtime of Friday, June 7th . Departure day is Friday, June 7th  after lunch.

The cost for the full board and lodging at the Hotel Villa Madruzzo is:

  • 106,50 EUR per night in a single bedroom (tourist tax included),
  • 81,50 EUR per night per person in a double bedroom (tourist tax included).

Please click here to find Information to reach the Hotel Villa Madruzzo.