## Deligne on cohomology of algebraic varieties

Posted by Andreas Holmstrom on August 12, 2009

A week ago, Deligne gave a talk at the Newton Institute on cohomology of algebraic varieties, and this is now available to watch here in a number of formats. The talk is a very nice and accessible introduction to the classical cohomology theories in algebraic geometry, and gives some hint of the rich arithmetic structures they carry, such as Galois actions and Hodge structures. He also touches at things like periods and derived categories, while all the time staying at a fairly elementary level.

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## T. said

Illusie on Grothendieck and his seminar: http://www.math.uchicago.edu/~mitya/langlands/Illusie.wav

## EE said

This may be a stupid question. But I didn’t quite get what he mentions about derived categories in the talk. It almost seems he didn’t mention anything beyond a sideways reference to the phrase.

## homotopical said

In some sense he talks indirectly about derived categories when he talks about the significance of having explicit complexes which compute the cohomology groups, rather than having just the cohomology groups themselves. The most natural language for studying such complexes would be derived categories.

## EE said

Another question: What do you mean by a period? Whenever you integrate a differential form on a circle or a cycle, do you call the resulting thing a period? In Ahlfors’ book, he seems to call any integral of a meromorphic function on a close path as a period. Then there are periods of modular forms, and periods in the sense of Kontsevich-Zagier. What exactly is the meaning of the term period?

## EE said

I mean, integral of meromorphic function on a *closed* path. Not close path. Sorry.

## homotopical said

Ah, good question. In the setting of Deligne’s talk I think a period (of a variety V) would be any complex number occurring as an entry in the matrix representing the comparison isomorphism between de Rham and Betti cohomology (of this variety). This matrix must be taken with respect to some choice of bases for the original Q-vector spaces. I believe the Kontsevich-Zagier definition of a period includes all such matrix entries, and also all periods of modular forms. I am not sure about the precise relation between the different concepts though. I had a quick look at the original preprint of Kontsevich and Zagier, available here – it looks very readable and I hope to read it in more detail as soon as I have time.

## homotopical said

For periods in the sense of such matrix entries a good reference is the last part of the book Une introduction aux motifs, by André (assuming you read French).

## Thomas said

The famous survey by Kontsevich and Zagier is

here, a detailed exposition of periods and deRahm cohomologyhere, a further survey by Andréhere.## EE said

This video was a good one. There seem to be a good number of other videos in the Cambridge website. I watched the other video of Deligne, on Drinfel’d counting. Do you have any other video suggestions, which are especially noteworthy among the long list there?

## homotopical said

I haven’t had time to watch any others, but there are several that I would like to watch. If you like Deligne, there was a talk by Knop about “Representations of when t is not a natural number”, link here. As discussed on the Secret Blogging Seminar already, this is something Deligne has been thinking about recently.

There were also lots of interesting talks in the recent workshop on nonabelian fundamental groups in arithmetic geometry, all available here. Unfortunately I was away from Cambridge that week at another conference in Germany, so I missed all these talks. I hope to see Minhyong Kim’s talk on Galois theory and Diophantine geometry some time, but he also wrote up a really nice 19-page piece of exposition – see the 2nd of the 2 pdf files here.

## Thomas said

There was a

thread in the n-category cafeon Minhyong Kim’s ideas.## EE said

It seems that much of this workshop you mention is oriented towards the “Grothendieck’s section conjecture”.

## Thomas said

Hereis a description of that conjecture.## EE said

Oh, this conjecture is already proved by the three authors, according to the source you mention. Then why so much fuss about it?

## Thomas said

The article is about several conjectures, the

section conjectureis still unsolved.