Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Toen on homotopy types of algebraic varieties

Posted by Andreas Holmstrom on October 4, 2009

Two recent conversations both reminded me of a short note of Toen, with the title Homotopy types of algebraic varieties. This note explains in only eight pages several exciting ideas, which I find interesting especially because they point towards some possible future interactions between homotopy theory and arithmetic geometry.

He starts out by a conceptual discussion of classical Weil cohomology theories, which were discussed in this earlier post. The idea is that the cohomological invariants should be refined into some notion of “homotopy type”, the relation being somewhat analogous to the relation in algebraic topology, between the cohomology and the homotopy type of, say, a CW complex. He then goes on to sketch how this can be made precise, using the language of stacks and schematic homotopy types.

Towards the end of the paper, he speculates about a possible connection between the homotopy types of a variety and rational points on the variety. The study of rational points is one of the main themes of arithmetic geometry, as they correspond to integer or rational solutions of (systems of) polynomial equations. The famous section conjecture of Grothendieck, explained in these notes of Kim, is supposed to give a conceptual proof of Faltings’ theorem, aka the Mordell conjecture. Faltings’ theorem says that a curve of genus at least 2, defined over \mathbb{Q}, only has a finite number of rational points. Toen suggests a generalization of the section conjecture to higher-dimensional varieties, using his notion of homotopy types.

Another main theme of arithmetic geometry is L-functions of various kinds. To any variety over \mathbb{Q}, one can attach an L-function, which encodes lots of information about the arithmetic properties of the variety. Many outstanding conjectures in number theory are formulated in terms of these functions, for example the Riemann hypothesis and the Birch-Swinnerton-Dyer conjecture, as well as many other more accessible conjectures. The building blocks of an L-function are precisely the various Weil cohomology groups, and one could speculate about the significance of Toen’s conceptual approach to Weil cohomologies. Could it give us some new tools for approaching questions about L-functions? Or could it be that L-functions are not the right thing to consider, but that the notion of homotopy types could lead us to some better objects of study?

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