Standard conjectures in model theory, and categoricity of comparison isomorphisms

TL;DR

This paper proposes two conjectures linking etale cohomology and fundamental groups in algebraic geometry to categoricity principles in model theory, suggesting uniqueness and factorization properties of these invariants.

Contribution

It introduces novel conjectures connecting model-theoretic categoricity with algebraic geometry invariants like etale cohomology and fundamental groups, and explores related evidence.

Findings

Some special cases relate to Grothendieck standard conjectures.

Evidence suggests connections to motivic Galois groups.

Conjectures imply uniqueness of comparison isomorphisms.

Abstract

We formulate two conjectures about etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique Z-form of the etale cohomology of complex algebraic varieties, up to Aut(C)-action on the source category; put differently, each comparison isomorphism between Betti and etale cohomology comes from a choice of a topology on C. Another conjecture says that each functor to groupoids from the category of complex algebraic varieties which is similar to the topological fundamental groupoid functor, in fact factors through it, up to a field automorphism of the complex numbers acting on the category of complex algebraic varieties. We also try to present some evidence towards these conjectures, and show that some special cases seem related to Grothendieck standard conjectures and conjectures about motivic Galois group.