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Register a new userOn the homotopy exact sequence for Noris fundamental group
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Unlike Grothendiecks etale fundamental group, Noris fundamental group does not fulfill the homotopy exact sequence in general. We give necessary and sufficient conditions which force exactness of the sequence.
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We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of Fulton-MacPherson. We import the tools of Fultons intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess intersections, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor-Witt spectrum recently constructed by Deglise-Fasel, we get a bivariant theory extending the Chow-Witt groups of Barge-Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss-Bonnet formula, computing Euler characteristics in the motivic homotopy category.
Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.
Given a smooth and separated K(pi,1) variety X over a field k, we associate a cycle class in etale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute Galois group of k. We discuss the algebraicity of this class in the case of curves over p-adic fields, and deduce in particular a new proof of Stixs theorem according to which the index of a curve X over a p-adic field k must be a power of p as soon as the natural map from the arithmetic fundamental group of X to the absolute Galois group of k admits a section. Finally, an etale adaptation of Beilinsons geometrization of the pronilpotent completion of the topological fundamental group allows us to lift this cycle class in suitable cohomology groups.
Let $X$ and $Y$ be nonsingular projective varieties over an algebraically closed field $k$ of positive characteristic. If $X$ and $Y$ are birational, we show their $S$-fundamental group schemes are isomorphic.
Relying on a notion of numerical effectiveness for Higgs bundles, we show that the category of numerically flat Higgs vector bundles on a smooth projective variety $X$ is a Tannakian category. We introduce the associated group scheme, that we call the Higgs fundamental group scheme of $X$, and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.
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