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We describe singular homology of a manifold $X$ via simplices $\sigma:\Delta_d\to X$ that satisfy Stokes' formula with respect to all differential forms. The notion is geared to the case of tame geometry (definable manifolds with respect to an o-minimal structure), where it gives a description of the period pairing with de Rham cohomology via definable $\sigma$'s., Comment: 22 pages

We define a category of motives for semi-algebraic spaces and show that it is trivial. This implies that there is no good extension of algebraic de Rham cohomology to semi-algebraic spaces compatible with the period isomorphism.

In this mostly expository note we explain how Nori's theory of motives achieves the aim of establishing a Galois theory of periods, at least under the period conjecture. We explain and compare different notions periods, different versions of the period conjecture and view the evidence by explaining the examples of Artin motive, mixed Tate motives and 1-motives.

We study four fundamental questions about $1$-periods and give complete answers. 1) We give a necessary and sufficient for a period integral to be transcendental. 2) We give a qualitative description of all $\overline{\mathbf{Q}}$-linear relations between $1$-periods, establishing Kontsevich's period conjecture in this case. 3) Periods may vanish and we determine all cases when this happens. 4) For a fixed $1$-motive, we derive a general formula for the dimension of its space of periods in the spirit of Baker's theorem., Comment: This is the final draft and very close to the published version of the monograph, but not identical. In particular layout and page numbers do not agree. To appear: Cambridge Tracts in Mathematics 227, Cambridge University Press, May 2022. Copyright Annette Huber and Gisbert W\"ustholz

We construct a tensor product on Freyd's universal abelian category attached to an additive tensor category or a tensor quiver and establish a universal property. This is used to give an alternative construction for the tensor product on Nori motives., Comment: Revised & updated version, 23 pages

This paper continues our study of the sheaf associated to K\"ahler differentials in the cdh-topology and its cousins, in positive characteristic, without assuming resolution of singularities. The picture for the sheaves themselves is now fairly complete. We give a calculation $\mathcal{O}_{cdh}(X) \cong \mathcal{O}(X^{sn})$ in terms of the seminormalisation. We observe that the category of representable cdh-sheaves is equivalent to the category of seminormal varieties. We conclude by proposing some possible connections to Berkovich spaces, and $F$-singularities in the last section. The tools developed for the case of differential forms also apply in other contexts and should be of independent interest., Comment: 37 pages

We generalize the definition of the polylogarithm classes to the case of commutative group schemes, both in the sheaf theoretic and the motivic setting. This generalizes and simplifies the existing cases., Comment: 36 pages

We prove a canonical Kuenneth decomposition of the relative motive with rational coefficients of a smooth commutative group scheme over a noetherian finite dimensional base. This paper is a follow-up of "On the motive of a commutative algebraic group" arXiv:1312.4171, Comment: final version, minor modifications

This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than K\"ahler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdh-topology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with K\"ahler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities. A number of examples highlight the difficulties that arise with torsion forms and with alternative candiates., Comment: The new version contains a considerable improvement on the earlier version. Following the remark of a referee, we are now able to show cdh-descent for differential forms on smooth varieties

We prove a canonical Kunneth decomposition for the motive of a commutative group scheme over a field. Moreover, we show that this decomposition behaves under the group law just as in cohomology. We also deduce applications of the decomposition to the existence of a weight filtration, computation of any Weil cohomology theory and study of 1-motives., Comment: final version, minor modifications

We study sheaves of differential forms and their cohomology in the h-topology. This allows to extend standard results from the case of smooth varieties to the general case. As a first application we explain the case of singularities arising in the Minimal Model Program. As a second application we consider de Rham cohomology., Comment: 29 pages, added references

These are extended abstracts from an series of lectures in 1997. The text has not been updated since then. We explain the construction of the motivic polylog as published in Annette Huber, J\"org Wildeshaus, Classical Motivic Polylogarithm According to Beilinson and Deligne, Doc.Math.J.DMV 3 (1998) 27-133. The main application is a comparison result for cyclotomic elements needed in the proof of the Tamagawa number conjecture of Bloch and Kato for Dirichlet characters. The exposition concentrates on the Hodge theoretic part of the story.

We show that the spectrum of Kontsevich's algebra of formal periods is a torsor under the motivic Galois group for mixed motives over the rational numbers. This assertion is stated without proof by Kontsevich and originally due to Nori. In a series of appendices, we also provide the necessary details on Nori's category of motives.

This is a contribution to the ICM 2002. We explain the relation between the (equivariant) Bloch-Kato conjecture for special values of L-functions and the Main Conjecture of (non-abelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how "twisting" in the non-abelian case would allow to reduce the general conjecture to the case of number fields. This is one the main motivations for a non-abelian Main Conjecture., Comment: 13 pages

This is the completely revised version of math.AG/0101071., Comment: 56 pages, latex2e

The main new result is the computation of the degeneration of l-adic Eisenstein classes at the cusps. This is done by relating it to the degeneration of the elliptic polylog. These classes come from K-theory and their Hodge regulator can also be computed (see: Dirichlet motives via modula curves, on the K-theory server). This gives a new proof of a comparison conjecture of Bloch and Kato which was used in the proof of their Tamagawa number conjecture for the Riemann zeta-function. The paper contains appendices on the definition of the classical and elliptic polylog, its degeneration and the comparison to Eisenstein classes.