Zero-cycles on varieties over p-adic fields and Brauer groups

In this paper, we study the Brauer-Manin pairing of smooth proper varieties over local fields, and determine the $p$-adic part of the kernel of one side. We also compute the $A_0$ of a potentially rational surface which splits over a wildly ramified extension.Comment: 31 pages. The proof of the main result in the local case has been much simplified. The computations of 0-cycles on cubic surfaces have been modifie

Relative cycles with moduli and regulator maps

Let X be a separated scheme of finite type over a field k and D a non-reduced effective Cartier divisor on it. We attach to the pair (X, D) a cycle complex with modulus, whose homotopy groups - called higher Chow groups with modulus - generalize additive higher Chow groups of Bloch-Esnault, R\"ulling, Park and Krishna-Levine, and that sheafified on $X_{Zar}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1. When X is smooth over k and D is such that $D_{red}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El-Zein's explicit construction. This is used to define a natural regulator map from the relative motivic complex of (X,D) to the relative de Rham complex. When X is defined over $\mathbb{C}$, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch's regulator from higher Chow groups. Finally, when X is moreover connected and proper over $\mathbb{C}$, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J^r_{X|D}$ of the pair (X,D). For r= dim X, we show that $J^r_{X|D}$ is the universal regular quotient of the Chow group of 0-cycles with modulus.Comment: 46 pages. Final version: Section 9 added and material rearranged. To appear in Journal of the Inst. of Math. Jussie

Lefschetz theorem for abelian fundamental group with modulus

We prove a Lefschetz hypersurface theorem for abelian fundamental groups allowing wild ramification along some divisor. In fact, we show that isomorphism holds if the degree of the hypersurface is large relative to the ramification along the divisor.Comment: 10 pages, final versio