Syntomic cohomology and p-adic regulators for varieties over p-adic fields

We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over p-adic rings extends uniquely to a cohomology theory for varieties over p-adic fields that satisfies h-descent. This new cohomology -syntomic cohomology -is a Bloch-Ogus cohomology theory, admits a period map to tale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the tale side and is related to the Bloch-Kato exponential map. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soul's tale regulators land in the potentially semistable Selmer groups.