The Gauß–Manin connection on the Hodge–Tate structures

Abstract

For each simplicial complex scheme X • the singular cohomology H ∗(X •) carries a canonical mixed Hodge structure. There exists a canonical homomorphism ∇ :H ∗(X •)→Ω 1 C/ Q ⊗H ∗(X •) , the Gauß–Manin connection. We show that there is a unique functorial connection on each mixed Hodge–Tate structure having certain properties of the Gauß–Manin connection. This connection is non-integrable in general, and therefore, its integrability is a non-trivial condition on the Hodge structure to be geometric.