Hodge theory over $\mathbf{C}((t))$

Abstract: I will describe some ways in which Hodge theory makes its way into the geometry of rigid-analytic varieties over $\mathbf{C}((t))$. Namely, such spaces have a "Betti realization", well-defined up to homotopy (joint work with Talpo), and their cohomology carries a mixed Hodge Structure (Steenbrink, Stewart-Vologodsky, Berkovich). The notion of "projective reduction" introduced by Li and studied by Hansen-Li is a good working analog of the Kaehler condition. In this case, Hodge symmetry holds, even though it fails in some cases over the $p$-adic numbers (Petrov). Moreover, there is a Riemann-Hilbert correspondence (work in progress), which should allow us to define variations of mixed Hodge structure in this context. All of these analogs rely on corresponding statements regarding the logarithmic special fiber of a semistable model. Open problems abound.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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