Periods (ongoing casual writing)

This document is primarily meant to keep track of my organized thoughts in combination with my learning process. I am exploring the geometric interpretations of periods in both classical and p-adic Hodge theory. My ultimate goal is to give a coherent story of p-adic period integrals, period rings, periods as functors, and periods in geometrization (including period domains and period maps). I will subsequently discuss how deformations and local models fit into the picture.

Section 1 currently surveys periods in the classical complex setting. It explicitly details the Rees construction for filtered vector spaces and shows how a filtration is equivalent to a modification of a trivial bundle at a boundary point. While the Rees construction is widely known for algebras, its specific application to filtered vector spaces is a foundational idea in Simpson's nonabelian Hodge theory, which I will build upon here.

Section 2 transitions into the p-adic setting. Moving forward, I plan to introduce the D_B functor by pairing. I will also talk about how Galois Tate twists Q_p(n) translate into geometric Serre twists O(n) on the Fargues-Fontaine curve.

Section 3 will explore heuristics for global deformations and integral liftability. I am hoping to see to what extent the discrepancy between generic existence and integral realizability can provide a geometric way to isolate singular deformation directions.