Comparison theorems in p-adic Hodge theory

Comparison theorems in p-adic Hodge theory

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Bhargav Bhatt, University of Michigan
Fine Hall 322

A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine's conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry --- specifically, derived de Rham cohomology --- and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way.