Scholze's approach to the weight-monodromy conjecture

Justin Asher, March 5, 2024

Abstract.   In his PhD thesis, Peter Scholze introduced perfectoid spaces as a tool to prove cases of Pierre Deligne's weight–monodromy conjecture in mixed characteristic. Deligne had established the equal-characteristic case, and Scholze used Fontaine's tilting idea to transfer the result to geometrically connected, proper smooth varieties that are set-theoretic complete intersections in smooth projective toric varieties. More recently, Peter Wear proved the conjecture for smooth complete intersections in abelian varieties; this is not a proof for arbitrary abelian varieties or arbitrary subvarieties of them.