Perfectoid Spaces and the Weight-Monodromy Conjecture for Complete Intersections in Toric Varieties

The Weight-Monodromy Conjecture, formulated by Deligne, posits a fundamental purity property for the -adic étale cohomology of algebraic varieties over local fields with bad reduction. Specifically, it asserts that the monodromy filtration on the cohomology groups coincides with the weight filtration up to a shift. While established in equal characteristic p by Deligne himself, the characteristic 0 (mixed characteristic) case remained largely open until the advent of perfectoid geometry. This monograph details the resolution of the Weight-Monodromy Conjecture for complete intersections in proper smooth toric varieties, following the breakthrough methods of Peter Scholze. We systematically develop the theory of perfectoid spaces, the tilting equivalence between perfectoid fields of mixed characteristic and characteristic p, and the almost purity theorem. We then demonstrate how these tools allow for the transfer of the conjecture from p-adic fields to local fields of characteristic p, where Deligne's results apply. Central to this exposition is the approximation lemma, which permits the deformation of hypersurfaces in characteristic 0 to their tilted counterparts in characteristic p without altering their étale cohomology.