Singular Supports in Equal and Mixed Characteristics - 4/4

Explore advanced concepts in algebraic geometry through this comprehensive lecture series on singular supports of constructible sheaves. Delve into Beilinson's foundational work defining singular support as closed conical subsets on cotangent bundles of smooth schemes over fields, examining his proof techniques utilizing Radon transforms. Study the reformulation using Braverman-Gaitsgory's interpretation of local acyclicity and understand the fundamental properties and existence proofs in equal characteristic settings. Investigate the challenges and developments in mixed characteristic theory, where complete understanding remains elusive. Learn about the innovative introduction of Frobenius-Witt cotangent bundles as replacements for traditional cotangent bundles in characteristic p fibers, despite their limited definition scope. Examine the construction of singular support and its relative variants in this mixed characteristic context. Analyze how Beilinson's Radon transform methodology provides existence proofs for the saturation of relative variants, bridging classical techniques with modern mixed characteristic challenges in algebraic geometry.