Singular Supports in Equal and Mixed Characteristics (1/4)

by Takeshi Saito (The University of Tokyo & IHES)

Thursday Sep 18, 2025, 10:30 AM → 12:30 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Beilinson defined the singular support of a constructible sheaf on a smooth scheme over a field as a closed conical subset on the cotangent bundle. He further proved its existence and fundamental properties, using Radon transform as a crucial tool. In first lectures, we formulate the definition in a slightly different but equivalent way, using an interpretation by Braverman--Gaitsgory of the local acycliciity. We also recall Beilinson's proof of existence.

In mixed characteristics, the theory is still far from complete. As a replacement of the cotangent bundle, we introduce the Frobenius--Witt cotangent bundle, that has the correct rank but defined only on the characteristic p fiber. Using it, we define  the singular support and its relative variant. Finally, we show that Beilinson's argument using the Radon transform gives a proof of the existence of the saturation of the relative variant.