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Unipotent nearby cycles and nearby cycles over general bases

Andrew Salmon

Abstract

We show that under some conditions, two constructions of nearby cycles over general bases coincide. More specifically, we show that under the assumption of $Ψ$-factorizability, the constructions of unipotent nearby cycles over an affine space can be described using the theory of nearby cycles over general bases via the vanishing topos. In particular, this applies to nearby cycles of Satake sheaves on Beilinson-Drinfeld Grassmannians with parahoric ramification.

Unipotent nearby cycles and nearby cycles over general bases

Abstract

We show that under some conditions, two constructions of nearby cycles over general bases coincide. More specifically, we show that under the assumption of -factorizability, the constructions of unipotent nearby cycles over an affine space can be described using the theory of nearby cycles over general bases via the vanishing topos. In particular, this applies to nearby cycles of Satake sheaves on Beilinson-Drinfeld Grassmannians with parahoric ramification.
Paper Structure (8 sections, 14 theorems, 64 equations)

This paper contains 8 sections, 14 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Nearby cycles over general bases
  3. Review of nearby cycles over general bases
  4. Nearby cycles over an affine space
  5. Relation between unipotent nearby cycles and nearby cycles via the vanishing topos
  6. The definition of Achar and Riche
  7. Tame and unipotent nearby cycles via the vanishing topos
  8. Relation of the two definitions

Key Result

Proposition 2

For a map $f \colon X \rightarrow S$ be locally of finite type whose fibers have dimension at most $N$. Let $K$ be an étale, resp. pro-étale sheaf of $\Bbbk$-modules. Then $R^i(\Psi)_{t}^{s} K = 0$ for $i > 2N$.

Theorems & Definitions (32)

  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 4
  • Proposition 5
  • proof
  • Proposition 6
  • Conjecture 7
  • ...and 22 more