Filtrations of the perverse sheaf of nearby cycles in the semi stable situation

TL;DR

The paper analyzes the perverse sheaf of nearby cycles $\Psi(\Lambda)$ in the strict semi-stable setting, describing its irreducible constituents and constructing filtrations induced by the stratification of the special fiber. It provides explicit Grothendieck-group formulas for the constituents, develops weight filtrations for the sheaves $j_{I,!}\Lambda_I$ and for $\Psi(\Lambda)$, and establishes the (non)split behavior of extensions within these filtrations. A nilpotent monodromy operator $N$ is constructed and studied, with kernels and socles tied to the stratification filtrations, and integral and mod-$l$ realizations are treated to show $N$ is nilpotent of order $r$ across $\overline{\mathbb{Q}}_l$, $\overline{\mathbb{Z}}_l$, and $\overline{\mathbb{F}}_l$. The results are framed as a bridge between semi-stable geometry and torsion phenomena, with connections to cohomology calculations in Lubin-Tate and related Shimura-type settings.

Abstract

In the strict semi stable reduction situation, we describe the various filtrations of the perverse sheaf of nearby cycles coming from the stratification of the special fiber and determine which extensions given by these filtrations, are split or not. Considering the similarity with the results of my paper proving the torsion freeness of the cohomology of Lubin-Tate spaces, it could be a good introduction before reading it.

Figures (3)

• Figure 1: Filtrations of $\Psi(\Lambda)$ when $r=4$.
• Figure 2: Filtration of stratification of $\Psi(\Lambda)$.
• Figure 3: Filtration (fig. on the left) and cofiltration (fig. on the right) of stratification of $\Psi(\Lambda)$ with $r=3$: the number correspond to the weights.

Theorems & Definitions (34)

• Theorem 2.1
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Lemma 3.5
• proof
• Lemma 3.6