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The Intermediate Extension, Vanishing Cycles, and Perverse Eigenspace of One

David B. Massey

Abstract

We prove a number of results involving the kernel of the identity minus the monodromy on the vanishing cycles.

The Intermediate Extension, Vanishing Cycles, and Perverse Eigenspace of One

Abstract

We prove a number of results involving the kernel of the identity minus the monodromy on the vanishing cycles.
Paper Structure (7 sections, 11 theorems, 47 equations)

This paper contains 7 sections, 11 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminary notation, definitions and results
  3. A Fundamental Exact Sequence
  4. Splitting
  5. The intermediate extension to $V(f)$
  6. The constant sheaf on affine space
  7. A question

Key Result

Proposition 3.1

${}^{\mu} H^k(j^![1]\mathbf P^\bullet)$ is possibly non-zero only for $k=-1, 0$, ${}^{\mu} H^k(j^*[-1]\mathbf P^\bullet)$ is possibly non-zero only for $k=0, 1$, and there are exact sequences and Thus, ${}^{\mu} H^{0}(j^*[-1]\mathbf P^\bullet)\cong\ker\{\operatorname{can}\}$, ${}^{\mu} H^{1}(j^*[-1]\mathbf P^\bullet)\cong{\operatorname{coker}}\{\operatorname{can}\}$, ${}^{\mu} H^{-1}(j^![1]\math

Theorems & Definitions (25)

  • Definition 2.1
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Proposition 5.1
  • proof
  • Remark 5.2
  • ...and 15 more