Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An ideal proof for Fujisawa's result and its generalization

Yukiyoshi Nakkajima

TL;DR

This work provides a purely algebraic generalization of Fujisawa's theorem for log de Rham cohomology in SNCL settings, avoiding analytic constraints like properness and Kählerness. By developing and leveraging Hirsch (and PD-Hirsch) extensions, the authors construct a Hirsch-extended complex that can be compared to the log de Rham complex. They prove a filtered isomorphism between the Hirsch-extended complex and the log de Rham complex, generalizing to coefficients via locally nilpotent integrable connections, and establish functoriality and pushforward results in the derived category. The results yield a simpler, more general proof of the Fujisawa-type isomorphism and extend its applicability to analytic families of log points and various coefficient systems, with contravariant functoriality and base-change compatibility.

Abstract

We give a generalization of Fujisawa's theorem in [F]. Our proof of the generalized theorem is purely algebraic and it is simpler than his proof.

An ideal proof for Fujisawa's result and its generalization

TL;DR

This work provides a purely algebraic generalization of Fujisawa's theorem for log de Rham cohomology in SNCL settings, avoiding analytic constraints like properness and Kählerness. By developing and leveraging Hirsch (and PD-Hirsch) extensions, the authors construct a Hirsch-extended complex that can be compared to the log de Rham complex. They prove a filtered isomorphism between the Hirsch-extended complex and the log de Rham complex, generalizing to coefficients via locally nilpotent integrable connections, and establish functoriality and pushforward results in the derived category. The results yield a simpler, more general proof of the Fujisawa-type isomorphism and extend its applicability to analytic families of log points and various coefficient systems, with contravariant functoriality and base-change compatibility.

Abstract

We give a generalization of Fujisawa's theorem in [F]. Our proof of the generalized theorem is purely algebraic and it is simpler than his proof.

Paper Structure

This paper contains 4 sections, 20 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Recall on the Hirsch extension
  3. The key result
  4. Main Result

Key Result

Theorem 1.1

Assume that the irreducible components of $\overset{\circ}{X}$ are proper and Kähler. Then the morphism (ali:kckffsx) induces the following isomorphism of cohomologies with Hodge filtrations:

Theorems & Definitions (39)

  • Theorem 1.1: fut
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1: nhi
  • Proposition 2.2: nhi
  • Corollary 2.3: nhi
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 29 more