On Linear Part of Filled-Section in Splicing

TL;DR

This paper introduces a new filled-section in the splicing for Gromov-Witten theory, demonstrating its $C^1$ regularity using classical Banach analysis instead of polyfold theory.

Contribution

It defines a novel filled-section in the smooth model for GW theory and proves its $C^1$ regularity with traditional Banach analysis methods.

Findings

Established $C^1$ regularity of the new filled-section.

Provided an alternative proof framework avoiding polyfold theory.

Enhanced understanding of the smooth structure in GW theory.

Abstract

In this paper and its companion, we define a new filled-section in the splicing in the smooth model for GW theory, and prove that it is of class $C^1$ in the usual Banach analysis rather than in the frame work of polyfold theory.