Counting bare curves

TL;DR

This paper develops a new perturbation scheme for counting curves in Calabi-Yau threefolds, leading to a reduced Gromov-Witten theory that isolates zero-area components and relates to classical invariants.

Contribution

It introduces a novel perturbation construction that ensures compactness and transversality for solution spaces, enabling a well-defined reduced Gromov-Witten invariant and skein-valued curve counting.

Findings

Constructed perturbations vanish on zero symplectic area components.

Proved the compactness and transversality of solution loci for generic perturbations.

Established the relation between the reduced and classical Gromov-Witten invariants.

Abstract

We construct a class of perturbations of the Cauchy-Riemann equations for maps from curves to a Calabi-Yau threefold. Our perturbations vanish on components of zero symplectic area. For generic 1-parameter families of perturbations, the locus of solution curves without zero-area components is compact, transversely cut out, and satisfies certain natural coherence properties. For curves without boundary, this yields a reduced Gromov-Witten theory in the sense of Zinger. That is, we produce a well defined invariant given by counting only maps without components of zero symplectic area, and we show that this invariant is related to the usual Gromov-Witten invariant by the expected change of variables. For curves with boundary on Maslov zero Lagrangians, our construction provides an `adequate perturbation scheme' with the needed properties to set up the skein-valued curve counting, as axiomatized in our previous work. The main technical content is the construction, over the Hofer-Wysocki-Zehnder Gromov-Witten configuration spaces, of perturbations to which the `ghost bubble censorship' argument can be applied. Certain local aspects of this problem were resolved in our previous work. The key remaining difficulty is to ensure inductive compatibilities, despite the non-existence of marked-point-forgetting maps for the configuration spaces.