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Counting bare curves

Tobias Ekholm, Vivek Shende

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.

Counting bare curves

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.
Paper Structure (43 sections, 43 theorems, 129 equations, 2 figures)

This paper contains 43 sections, 43 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Methods and difficulties
  3. Curves, maps, and regularity conditions
  4. Notation for functional analytic norms
  5. Bordered, noded, surfaces
  6. Stable curves and maps
  7. Ghosts and bare maps
  8. Holomorphic polar coordinates
  9. Regularity conditions
  10. The configuration space of maps from curves with boundary
  11. Gluing
  12. Hyperbolic nodes
  13. Elliptic nodes
  14. Stable map moduli
  15. The bundle of complex anti-linear differentials and the $\bar{\partial}_{J}$-section
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $X$ be a Calabi-Yau 3-fold, and $L \subset X$ a (possibly empty) Lagrangian submanifold of Maslov index zero. Fix a class $d \in H_2(X, L)$, and choose some integer $\chi_{\min}$. We write $\mathbf{Z}$ for the HWZ-GW configuration space of maps realizing class $d$ with possibly disconnected doma

Figures (2)

  • Figure 1: Induced perturbations in (1c). Left, smoothing of a rational bubble with two tori attached. Right, the same domain appearing when tori are attached directly. The induced perturbations are different, supports are indicated in grey, but close provided the gluing parameter cut off is supported at large gluing parameter.
  • Figure 2: The different contributions for (2). For $\epsilon_2^\beta$ sufficiently small the perturbation on the domain to the right is cut off and only the perturbation from the domain on the left contributes.

Theorems & Definitions (104)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4: \ref{['c : bare to GW']}
  • Remark 1.5
  • Remark 1.6
  • Example 2.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • ...and 94 more