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Hodge-Gromov-Witten theory

Jérémy Guéré

Abstract

We determine the all-genus Hodge-Gromov-Witten theory of a smooth hypersurface in weighted projective space defined by a chain or loop polynomial. In particular, we obtain the first genus-zero computation of Gromov-Witten invariants for hypersurfaces in non-Gorenstein ambiant spaces, where the convexity property fails. We extend it to any weighted projective hypersurface defined by an invertible polynomial.

Hodge-Gromov-Witten theory

Abstract

We determine the all-genus Hodge-Gromov-Witten theory of a smooth hypersurface in weighted projective space defined by a chain or loop polynomial. In particular, we obtain the first genus-zero computation of Gromov-Witten invariants for hypersurfaces in non-Gorenstein ambiant spaces, where the convexity property fails. We extend it to any weighted projective hypersurface defined by an invertible polynomial.

Paper Structure

This paper contains 16 sections, 21 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Hodge--Gromov--Witten theory
  3. (Relative) perfect obstruction theory for $\boldsymbol{X_1}$
  4. Regular specialization theorem
  5. Pull-backs from the regular family
  6. Ambient space
  7. Gromov--Witten cycles
  8. Torus action
  9. Equivariant quantum Lefschetz theorem
  10. $\boldsymbol{\mathbb{C}^*}$-localization versus torus localization
  11. Equivariant quantum Lefschetz formula
  12. Smooth hypersurfaces in weighted projective spaces
  13. Hodge--Gromov--Witten theory for chain polynomials
  14. Hodge--Gromov--Witten theory for loop polynomials
  15. Hodge--Gromov--Witten theory for invertible polynomials
  16. ...and 1 more sections

Key Result

Lemma 2.7

The regularized virtual cycle equals the Hodge--Gromov--Witten virtual cycle up to a sign. Precisely, we have the relation where $\lambda_g := c_\mathrm{top}(\mathbb{E})$ is the top Chern class of the Hodge bundle.

Theorems & Definitions (62)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 52 more