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Gromov-Witten theory of root gerbes I: structure of genus $0$ moduli spaces

Elena Andreini, Yunfeng Jiang, Hsian-Hua Tseng

Abstract

Let $X$ be a smooth complex projective algebraic variety. Given a line bundle $\mathcal{L}$ over $X$ and an integer $r>1$ one defines the stack $\sqrt[r]{\mathcal{L}/X}$ of $r$-th roots of $\mathcal{L}$. Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L}/X}$. Our main results are explicit constructions of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L}/X}$ starting from moduli stack of genus $0$ stable maps to $X$. As a consequence, we prove an exact formula expressing genus $0$ Gromov-Witten invariants of $\sqrt[r]{\mathcal{L}/X}$ in terms of those of $X$.

Gromov-Witten theory of root gerbes I: structure of genus $0$ moduli spaces

Abstract

Let be a smooth complex projective algebraic variety. Given a line bundle over and an integer one defines the stack of -th roots of . Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus twisted stable maps to . Our main results are explicit constructions of moduli stacks of genus twisted stable maps to starting from moduli stack of genus stable maps to . As a consequence, we prove an exact formula expressing genus Gromov-Witten invariants of in terms of those of .

Paper Structure

This paper contains 22 sections, 28 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Twisted stable maps
  4. Root gerbes
  5. Line bundles over twisted curves
  6. Normalization of twisted curves
  7. Logarithmic geometry and twisted curves
  8. The stack of twisted curves
  9. Moduli of twisted stable maps to root gerbes
  10. Components of moduli stack
  11. Admissible vectors
  12. The stacks $\mathfrak{M}_{g,n,\beta}$ and $\mathfrak{M}^{tw}_{g,n,\beta}$
  13. Boundary of $\mathfrak{M}_{0,n,\beta}$
  14. Log structure on ${\mathfrak M}_{0,n,\beta}$.
  15. The stack $\mathfrak{Y}_{0,n,\beta}^{\vec{g}}$
  16. ...and 7 more sections

Key Result

Proposition 2.3

The stack $\sqrt[r]{{\mathcal{L}}/X}$ is the quotient stack $[{\mathcal{L}}^{\times}/\mathbb{C}^*]$, where ${\mathcal{L}}^{\times}$ is the principal $\mathbb{C}^*$-bundle obtained by deleting the zero section of ${\mathcal{L}}$, and $\mathbb{C}^*$ acts on ${\mathcal{L}}^\times$ via $\lambda \cdot z=

Theorems & Definitions (47)

  • Definition 2.1: AV, Definition 4.1.2
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6: Ca, Corollary 2.12
  • Lemma 2.7: Ca, Corollary 2.13
  • Remark 2.8
  • Lemma 2.9: See Ch06, Theorem 3.2.3
  • Remark 2.10
  • ...and 37 more