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Orbifold theta functions and mid-age invariants

Fenglong You

Abstract

We use the orbifold approach to study theta functions in intrinsic mirror symmetry. We introduce a new type of orbifold invariants for snc pairs, called mid-age invariants, and use these invariants to define orbifold invariants associated with the broken line type. Then, we define the orbifold theta functions as generating functions of orbifold invariants with mid-ages. We show that these orbifold theta functions are well-defined and satisfy the multiplication rule.

Orbifold theta functions and mid-age invariants

Abstract

We use the orbifold approach to study theta functions in intrinsic mirror symmetry. We introduce a new type of orbifold invariants for snc pairs, called mid-age invariants, and use these invariants to define orbifold invariants associated with the broken line type. Then, we define the orbifold theta functions as generating functions of orbifold invariants with mid-ages. We show that these orbifold theta functions are well-defined and satisfy the multiplication rule.
Paper Structure (20 sections, 17 theorems, 153 equations)

This paper contains 20 sections, 17 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Further directions
  5. The tropicalization of an snc pair
  6. Notion in tropical geometry
  7. Preliminary on orbifold Gromov--Witten theory
  8. The orbifold pairing
  9. Orbifold Gromov--Witten theory of multi-root stacks
  10. The degeneration formula for orbifold Gromov--Witten theory
  11. General case
  12. The degeneration formula for multi-root stacks
  13. Special cases
  14. Orbifold invariants with mid-ages
  15. Mid-age invariants for smooth pairs
  16. ...and 5 more sections

Key Result

Theorem 1.1

[=Theorem thm-mid-age-c] For $r\gg 1$ and a pair of positive integers $\{k_a,k_b\}$ that satisfies where $c\in \mathbb Z$. The genus zero cycle class of the root stack $X_{D,r}$ is constant in $r$ for sufficiently large $r$. Furthermore, there exists a positive integer $d_0$ such that this cycle class does not depend on the pair $(k_a,k_b)$ as long as $d_0<k_a\leq k_b$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2: =Theorem \ref{['thm-mid-age-mul']}
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: =Theorem \ref{['thm-indep-mid']}
  • Definition 1.6: =Definition \ref{['def-theta-mid-age']}
  • Theorem 1.7: =Theorem \ref{['thm-wdvv']}
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 32 more