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Gopakumar-Vafa invariants associated to $cA_n$ singularities

Hao Zhang

Abstract

This paper describes Gopakumar-Vafa (GV) invariants associated to $cA_n$ singularities. We (1) generalize GV invariants to crepant partial resolutions of $cA_n$ singularities, (2) show that generalized GV invariants also satisfy Toda's formula and are determined by their associated contraction algebra, (3) give filtration structures on the parameter space of contraction algebras associated to $cA_n$ crepant resolutions with respect to generalized GV invariants, and (4) numerically constrain the possible tuples of GV invariants that can arise. We further give all the tuples that arise from GV invariants of $cA_2$ crepant resolutions.

Gopakumar-Vafa invariants associated to $cA_n$ singularities

Abstract

This paper describes Gopakumar-Vafa (GV) invariants associated to singularities. We (1) generalize GV invariants to crepant partial resolutions of singularities, (2) show that generalized GV invariants also satisfy Toda's formula and are determined by their associated contraction algebra, (3) give filtration structures on the parameter space of contraction algebras associated to crepant resolutions with respect to generalized GV invariants, and (4) numerically constrain the possible tuples of GV invariants that can arise. We further give all the tuples that arise from GV invariants of crepant resolutions.

Paper Structure

This paper contains 26 sections, 44 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Singular Invariants
  3. Restriction to Smooth Case
  4. Filtration
  5. Obstructions
  6. Preliminaries and Recap
  7. Algebraic Preliminaries
  8. Geometric Preliminaries
  9. Contraction Algebras
  10. Gopakumar--Vafa invariants
  11. Recap
  12. generalised GV Invariants of Crepant Partial Resolutions
  13. Generalised GV invariants
  14. Contraction algebra determines generalised GV invariants
  15. Classical case: known facts
  16. ...and 11 more sections

Key Result

Proposition 1.1

Let $\uppi$ be a crepant partial resolution of a $cA_n$ singularity with $m$ exceptional curves. For any $1 \leq s \leq t \leq m$, the following equality holds. In particular, $\operatorname{dim}_{\mathbb{C}}\Lambda_{\mathrm{con}}(\uppi)=\sum_{\upbeta} |\upbeta|^2 N_{\upbeta}(\uppi)$ where $|\upbeta| = \upbeta_1 + \dots + \upbeta_m$.

Theorems & Definitions (98)

  • Proposition 1.1: \ref{['thm:Toda']}, \ref{['rmk:ijtobeta']}
  • Theorem 1.2: \ref{['371']}, \ref{['rmk:ijtobeta']}
  • Theorem 1.3: \ref{['37']}, \ref{['rmk:ijtobeta']}
  • Corollary 1.4: \ref{['371']}, \ref{['rmk:ijtobeta']}
  • Theorem 1.5: \ref{['stra']}
  • Theorem 1.7: \ref{['thm:obs']}
  • Corollary 1.8: \ref{['prop:gv_cA2']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 88 more