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DT-GV correspondence on the Mukai-Umemura variety

Kiryong Chung, Joonyeong Won

Abstract

We compute Donaldson-Thomas(DT) invariants and their descendant invariants for the local Calabi-Yau 4-fold over the Mukai-Umemura variety via several localization formulas. Assuming that the genus-one Gopakumar-Vafa(GV) type invariants vanish, our computations verify the predictions of Cao, Maulik, and Toda.

DT-GV correspondence on the Mukai-Umemura variety

Abstract

We compute Donaldson-Thomas(DT) invariants and their descendant invariants for the local Calabi-Yau 4-fold over the Mukai-Umemura variety via several localization formulas. Assuming that the genus-one Gopakumar-Vafa(GV) type invariants vanish, our computations verify the predictions of Cao, Maulik, and Toda.
Paper Structure (13 sections, 17 theorems, 56 equations, 6 tables)

This paper contains 13 sections, 17 theorems, 56 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Motivation and result
  3. Preliminary
  4. Localization formulas for varieties with isolated fixed points
  5. Some facts in equivariant K-theory
  6. Geometric data of the variety $X$
  7. Local charts of the variety $X$ and its weights
  8. Torus fixed curves and its local equations
  9. Computation of DT-invariants
  10. Computation of the integration via localization formulas
  11. Equivariant Euler characteristics of fixed curves
  12. Integration of the primary and descendent insertions
  13. Proof of Conjecture \ref{['conjor']}

Key Result

Theorem 1.2

For $Y=\mathrm{Tot}(K_X)$ with the Mukai-Umemura variety $X$, the equality eq:mov1 in Conjecture conjor holds for $\beta = 4[\mathrm{line}]$ whenever we assume that $n_{1, d}=0$ for $1\leq d\leq 4$.

Theorems & Definitions (38)

  • Conjecture 1.1: CMT18 and CT21
  • Theorem 1.2: $=$Theorem \ref{['pfmeeting1']}
  • Theorem 2.1: Virtual localization GP99
  • Remark 2.2: Smooth case AB84BV82
  • Theorem 2.3: Theorem 3.5 Tho92
  • Theorem 2.4: Tho93
  • Remark 2.5
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • ...and 28 more