Orbifold Gromov–Witten theory of the symmetric product of 𝒜<sub><i>r</i></sub>

Cheong, Wan Keng; Gholampour, Amin

doi:10.2140/gt.2012.16.475

Cited by 4 publications

(2 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [58,59], the authors compute the relative orbifold DT invariants and relative orbifold GW invariants of [C 2 /Z n+1 ]× P 1 respectively and prove the crepant resolution conjecture relating with their corresponding theories of A n ×P 1 , where A n → C 2 /Z n+1 is a crepant resolution. The GW/DT/Hilb/Sym correspondence is proved for the surface A n in [35,38,39,11] and also for the stacky quotient [C 2 /Z n+1 ] in [58,59], see the beautiful diagram in [59, Section 6.2] for more details. We will explore the computation of the relative orbifold PT invariants of [C 2 /Z n+1 ] × P 1 elsewhere as in DT case [58] and consider putting this orbifold PT theory in the GW/DT/Hilb/Sym correspondence for…”

Section: ̺+̺0mentioning

confidence: 99%

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Lin

2021

Preprint

View full text Add to dashboard Cite

We construct relative moduli spaces of semistable pairs on a family of projective Deligne-Mumford stacks. We define moduli stacks of stable orbifold Pandharipande-Thomas pairs on stacks of expanded degenerations and pairs, and then show they are separated and proper Deligne-Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande-Thomas invariants. Contents 1. Introduction 1 2. Preliminary 5 2.1. Projective Deligne-Mumford stacks 5 2.2. Stacks of expanded degenerations and pairs 7 2.3. Admissibility of sheaves and its numerical criterion 12 3. Relative moduli spaces of semistable pairs 13 3.1. Semistable pairs on the family of projective Deligne-Mumford stacks 13 3.2. Boundedness of the family of semistable pairs 14 3.3. Construction of relative moduli spaces of semistable pairs 16 4. Perfect relative obstruction theory and virtual fundamental class 20 5. Moduli of stable orbifold Pandharipande-Thomas pairs 21 5.1. Stable orbifold PT pairs 21 5.2. Moduli of stable orbifold PT pairs 23 5.3. Moduli of stable orbifold PT pairs with fixed Hilbert homomorphism 24 5.4. Properties of the moduli of stable orbifold PT pairs 25 6. Degeneration formula 29 6.1. Decomposition of central fibers 29 6.2. Absolute and relative orbifold Pandharipande-Thomas theory 32 6.3. Cycle version of degeneration formula 35 6.4. Numerical version of degeneration formula 37 References 39

show abstract

Section: ̺+̺0mentioning

confidence: 99%

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Lin

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Furthermore, we can even use the map L and the setting of this paper to compare the full Gromov-Witten theories of [Sym n (A r )] and Hilb n (A r ), where A r is the minimal resolution of the quotient variety C 2 /µ r+1 (µ r+1 is the group of (r + 1)-th roots of unity). We refer the reader to Cheong-Gholampour [10] and Maulik-Oblomkov [28] for more details.…”

Section: The Following Diagrammentioning

confidence: 99%

Strengthening the cohomological crepant resolution conjecture for Hilbert–Chow morphisms

Cheong

2012

Math. Ann.

Self Cite

View full text Add to dashboard Cite

Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym n (S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb n (S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism Hilb n (S) → Sym n (S) and yields a method of reconstructing the cup product for Hilb n (S) from the orbifold invariants of [Sym n (S)].

show abstract

Relative Orbifold Pandharipande–Thomas Theory and the Degeneration Formula

Lin

2022

International Mathematics Research Notices

View full text Add to dashboard Cite

We construct relative moduli spaces of semistable pairs on a family of projective Deligne–Mumford stacks. We define moduli stacks of stable orbifold Pandharipande–Thomas pairs on stacks of expanded degenerations and pairs and then show they are separated and proper Deligne–Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande–Thomas invariants.

show abstract

Orbifold Gromov–Witten theory of the symmetric product of 𝒜_r

Cited by 4 publications

References 18 publications

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Strengthening the cohomological crepant resolution conjecture for Hilbert–Chow morphisms

Relative Orbifold Pandharipande–Thomas Theory and the Degeneration Formula

Contact Info

Product

Resources

About

Orbifold Gromov–Witten theory of the symmetric product of 𝒜r

Cited by 4 publications

References 18 publications

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

Strengthening the cohomological crepant resolution conjecture for Hilbert–Chow morphisms

Relative Orbifold Pandharipande–Thomas Theory and the Degeneration Formula

Contact Info

Product

Resources

About

Orbifold Gromov–Witten theory of the symmetric product of 𝒜_r