Virtual Moduli Cycles and Gromov-Witten Invariants of Noncompact Symplectic Manifolds

By Guangcun Lu. 2003

150 citations recorded by the source corpus.

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their variants. The more solutions of WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To realize these purposes we further develop the language introduced by Liu-Tian to describe the virtual moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ddf410ab98495d3e961ce6e18c1e516afb25b8f6

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In Applied Mathematics

1. J. Pardon
2. Junho Lee
3. Jianxun Hu
4. Mark McLean
5. Y. Ruan
6. Thomas H. Parker
7. Tian-Jun Li
8. Christopher Woodward

In Geometry and Topology

1. Christopher Woodward
2. J. Pardon
3. Junho Lee
4. Jianxun Hu
5. Mark McLean
6. Y. Ruan
7. Thomas H. Parker
8. Tian-Jun Li

Citations in the corpus

• A structure theorem for the Gromov-Witten invariants of Kähler surfaces (2006, methodology): “We will use the definitions and notation of [14] for stable maps and the Gromov-Witten invariants; these are based on the approach developed by Ruan-Tian [22] and Li-Tian [20].”
• QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I (2012, methodology): “We then use the theory of virtual fundamental classes developed by Behrend–Fantechi [3], based on earlier work of Li–Tian [30], to define gauged Gromov–Witten invariants.”
• Birational cobordism invariance of uniruled symplectic manifolds (2006, methodology): “There are several ways to extract invariants [FO, LT ,S1,R1], the one we use is the virtual neighborhood method in [R1].”
• Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms (2010, unknown): “Theorem 1.2 ([Lu06, Corollary 1.19]).”
• Reeb orbits and the minimal discrepancy of an isolated singularity (2014, methodology): “Lemma 2.14 hold? The idea here is to use GromovWitten theory. Genus 0 Gromov-Witten invariants for general symplectic manifolds have now been defined in many different ways: [FO99], [CM07], [Hof11] and [LT98b]. Earlier work for special symplectic manifolds such as symplectic manifolds of real dimensio”
• Contact homology and virtual fundamental cycles (2015, methodology): “ensional reductions to construct virtual fundamental cycles has been applied to moduli spaces of pseudo-holomorphic curves by many authors, including Fukaya–Ono [FO99] (Kuranishi structures), Li–Tian [LT98a], Liu–Tian [LT98b], Ruan [Rua99], Fukaya–Oh– Ohta–Ono[FOOO09a,FOOO09b,FOOO12,FOOO16b,FOOO15,F”