Gromov-Witten invariants and pseudo symplectic capacities

By Guangcun Lu. 2001

65 citations recorded by the source corpus.

We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic uniruled manifolds (including the uniruled manifolds in algebraic geometry) and also show that any closed symplectic submanifold of codimension two in any symplectic manifold has a small neighborhood whose Hofer-Zehnder capacity is less than a given positive number. Finally, we give two results on symplectic packings in Grassmannians and on Seshadri constants.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cf8efdf279c335341582fb617dc237f9fed8cf35

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In Geometry and Topology

1. F. Schlenk
2. D. Mcduff
3. Michael Usher
4. E. Shelukhin
5. A. Kislev
6. J. Latschev
7. Jianxun Hu
8. Y. Ruan

In Mathematical Physics

1. E. Shelukhin
2. Michael Usher
3. A. Kislev
4. D. Mcduff
5. F. Schlenk
6. Tian-Jun Li
7. J. Latschev
8. Y. Ruan

Citations in the corpus

• Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms (2010, unknown): “Theorem 1.2 ([Lu06, Corollary 1.19]).”
• Birational cobordism invariance of uniruled symplectic manifolds (2006, background): “There is also the notion of genus g uniruled in [ Lu2 ].”
• The Gromov width of 4-dimensional tori (2011, background): “Computations and estimates of the Gromov width for various examples can be found in [4, 5, 6, 15, 18, 19, 21, 22, 29, 34, 38, 41, 42].”
• Bounds on spectral norms and barcodes (2018, background): “the argument for case n= 1 of Lemma 45, is 1 4 (M;L); and 1 4 is strictly smaller than the upper bound n 2n+2 for n>1:Somewhat similarly, in the case of HPnˆGr(2;2n+ 2) the explicit embedding of [72] of a ball of capacity A= !([CP1]) adapted in such a way as to intersect L= HPnprecisely along its”
• Hofer's metrics and boundary depth (2011, methodology): “It is shown in [Lu06, Corollary 1.19] (using an argument that essentially dates back to [HV92]) that any closed symplectic mani- fold which admits a nonvanishing genus-zero Gromov–Witten invariant counting pseudoholomorphic spheres that pass through two generic points has finite π1-sensitive…”
• THE SHARP ENERGY-CAPACITY INEQUALITY (2008, unknown): “30 of [14], which establishes Corollary 1.”