Symplectic genus, minimal genus and diffeomorphisms

By Banghe Li, Tian-Jun Li. 2001

77 citations recorded by the source corpus.

In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least -1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine the orbits of the diffeomorphism group on the set of classes represented by embedded spheres of square at least -1 in any 4-manifold admitting a symplectic structure.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ecebda18608f2c800a5cbfb14d612903e33b9c5e

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Symplectic embeddings of 4‐dimensional ellipsoids (2008, background): “By Theorems B and C in Li–Li [12], given any symplectic form ω with canonical class K, each E ∈ EK(Xk) can be represented by an ω-symplectically embedded −1 sphere.”
The embedding capacity of 4-dimensional symplectic ellipsoids (2009, methodology): “After further work by Biran [2] and McDuff [14], the structure of this cone was finally elucidated in Li–Liu [13] and Li– Li [12].”
The Hofer conjecture on embedding symplectic ellipsoids (2010, background): “Following Li–Li [8], consider the Cremona transformation Cr that acts on tuples by Cr(d;m) = (d′;m′), where m′j = mj for j ≥ 4 and d′ = 2d− (m1 +m2 +m3), m′i = d− (mj +mk) for {i, j, k} = {1, 2, 3}.”
Symplectic embeddings from concave toric domains into convex ones (2014, unknown): “Then by a result of Li-Li [19], E is an exceptional class for any symplectic form on CP 2#pm`nqCP 2.”
Quantitative embedded contact homology (2010, background): “2], results of [15, 18] imply that one can replace the inequalities (ii) above by a certain subset thereof.”