Mathematics

Let (M, ω M ) be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian S 1 -action. We show that (M, ω M ) is S 1 -equivariant symplectomorphic to some Kähler Fano manifold (X, ω X ,J) with a certain holomorphic C ∗ -action. We also give a complete list of all such Fano manifolds and describe all semifree C ∗ -actions on them specifically.

Let (M, ω M ) be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian S 1 -action. We show that if the maximal and the minimal fixed component are both two dimensional, then (M, ω M ) is S 1 -equivariantly symplectomorphic to some Kähler Fano manifold (X, ω X ,J) equipped with a certain holomorphic Hamiltonian S 1 -action. We also give a complete list of all such Fano manifolds together with an explicit description of the corresponding S 1 -actions.

In this paper, we complete the classification of six-dimensional closed monotone symplectic manifolds admitting semifree Hamiltonian S 1 -actions. We also show that every such manifold is S 1 -equivariantly symplectomorphic to some Käahler Fano manifold with a certain holomorphic Hamiltonian circle action.

Given an exact symplectic cobordism (X,λ) between contact 3 -manifolds ( Y + , λ + ) and ( Y − , λ − ) with no elliptic Reeb orbits up to a certain action, we define a chain map from the embedded contact homology (ECH) chain complex of ( Y + , λ + ) to that of ( Y − , λ − ) , both taken with coefficients in Z/2Z . The map is defined by counting punctured holomorphic curves with ECH index 0 in the completion of the cobordism and new objects that we call ECH buildings, answering a question of Hutchings.

For subsets in the standard symplectic space ( R 2n , ω 0 ) whose closures are intersecting with coisotropic subspace R n,k we construct relative versions of their Ekeland-Hofer capacities with respect to R n,k , establish representation formulas for such capacities of bounded convex domains intersecting with R n,k , and also prove a product formula and a fact that the value of this capacity on a hypersurface S of restricted contact type containing the origin is equal to the action of a generalized leafwise chord on S .

We prove representation formulas for the coisotropic Hofer-Zehnder capacities of bounded convex domains with special coisotropic submanifolds and the leaf relation (introduced by Lisi and Rieser recently), study their estimates and relations with the Hofer-Zehnder capacity, give some interesting corollaries, and also obtain corresponding versions of a Brunn-Minkowski type inequality by Artstein-Avidan and Ostrover and a theorem by Evgeni Neduv.

We study coisotropic submanifolds of b -symplectic manifolds. We prove that b -coisotropic submanifolds (those transverse to the degeneracy locus) determine the b -symplectic structure in a neighborhood, and provide a normal form theorem. This extends Gotay's theorem in symplectic geometry. Further, we introduce strong b -coisotropic submanifolds and show that their coisotropic quotient, which locally is always smooth, inherits a reduced b -symplectic structure.

We compute the contribution of all multiple covers of an isolated rigid embedded holomorphic annulus, stretching between Lagrangians, to the skein-valued count of open holomorphic curves in a Calabi-Yau 3-fold. The result agrees with the predictions from topological string theory and we use it to prove the Ooguri-Vafa formula that identifies the colored HOMFLYPT invariants of a link with a count of holomorphic curves ending on the conormal Lagrangian of the link in the resolved conifold. This generalizes our previous work which proved the result for the fundamental color.

Let Λ ± = Λ + ∪ Λ − ⊂( R 3 , ξ std ) be a contact surgery diagram determining a closed, connected contact 3 -manifold ( S 3 Λ ± , ξ Λ ± ) and an open contact manifold ( R 3 Λ ± , ξ Λ ± ) . Following arXiv:0911.0026 and arXiv:1906.07228 we demonstrate how Λ ± determines a family α ϵ of standard-at-infinity contact forms on ( R 3 Λ ± , ξ Λ ± ) whose closed Reeb orbits are in one-to-one correspondence with cyclic words of composable Reeb chords on Λ ± . We compute the homology classes and integral Conley-Zehnder indices of these orbits diagrammatically using a simultaneous framing of all orbits naturally determined by the surgery diagram, providing a (typically non-canonical) Z -grading on the chain complexes underlying the "hat" version of contact homology as defined in arXiv:1004.2942. Using holomorphic foliations, algebraic tools for studying holomorphic curves in symplectizations of and surgery cobordisms between the ( R 3 Λ ± , ξ Λ ± ) are developed. We use these computational tools to provide the first examples of closed, tight, contact manifolds with vanishing contact homology -- contact 1 k surgeries along the right-handed, tb=1 trefoil for k>0 , which are known to have non-zero Heegaard-Floer contact classes by arXiv:math/0404135.

Motivated by Pazit Haim-Kislev's combinatorial formula for the Ekeland-Hofer-Zehnder capacities of convex polytopes, we give corresponding formulas for Ψ -Ekeland-Hofer-Zehnder and coisotropic Ekeland-Hofer-Zehnder capacities of convex polytopes introduced by the second named author and others recently. Contrary to Pazit Haim-Kislev's subadditivity result for the Ekeland-Hofer-Zehnder capacities of convex domains, we show that the coisotropic Hofer-Zehnder capacities satisfy the subadditivity for suitable hyperplane cuts of two-dimensional convex domains in the reverse direction.