Mathematics

Let (S,?) be a closed orientable surface whose genus l is at least two. Then we provide an obstruction for commuting symplectomorphisms in terms of the flux homomorphism. More precisely, we show that for every non-negative integer n and for every homomorphism α: Z n ??Symp 0 (S,?) , the image of Flux ? ?��? Z n ??H 1 (S;R) is contained in an l -dimensional real linear subspace of H 1 (S;R) . For the proof, we show the following two keys: a refined version of the non-extendability of Py's Calabi quasimorphism μ P on Ham(S,?) , and an extension theorem of G ^ -invariant quasimorphisms on G for a group G ^ and a normal subgroup G with certain conditions. We also pose the conjecture that the cup product of the fluxes of commuting symplectomorphisms is trivial.

Polishchuk-Zaslow explained the homological mirror symmetry between Fukaya category of symplectic torus and the derived category of coherent sheaves of elliptic curves via Lagrangian torus fibration. Recently, Cho-Hong-Lau found another proof of homological mirror symmetry using localized mirror functor, whose target category is given by graded matrix factorizations. We find an explicit relation between these two approaches.

We present the various constructions of new symplectic 4 -manifolds with non-negative signatures using the complex surfaces on the BMY line c 2 1 =9 ? h , the Cartwright-Steger surfaces, the quotients of Hirzebruch's certain line-arrangement surfaces, along with the exotic symplectic 4 -manifolds constructed in \cite{AP2, AS}. In particular, our constructions yield to (i) an irreducible symplectic and infinitely many non-symplectic 4 -manifolds that are homeomorphic but not diffeomorphic to (2n??)C P 2 #(2n??) CP ¯ 2 for each integer n?? , (ii) the families of simply connected irreducible nonspin symplectic 4 -manifolds that have the smallest Euler characteristics among the all known simply connected 4 -manifolds with positive signatures and with more than one smooth structure. We also construct a complex surface with positive signature from the Hirzebruch's line-arrangement surfaces, which is a ball quotient.

We formulate some conjectures about the K-theory of symplectic manifolds and their Fukaya categories, and prove some of them in very special cases.

We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio 1 .

We present recursive formulas which compute the recently defined "higher symplectic capacities" for all convex toric domains. In the special case of four-dimensional ellipsoids, we apply homological perturbation theory to the associated filtered L-infinity algebras and prove that the resulting structure coefficients count punctured pseudoholomorphic curves in cobordisms between ellipsoids. As sample applications, we produce new previously inaccessible obstructions for stabilized embeddings of ellipsoids and polydisks, and we give new counts of curves with tangency constraints.

We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids Σ≃ S n+k−1 × R n−k . Using an embedding of a compact sphere Σ 0 ≃ S 2k−1 into the hypersurface Σ , we construct a chain map from the Floer complex of Σ to the Floer complex of Σ 0 . In contrast to the compact case, the Rabinowitz Floer homology groups of Σ are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.

We study here, from the Gromov-Witten theory point of view, some aspects of rigidity of locally conformally symplectic manifolds, or lcs manifolds for short, which are a natural generalization of both contact and symplectic manifolds. In particular, we give a first known analogue of the classical Gromov non-squeezing in lcs geometry. Another possible version of non-squeezing related to contact non-squeezing is also discussed. In a different direction we study Gromov-Witten theory of the lcs manifold C? S 1 induced by a contact form λ on C , and show that the extended Gromov-Witten invariant counting certain charged elliptic curves in C? S 1 is identified with the extended classical Fuller index of the Reeb vector field R λ , by extended we mean that these invariants can be ±??-valued. Partly inspired by this, we conjecture existence of certain 1-d curves we call Reeb curves in certain lcs manifolds, which we call conformal symplectic Weinstein conjecture, and this is a direct extension of the classical Weinstein conjecture. Also using Gromov-Witten theory, we show that the CSW conjecture holds for a C 3 - neighborhood of the induced lcs form on C? S 1 , for C a contact manifold with contact form whose Reeb flow has non-zero extended Fuller index, e.g. S 2k+1 with standard contact form, for which this index is ±??. We also show that in some cases the failure of this conjecture implies existence of sky catastrophes for families of holomorphic curves in a lc manifold. The latter phenomenon is not known to exist, but if it does, would be analogous to sky catastrophes in dynamical systems discovered by Fuller.

Suppose given a Hamiltonian and holomorphic action of G=U(2) on a compact Kähler manifold M , with nowhere vanishing moment map. Given an integral coadjoint orbit O for G , under transversality assumptions we shall consider two naturally associated 'conic', reductions. One, which will be denoted M ¯ ¯ ¯ ¯ ¯ G O , is taken with respect to the action of G and the cone over O ; another, which will be denoted M ¯ ¯ ¯ ¯ ¯ T ν , is taken with respect to the action of the standard maximal torus T⩽G and the ray R + ıν along which the cone over O intersects the positive Weyl chamber. These two reductions share a common 'divisor', which may be viewed heuristically as bridging between their structures. This point of view motivates studying the (rather different) ways in which the two reductions relate to the the latter divisor. In this paper we provide some results in this directions. Furthermore, we give explicit transversailty criteria for a large class of such actions in the projective setting, as well as a description of corresponding reductions as weighted projective varieties, depending on combinatoric data associated to the action and the orbit.

In \cite{btoric}, Guillemin et al. proved a Delzant-type theorem which classifies b -symplectic toric manifolds. More generally, in \cite{torus} they proved a similar convexity result for general Hamiltonian torus action on b -symplectic manifolds. In this paper, we provide a new way to construct b -symplectic toric manifolds from usual toric manifolds. Conversely, through this way, we can also decompose a b -symplectic toric manifolds to usual toric manifolds. Finally, we will try to prove that this kind of decomposition is useful, although the symplectic structure for our decomposition or construction is not canonical.