Showing papers in "Journal of Symplectic Geometry in 2011"

Differentiable stacks and gerbes

Abstract: We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions. We define connections and curvings for groupoid $S^1$-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for $S^1$-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both $S^1$-bundles and $S^1$-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of $S^1$-central extensions with prescribed curvature-like data.

Symplectic mapping class groups of some Stein and rational surfaces

Abstract: In this paper we compute the homotopy groups of the symplectomorphism groups of the three-, four- and five-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundle of $RP^2$ and of $C^∗ ×C$. We also make progress in the case of the $A_n$-Milnor fibres: here we can show that the (compactly supported) Hamiltonian group is contractible and that the symplectic mapping class group embeds in the braid group on n-strands.

On the extistence of symplectic realizations

Abstract: We give a direct global proof for the existence of symplectic realizations of arbitrary Poisson manifolds.

Maslov index formulas for Whitney $n$-gons

Abstract: In this short article, we find an explicit formula for Maslov index of Whitney $n$-gons joining intersections points of $n$ half-dimensional tori in the symmetric product of a surface. The method also yields a formula for the intersection number of such an $n$-gon with the fat diagonal in the symmetric product.

Toric geometry of convex quadrilaterals

Abstract: We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kahler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kahler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kahler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.

On cohomological decomposition of almost-complex manifolds and deformations

Abstract: While small deformations of K\"ahler manifolds are K\"ahler too, we prove that the cohomological property to be $\mathcal{C}^\infty$-pure-and-full is not a stable condition under small deformations. This property, that has been recently introduced and studied by T.-J. Li and W. Zhang in [Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.] and T. Dr\v{a}ghici, T.-J. Li, and W. Zhang in [Symplectic forms and cohomology decomposition of almost complex four-manifolds, Int. Math. Res. Not.], [On the J-anti-invariant cohomology of almost complex 4-manifolds, to appear in Q. J. Math.], is weaker than the K\"ahler one and characterizes the almost-complex structures that induce a decomposition in cohomology. We also study the stability of this property along curves of almost-complex structures constructed starting from the harmonic representatives in special cohomology classes.

Tamed to compatible: symplectic forms via moduli space integration

Abstract: Fix a compact 4-dimensional manifold with self-dual second Betti number one and with a given symplectic form. This article proves the following: The Frechet space of tamed almost complex structures as defined by the given symplectic form has an open and dense subset whose complex structures are compatible with respect to a symplectic form that is cohomologous to the given one. The theorem is proved by constructing the new symplectic form by integrating over a space of currents that are defined by pseudo-holomorphic curves.

Negative inflation and stability in symplectomorphism groups of ruled surfaces

Abstract: Consider symplectic ruled surfaces M λ = (Σg ×S2, λσΣg ⊕σS2) such that Σg has area λ and S has area 1. We show that for k ≥ g/2 the homotopy type of the symplectomorphism groups Ggλ of M g λ is constant as λ increases in the interval (k, k + 1], thus generalizing an existent result of Abreu–McDuff for the rational ruled surfaces with g = 0. We also investigate the changes in the groups π∗G g λ as λ passes an integer k and show the existence of higher Samelson products in π4k+2gG g λ that exist only for λ in the range (k, k + 1]. To prove these results we introduce a refinement of the negative inflation technique introduced by Li–Usher.

Floer trajectories with immersed nodes and scale-dependent gluing

Abstract: Development of pseudo-holomorphic curves and Floer homology in symplectic topology has led to moduli spaces of pseudo-holomorphic curves consisting of both “smooth elements” and “spiked elements”, where the latter are combinations of $J$-holomorphic curves (or Floer trajectories) and gradient flow line segments. In many cases the “spiked elements” naturally arise under adiabatic degeneration of “smooth elements” which gradually go through thick–thin decomposition. The reversed process, the recovering problem of the “smooth elements” from “spiked elements” is recently of much interest. In this paper, we define an enhanced compactification of the moduli space of Floer trajectories under Morse background using the adiabatic degeneration and the scale-dependent gluing techniques. The compactification reflects the one-jet datum of the smooth Floer trajectories nearby the limiting nodal Floer trajectories arising from adiabatic degeneration of the background Morse function. This paper studies the gluing problem when the limiting gradient trajectories has length zero through a renomalization process. The case with limiting gradient trajectories of nonzero length will be treated elsewhere. An immediate application of our result is a complete proof of the isomorphism property of the PSS map: a proof of this isomorphism property was outlined by Piunikhin–Salamon–Schwarz in a way somewhat different from the current proof in its details. This kind of scale-dependent gluing techniques was initiated in "Lagrangian intersection Floer theory-anomaly and obstruction," in relation to the metamorphosis of holomorphic polygons under Lagrangian surgery and is expected to appear in other gluing and compactification problem of pseudo-holomorphic curves that involves ‘adiabatic’ parameters or rescaling of the targets.

Equivariant homology for generating functions and orderability of lens spaces

Abstract: In her PhD thesis [M08] Milin developed a Z k-equivariant version of the contact homology groups constructed in [EKP06] and used it to prove a Z k-equivariant contact non-squeezing theorem. In this article, we re-obtain the same result in the setting of generating functions, starting from the homology groups studied in [S11]. As Milin showed, this result implies orderability of lens spaces.

Cohomologically symplectic solvmanifolds are symplectic

Abstract: We consider aspherical manifolds with torsion-free virtually polycyclic fundamental groups, constructed by Baues. We prove that if those manifolds are cohomologically symplectic then they are symplectic. As a corollary we show that cohomologically symplectic solvmanifolds are symplectic.

The Calabi invariant for some groups of homeomorphisms

Abstract: We show that the Calabi homomorphism extends to some groups of homeomorphisms on exact symplectic manifolds. The proof is based on the uniqueness of the generating Hamiltonian (proved by Viterbo) of continuous Hamiltonian isotopies (introduced by Muller and Oh).

Non-displaceable contact embeddings and infinitely many leaf-wise intersections

Abstract: We construct using Lefschetz fibrations a large family of contact manifolds with the following properties: any bounding contact embedding into an exact symplectic manifold satisfying a mild topological assumption is non-displaceable and generically has infinitely many leafwise intersection points. Moreover, any Stein filling of dimension at least six has infinite-dimensional symplectic homology.

Immersions in a manifold with a pair of symplectic forms

Abstract: Let (N, σ) be a symplectic manifold with the symplectic form σ and M a manifold with a closed two-form ω. An immersion f : M → N is said to be a symplectic immersion if f pulls back the form σ onto ω. All manifolds and maps in this article are assumed to be smooth. The symplectic immersion theorem of Gromov states that the symplectic immersions f : M → N satisfy the C0 dense h-principle near the continuous maps f0 : M → N which pull back the deRham cohomology class of σ onto that of ω [8, 3.4.2(A)]. The aim of this paper is to generalize this theorem when the manifold N comes equipped with a pair of symplectic forms σ1 and σ2 and M has a pair of closed two-forms ω1 and ω2. An immersion f : M → N will be called a bisymplectic immersion if it satisfies the relations f∗σ1 = ω1 and f∗σ2 = ω2. The bisymplectic immersions are solutions to a system of first-order partial differential equations (PDEs) on a manifold. In fact, we can associate a first-order partial differential operator D defined on the space of C∞ maps from M to N such that the bisymplectic maps are solutions to the equation D = (ω1, ω2) for a given pair of closed two-forms ω1, ω2 on M . This takes us into the theory of C∞ operators. Generally, to solve a PDE we need to prove an appropriate implicit function theorem so as to obtain a local inversion of the operator D. The implicit function theorem in the present case should ensure the C∞-smoothness (regularity) of the inversions. Gromov proves in [8, 2.3] that if an rth-order C∞ operator D is infintesimally invertible on an open subset U in the space

On the anti-diagonal filtration for the heegaard floer chain complex of a branched double-cover

Abstract: Seidel and Smith introduced the graded fixed-point symplectic Khovanov cohomology group Khsymp,inv(K) for a knot K ⊂ S3, as well as a spectral sequence converging to the Heegaard Floer homology group HFd (Σ(K)#(S2 × S1)) with E1-page isomorphic to a factor of Khsymp,inv(K) [26]. There the authors proved that Khsymp,inv is a knot invariant. We show here that the higher pages of their spectral sequence are knot invariants also, and further define a reduced version of the spectral sequence which directly computes HFd (Σ(K)). Under some degeneration conditions, one obtains a new absolute Maslov grading on the group HFd (Σ(K)). This occurs when K is a two-bridge knot, and we compute the grading in this case.

Legendrian contact homology and nondestabilizability

Abstract: We provide the first example of a Legendrian knot with nonvanishing contact homology whose Thurston–Bennequin invariant is not maximal.

Extended flux maps on surfaces and the contracted Johnson homomorphism

Abstract: On a closed symplectic surface $\Sigma$ of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group $\operatorname{Symp}(\Sigma)$ to the cohomology group $H^1(\Sigma;\mathbb{R})$ that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface and a correction factor related to the Johnson homomorphism. For surfaces of genus three or more, we give another new construction of an extended flux map using hyperbolic geometry.