Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold(M,ω) into a smooth symplectic manifold . Then we study how the formality and the Lefschetz property of are compared with that of (M,ω). We also study the formality of the symplectic blow-up of (M,ω) along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.     

Examples of non-Kähler Hamiltonian circle manifolds with the strong Lefschetz property

In this paper we construct six-dimensional compact non-Kähler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known counterexamples to the question whether the strong Lefschetz property descends to the symplectic quotient. We also give examples of Hamiltonian strong Lefschetz circle manifolds which have a non-Lefschetz fixed point submanifold. In addition, we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a strong Lefschetz manifold. We then use it to show the existence of Lefschetz four-manifolds with non-Lefschetz finite covering spaces.     

An obstruction to quantizing compact symplectic manifolds

We prove that there are no nontrivial finite-dimensional Lie representations of certain Poisson algebras of polynomials on a compact symplectic manifold. This result is used to establish the existence of a universal obstruction to quantizing a compact symplectic manifold, regardless of the dimensionality of the representation.

     

Hodge structures for orbifold cohomology

We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology for projective -orbifolds satisfying a ``Hard Lefschetz Condition'. Furthermore, the total cohomology forms a mixed Hodge structure that is polarized by every element of the Kähler cone of . Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of .

This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of , in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

     

Symplectically hyperbolic manifolds

A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are:

• If a symplectic form represents a bounded cohomology class then it is hyperbolic.

• The symplectic hyperbolicity is equivalent to a certain isoperimetric inequality.

• The fundamental group of symplectically hyperbolic manifold is non-amenable.

We also construct hyperbolic symplectic forms on certain bundles and Lefschetz fibrations, discuss the dependence of the symplectic hyperbolicity on the fundamental group and discuss some properties of the group of symplectic diffeomorphisms of a symplectically hyperbolic manifold.

Symmetries of Lagrangian fibrations

We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the six-dimensional examples constructed by Castaño-Bernard and Matessi (2009) [8], which include a six-dimensional symplectic manifold homeomorphic to the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.     

The Picard–Lefschetz formula for p-adic cohomology

In this paper, we discuss a p-adic analogue of the Picard–Lefschetz formula. For a family with ordinary double points over a complete discrete valuation ring of mixed characteristic (0,p), we construct vanishing cycle modules which measure the difference between the rigid cohomology groups of the special fiber and the de Rham cohomology groups of the generic fiber. Furthermore, the monodromy operators on the de Rham cohomology groups of the generic fiber are described by the canonical generators of the vanishing cycle modules in the same way as in the case of the ℓ-adic (or classical) Picard–Lefschetz formula. For the construction and the proof, we use the logarithmic de Rham–Witt complexes and those weight filtrations investigated by Mokrane (Duke Math. J. 72(2):301–337, 1993).      

Lagrangian non-intersections

No Abstract. . This research was supported by the Israel Science Foundation (grant No. 205/02 *). Received: December 2004 Revision: March 2005 Accepted: March 2005     

Self-maps of the Grassmannian of Complex Structures Dedicated to Professor Boju Jiang on his 65th birthday

Let CS _n be the flag manifold SO(2n)/U(n). We give a partial classification for the endomorphisms of the cohomology ring H ^*(CS _n ; Z) which is very close to a homotopy classification of all selfmaps of CS _n . Applications concerning the geometry of the space are discussed.     

A convexity theorem for three tangled Hamiltonian torus actions,and super-integrable systems

A completely integrable system on a symplectic manifold is called super-integrable when the number of independent integrals of motion is more than half the dimension of the manifold. Several important completely integrable systems are super-integrable: the harmonic oscillators, the Kepler system, the non-periodic Toda lattice, etc. Motivated by an additional property of the super-integrable system of the Toda lattice (Agrotis et al., 2006) [2], we will give a generalization of the Atiyah and Guillemin–Sternberg?s convexity theorem.     

Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Given a normally hyperbolic invariant manifold Λ for a map f, whose stable and unstable invariant manifolds intersect transversally, we consider its associated scattering map. That is, the map that, given an asymptotic orbit in the past, gives the asymptotic orbit in the future.We show that when f and Λ are symplectic (respectively exact symplectic) then, the scattering map is symplectic (respectively exact symplectic). Furthermore, we show that, in the exact symplectic case, there are extremely easy formulas for the primitive function, which have a variational interpretation as difference of actions.We use this geometric information to obtain efficient perturbative calculations of the scattering map using deformation theory. This perturbation theory generalizes and extends several results already obtained using the Melnikov method. Analogous results are true for Hamiltonian flows. The proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems, hence the results can be used to obtain intersection properties of objects of any type.We also reexamine the calculation of the scattering map in a geodesic flow perturbed by a quasi-periodic potential. We show that the geometric theory reproduces the results obtained in [Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math. 202 (1) (2006) 64-188] using methods of fast-slow systems. Moreover, the geometric theory allows to compute perturbatively the dependence on the slow variables, which does not seem to be accessible to the previous methods.     

ON HARMONIC MAPS INTO SYMPLECTIC GROUPS Sp（N）

50. IntroductionThe construction and the factorization of harmonic maps from R2 (or its simPlyconnecteddomain) into the uIiltary group U(N) were firstly solved by K.Ulilenbeck in [11, wherethe conception of unitons was iniroduced. Since then various developmenis have beencoatributed[2--5]. Recently, by introducing (singular) Darboux transformations, a purelya1gebraic method to construct harmonic maPs and unitons illto U(N) has been shownin t6'7]. This method can be aIso aPplied to the ca…     

Künneth formulae and cross products for the symplectic Floer cohomology

For a symplectic monotone manifold (P,ω) and φSymp₀(P,ω), we define a -graded symplectic Floer cohomology (a local invariant) over integral coefficients. There is a spectral sequence which arises from a filtration on the -graded symplectic Floer cochain complex. The spectral sequence converges to the -graded symplectic Floer cohomology (a global invariant). We show that there are cross products on the -graded symplectic Floer cohomology and on the spectral sequence, hence on the usual -graded symplectic Floer cohomology. The Künneth formula for the -graded symplectic Floer cohomology is proved and similar results for the spectral sequence are obtained.     

On the bilinear and cubic forms of some symplectic connected sums

Let ( M1 , M2 , N ) be three symplectic manifolds and suppose that we can do the symplectic connected sum of M1 and M2 along their submanifold N to obtain M1#NM2 . In this paper, we consider the bilinear and cubic forms of H* (M1#NM2 , Z) when dim M1#NM2 = 4, 6. Under some conditions, we get some relations of the bilinear and the cubic forms between M1#NM2 and M1■M2 .     

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

A recursive basis for primitive forms in symplectic spaces and applications to Heisenberg groups

This paper is divided in two parts: in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space(V~(2n), ω). Successively, in Section 3, we apply our construction in the setting of Heisenberg groups H~n, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin's complex of differential forms in H~n.     

On the Existence of n-Harmonic Spheres

Using the bubbling argument of Sacks and Uhlenbeck, we prove the existence of n-harmonic maps from the n-sphere to Riemannian manifolds. An application is made to a problem concerning manifolds with strongly pth moment stable stochastic dynamical systems.