Andrés Viña

A characteristic number of Hamiltonian bundles over S2

Abstract Each loop ψ in the group Ham ( M ) of Hamiltonian diffeomorphisms of a symplectic manifold M determines a fibration E on S 2 , whose coupling class [V. Guillemin, L. Lerman, S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams, Cambridge U.P., Cambridge, 1996] is denoted by c . If VTE is the vertical tangent bundle of E , we relate the characteristic number ∫ E c 1 ( VTE ) c n to the Maslov index of the linearized flow ψ t ∗ and the Chern class c 1 ( TM ) . We give the value of this characteristic number for loops of Hamiltonian symplectomorphisms of Hirzebruch surfaces.

Hamiltonian symplectomorphisms and the Berry phase

Abstract On the space L , of loops in the group of Hamiltonian symplectomorphisms of a symplectic quantizable manifold, we define a closed Z -valued 1-form Ω . If Ω vanishes, the prequantization map can be extended to a group representation. On L one can define an action integral as an R / Z -valued function, and the cohomology class [Ω] is the obstruction to the lifting of that action integral to an R -valued function. The form Ω also defines a natural grading on π 1 ( L ) .

Identification of Kähler quantizations and the Berry phase

Abstract For a given symplectic manifold M we consider the bundle whose base is the space of Kahler structures on M , and whose fibers are the corresponding Kahler quantizations of M . We analyse the possible parallel transports in that bundle and the relation between the holonomy of some of them and the Berry phase.

Bundle of quantizations of a symplectic torus

For a given symplectic torus (M=V/Λ,ω) we construct a bundle whose base is the space of complex structures onV, and whose fibres are the corresponding quantizations ofM. We prove that there is no trivializations of this bundle which allow us to define a continuous identification of the quantizations.

Continuous families of Hamiltonian torus actions

Abstract We determine conditions under which two Hamiltonian torus actions on a symplectic manifold M are homotopic by a family of Hamiltonian torus actions, when M is a toric manifold and when M is a coadjoint orbit.

Characteristic Number Associated to Mass Linear Pairs

Let Δ be a Delzant polytope in ℝ𝑛 and 𝐛∈ℤ𝑛. Let 𝐸 denote the symplectic fibration over 𝑆2 determined by the pair (Δ,𝐛). Under certain hypotheses, we prove the equivalence between the fact that (Δ,𝐛) is a mass linear pair (McDuff and Tolman, 2010) and the vanishing of a characteristic number of 𝐸. Denoting by Ham(𝑀Δ), the Hamiltonian group of the symplectic manifold defined by Δ, we determine loops in Ham(𝑀Δ) that define infinite cyclic subgroups in 𝜋1(Ham(𝑀Δ)) when Δ satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch sur-face, (ii) it is a Δ𝑝 bundle over Δ1, and (iii) Δ is the truncated simplex associated with the one point blowup of ℂ𝑃𝑛.

Stability of B-incompleteness

Abstract In the set M of lorentzian metrics on the manifold M a topology is defined, using the topology induced by the fibre bundle T 1 2 s ( M ) (of tensors of type (1,2 s ) in the set of its cross sections. For g ϵ M and γ a curve in M , there is a neighbourhood e of g , such that if g ϵ e , the generalized affine parameter on γ defined by means of g is bounded iff the affine parameter defined by using g is bounded.