We find an upper bound for the Gromov width of coadjoint orbits of U (n) with respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten invariants. The approach presented here is closely related to the one used by Gromov in his celebrated non-squeezing theorem.

Abstract. We show the existence of Calabi quasimorphisms on the universal covering Ham(O λ , ω λ ) of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit O λ of a compact Lie group. We show that this result follows from positivity results of Gromov-Witten invariants and the fact that the quantum product of Schubert classes can never be zero.

In this paper we pose the question of whether the (generalized) Mukai inequalities hold for compact, positive monotone symplectic manifolds. We first provide a method that enables one to check whether the (generalized) Mukai inequalities hold true. This only makes use of the almost complex structure of the manifold and the analysis of the zeros of the so-called generalized Hilbert polynomial, which takes into account the Atiyah-Singer indices of all possible line bundles.We apply this method to generalized flag varieties. In order to find the zeros of the corresponding generalized Hilbert polynomial we introduce a modified version of the Kostant game and study its combinatorial properties.

In this paper, we derive topological and number theoretical consequences of the rigidity of elliptic genera, which are special modular forms associated to compact almost complex manifolds. On the geometry side, we prove that rigidity implies relations between the Betti numbers and the index of a compact symplectic manifold admitting a Hamiltonian action of a circle with isolated fixed points. We investigate the case of maximal index and toric actions. On the number theoretical side, we prove that from each compact almost complex manifold of index greater than one, that can be endowed with the action of a circle with isolated fixed points, one can derive non-trivial relations among Eisenstein series. We give explicit formulas coming from the standard action on ${{\mathbb{C}}} P^n$.

No abstract