We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of harmonic one-forms. For example, this is the case for complete Kähler manifolds for which the symplectic form has an appropriate decay at infinity. This extends ...

In this paper, the symplectic genus for any 2−dimensional class in a 4−manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least −1 in such manif...

We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction becomes the moduli space of flat connections over the manifold. On the moduli space of flat connections we shall construct a hermitian line bundle with conne...

We consider canonical symplectic structure on the moduli space of flat G-connections on a Riemann surface of genus g with n marked points. For G being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum of n copies of Kirillov symplectic form on the orbit of dressing transformations in the Poisson-Lie group G an...

We analyze K3 surfaces admitting an elliptic fibration E and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration E/G comparing its properties to the ones of E. We show that if E admits an n-torsion section, its quotient by the group of automorphisms induced by this section admits again an n-torsion section. Considering au...

We classify the group of birational automorphisms Hilbert schemes points on algebraic K3 surfaces Picard rank one. study whether these are symplectic or non-symplectic and if there exists a hyperkähler model which they become biregular. also present new geometrical constructions for some automorphisms.

A remarkable theorem of Donaldson [D] says that a symplectic 4manifold admits a Lefschetz pencil by symplectic surfaces. Given a Lefschetz pencil on a 4-manifold X, one can blow up points at the base locus to get a Lefschetz fibration of X over S, which admits a nice handlebody decomposition and can be described by geometric monodromy representation into the mapping class group of a regular fib...

Given an abelian group G endowed with a T=R/Z-pre-symplectic form, we assign to it symplectic twisted ?-algebra WG and then provide criteria for the uniqueness of states invariant under ergodic action automorphism. As application, discuss notion natural in quantum Chern-Simons theory.

We propose a reduction procedure for symplectic connections with symmetry. This is applied to coadjoint orbits whose isotropy is reductive. Key-words: Marsden-Weinstein reduction, symplectic connections, Hamiltonian group actions MSC 2000: 53C15, 53D20 1 e-mail: [email protected] 2 Research supported by the Marie Curie Fellowship Nr. HPMF-CT-1999-00062 3 e-mail: [email protected] 4 Research supp...

In this paper we study Schur-Weyl duality between the symplectic group and Brauer’s centralizer algebra over an arbitrary infinite field K. We show that the natural homomorphism from the Brauer’s centralizer algebra Bn(−2m) to the endomorphism algebra of tensor space (K) as a module over the symplectic similitude group GSp2m(K) (or equivalently, as a module over the symplectic group Sp2m(K)) is...