Making use of the extended flux homomorphism defined in [13] on the group Symp§g of symplectomorphisms of a closed ori- ented surface §g of genus g ≥ 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly non- trivial. We also prove that the second homology of the group Ham§g of Hamiltonian symplectomorphisms of §g, equipped with the discrete topology, is very large for all g ≥ 2.

We show that in the generic case the smooth noncommutative tori associated with two$n$×$n$real skew-symmetric matrices are Morita equivalent if and only if the matrices are in the same orbit of the natural SO($n$,$n$∣$Z$) action.

We compute local Gromov–Witten invariants of cubic surfaces at all genera. We use a deformation a of cubic surface to a nef toric surface and the deformation invariance of Gromov–Witten invariants.

We prove the classical Yano-Obata conjecture by showing that the connected component of the group of h-projective transformations of a closed, connected Riemannian K¨ahler manifold consists of isometries unless the manifold is the complex projective space with the standard Fubini-Study metric (up to a constant).

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