Speaker: 

R. Inanc Baykur

Institution: 

Michigan State University

Time: 

Tuesday, March 7, 2006 - 2:00pm

Location: 

SSPA 1165

One possible strategy for understanding oriented smooth 4-manifolds is to break them up into more tractable classes of manifolds in a controlled manner. Situated in the intersection of complex, symplectic and Riemannian geometries, K\"ahler manifolds are the best known candidates to be pieces of such a decomposition. We have shown that this can be achieved for any closed oriented smooth 4-manifold X. To be precise, we can decompose X into two compact K\"ahler manifolds with strictly pseudoconvex boundaries, up to orientation, such that contact structures on the common boundary induced by the maximal complex distributions are isotopic. The decomposition gives rise to a folded K\"ahler structure on X, a globally defined 2-form, which is a particular generalization of a symplectic form. Moreover, folded Lefschetz fibrations, a certain analogue of Lefschetz fibrations, are shown to be the geometric counterpart of these structures. In this talk we would like to outline these existence results in the direction of generalizing the study of symplectic structures and Lefschetz fibrations on smooth 4-manifolds. Detailed proofs and examples can be found in the preprint at arXiv: “K\"ahler decomposition of 4-manifolds”, math.GT/0601396.