SeminarsFolded Holomorphic Maps
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The theory of folded holomorphic maps is a program to extend the tools of $J$--holomorphic curves to all oriented 4--manifolds by utilizing folded symplectic structures.
A folded symplectic structure on a manifold is a closed 2-form that is non-degenerate away from a real codimension 1 hypersurface ("fold"), on which its kernel gives a 1-dimensional foliation. Every oriented 4--manifold admits such a structure. Folded holomorphic maps are pairs of $J$--holomorphic maps from two halfs of a Riemann surfaces. The halfs are separated by a codimension 1 submanifold, which has image on the fold and is satisfying an appropriate boundary condition. This boundary condition constitutes the heart of the theory and can be visualized by "tunneling" in the fold, allowing maps to exit at a location that is different from where they enter the fold. We prove that under the simplifying assumption that the fold is "circle-invariant" this leads to a Fredholm problem. Such folds occur frequently. Under suitable genericity assumptions, moduli spaces of folded holomorphic maps have the expected dimension. In the case of pseudoconvex folds we show that they possess a natural compactification with boundary stratum of codimension at least 2. We will explain the theory and the mentioned results and give examples of moduli spaces of folded holomorphic maps, including moduli spaces of maps into the the 4-sphere.
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Jens von Bergmann
2008-06-03
11:00:00 - 12:10:00
308 , Mathematics Research Center Building (ori. New Math. Bldg.)
The theory of folded holomorphic maps is a program to extend the tools of $J$--holomorphic curves to all oriented 4--manifolds by utilizing folded symplectic structures.
A folded symplectic structure on a manifold is a closed 2-form that is non-degenerate away from a real codimension 1 hypersurface ("fold"), on which its kernel gives a 1-dimensional foliation. Every oriented 4--manifold admits such a structure. Folded holomorphic maps are pairs of $J$--holomorphic maps from two halfs of a Riemann surfaces. The halfs are separated by a codimension 1 submanifold, which has image on the fold and is satisfying an appropriate boundary condition. This boundary condition constitutes the heart of the theory and can be visualized by "tunneling" in the fold, allowing maps to exit at a location that is different from where they enter the fold. We prove that under the simplifying assumption that the fold is "circle-invariant" this leads to a Fredholm problem. Such folds occur frequently. Under suitable genericity assumptions, moduli spaces of folded holomorphic maps have the expected dimension. In the case of pseudoconvex folds we show that they possess a natural compactification with boundary stratum of codimension at least 2. We will explain the theory and the mentioned results and give examples of moduli spaces of folded holomorphic maps, including moduli spaces of maps into the the 4-sphere.