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You are here: Home en Open Course SGU Mathematics Kickoff Meeting video Rational curves and symplectic manifolds

Rational curves and symplectic manifolds

A symplectic manifold in the context of this talk is a compact complex manifold with the additional structure that each tangent space is equipped with a skew-symmetric bilinear form which, when expressed in terms of local coordinates, varies holomorphically. Whereas there are plenty of examples for the real version of this notion, compact complex symplectic manifolds are rather dicult to construct. A famous and by now classical paper of Mukai shows how such manifolds can be obtained as moduli spaces of vector bundles on a certain complex surface. The key insight is that geometric properties of a point in such a parameter space can be intrinsically expressed in terms of the properties of the object that the point represents. Taking this idea further we consider moduli spaces (=parameter spaces) of rational curves that lie on a fourdimensional hypersurface of degree 3 and show how new symplectic manifolds can be obtained from these by a contraction process. This relates to work of Kuznetsov on the derived category of cubic fourfolds and connects with another strand of Mukai’s early paper. In the talk I want to give an introduction to this circle of ideas and present some recent developments.