×

The moduli space of complex Lagrangian submanifolds. (English) Zbl 0958.53057

The first main result of this paper is on the moduli space \(M\) of local deformations of a complex Lagrangian submanifold \(Y\subset X\). It asserts that such a moduli space can be realized as a complex Lagrangian submanifold. This is done by embedding \(M\) in the product \(H^1(Y,\mathbf R)\times H^1(Y,\mathbf R)\). The author then studies submanifolds \(M\subset V\times V\), where \(V\) is a real symplectic vector space. Such a product admits two symplectic structures, \(\Omega_1\) and \(\Omega_2\), and one can consider “bi-Lagrangian” submanifolds. He shows how \(\Omega_1\) and \(\Omega_2\) can determine a metric on \(V\times V\). The second main result gives a complete characterization of special pseudo-Kähler manifolds. He shows that all such manifolds can be realized as bi-Lagrangian submanifolds \(M\subset V\times V\), with \(M\) transversal to the two projections onto \(V\). He also proves that the moduli spaces of complex special Lagrangian submanifolds admit special Kähler structures. The bi-Lagrangian picture is also used to explain the relation between a special Kähler structure on a manifold \(M\) and a Higgs bundle structure on the bundle \(T\oplus T^*\), where \(T\) is the tangent bundle of \(M\). Finally, the author looks at hyper-Kähler metrics on certain bundles over special Kähler manifolds. He shows how a construction due to Cecotti, Ferrara and Girardello can be regareded as a special case of a more general construction, called a Legendre transform.

MSC:

53D12 Lagrangian submanifolds; Maslov index
32G13 Complex-analytic moduli problems
PDFBibTeX XMLCite
Full Text: DOI