The moduli space of complex Lagrangian submanifolds

TL;DR

This paper studies the geometric structure of the moduli space of complex Lagrangian submanifolds in a complex symplectic manifold, revealing a special K"ahler metric and connecting it to hyperk"ahler geometry via Legendre transforms.

Contribution

It introduces a new perspective on the local differential geometry of complex Lagrangian moduli spaces using a submanifold structure in a symplectic vector space.

Findings

The moduli space admits a natural special K"ahler metric.

A new geometric viewpoint relates the moduli space to a Lagrangian submanifold in a product of symplectic vector spaces.

Connections to hyperk"ahler metrics via Legendre transforms are established.

Abstract

Following an earlier paper on the differential-geometric structure of the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold, we follow an analogous approach for compact complex Lagrangian submanifolds of a (K\"ahlerian) complex symplectic manifold. The natural geometric structure on the moduli space is a special K\"ahler metric, but we offer a different point of view on the local differential geometry of these, based on the structure of a submanifold of $V\times V$ (where $V$ is a symplectic vector space) which is Lagrangian with respect to two constant symplectic forms. As an application, we show using this point of view how the hyperk\"ahler metric of Cecotti, Ferrara and Girardello associated to a special K\"ahler structure fits into the Legendre transform construction of Lindstr\"om and Ro\v cek.