The Calabi homomorphism, Lagrangian paths and special Lagrangians

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Abstract

Let O be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold (X,ω) We define a functional C: O→ℝ for each differential form β of middle degree satisfying βΛω=0 and an exactness condition. If the exactness condition does not hold, C is defined on the universal cover of O. A particular instance of C recovers the Calabi homomorphism. If β is the imaginary part of a holomorphic volume form, the critical points of C are special Lagrangian submanifolds. We present evidence that C is related by mirror symmetry to a functional introduced by Donaldson to study Einstein-Hermitian metrics on holomorphic vector bundles. In particular, we show that C is convex on an open subspace O+⊂ O. As a prerequisite, we define a Riemannian metric on O+ and analyze its geodesics. Finally, we discuss a generalization of the flux homomorphism to the space of Lagrangian submanifolds, and a Lagrangian analog of the flux conjecture.

Original languageAmerican English
Pages (from-to)1389-1424
Number of pages36
JournalMathematische Annalen
Volume357
Issue number4
DOIs
StatePublished - Dec 2013

All Science Journal Classification (ASJC) codes

  • General Mathematics

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