Curvature of the space of positive Lagrangians

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Abstract

A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional curvatures are non-positive. The motivation for our calculation comes from mirror symmetry. Roughly speaking, an exact isotopy class of positive Lagrangians corresponds under mirror symmetry to the space of Hermitian metrics on a holomorphic vector bundle. The latter space is an infinite-dimensional analog of the non-compact symmetric space dual to the unitary group, and thus has non-positive curvature.

Original languageAmerican English
Pages (from-to)670-689
Number of pages20
JournalGeometric and Functional Analysis
Volume24
Issue number2
DOIs
StatePublished - Apr 2014

Bibliographical note

Funding Information:
The author is grateful to S. Donaldson, Y. Rubinstein, J. Streets, and G. Tian for helpful conversations. During the preparation of this manuscript, the author was partially supported by Israel Science Foundation grant 1321/2009 and Marie Curie grant No. 239381.

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