Curvature of the space of positive Lagrangians

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.

21 pages, added reference, corrected minor errors

References in corpus (2)

• Singularities of Lagrangian mean curvature flow: zero-Maslov class case
• The Calabi homomorphism, Lagrangian paths and special Lagrangians

Cited by in corpus (3)

• The degenerate special Lagrangian equation
• The space of almost calibrated $(1,1)$ forms on a compact Kähler manifold
• Geodesics of positive Lagrangians in Milnor fibers