Curvature of the space of positive Lagrangians

TL;DR

This paper computes the Riemann curvature of the space of positive Lagrangian submanifolds in an almost Calabi-Yau manifold, showing all sectional curvatures are non-positive, with implications for mirror symmetry.

Contribution

It provides the first explicit calculation of the Riemann curvature for the metric on the space of positive Lagrangians, linking geometric analysis with mirror symmetry.

Findings

All sectional curvatures are non-positive.

The space is analogous to a non-compact symmetric space.

Supports mirror symmetry predictions.

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.