期刊
GEOMETRIC AND FUNCTIONAL ANALYSIS
卷 24, 期 2, 页码 670-689出版社
SPRINGER BASEL AG
DOI: 10.1007/s00039-014-0267-6
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资金
- Israel Science Foundation grant [1321/2009]
- Marie Curie grant [239381]
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.
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