Curvature of the space of positive Lagrangians
Curvature of the space of positive Lagrangians
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正拉格朗日空间的曲率
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发表时间：
2013
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通讯作者：
Jake Solomon
中科院分区：
文献类型：
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作者：
Jake Solomon
几乎Calabi-Yau流形中的拉格朗日子流形称为正流形，如果限制于它的全纯体积形式的实部是正的。一类精确的正拉格朗日子流形具有自然黎曼度量。我们计算了该度量的黎曼曲率，并证明了所有的截面曲率都是非正的。我们计算的动机来自于镜像对称性。粗略地说，一类精确的正拉格朗日函数在镜像对称下对应于全纯向量丛上的厄米度量空间。后者是对偶于酉群的非紧对称空间的无限维模拟，因而具有非正曲率。
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.