Kähler 度量空间的 Chebyshev 位势、Fubini--Study 度量与几何
Chebyshev potentials, Fubini--Study metrics, and geometry of the space of Kähler metrics
AI总结：
本文反驳了关于 Chebyshev 位势线性与 Kähler 度量空间测地线关系的猜想，并通过显式求解 Monge--Ampère 方程刻画了射影空间上猜想成立的 Fubini--Study 测地线。
摘要：
Witt Nyström 引入的射影簇上 Kähler 位势的 Chebyshev 位势是定义在 Okounkov 体上的凸函数。它是 Guillemin 引入的环面不变 Kähler 位势在 Delzant 多面体上的辛位势的推广。一个民间猜想断言，与 Kähler 度量空间中某曲线相关联的 Chebyshev 位势曲线在时间变量上是线性的，当且仅当后者是 Mabuchi 度量下的测地线。这在特殊的环面情形下经典地成立，且 Witt Nyström 一般性地证明了充分性。本文旨在反驳该猜想。更一般地，我们刻画了猜想成立的射影空间上的 Fubini--Study 测地线。证明涉及显式求解描述 Fubini--Study 度量子空间上测地线的 Monge--Ampère 方程，并计算其 Chebyshev 位势。
英文摘要：
The Chebyshev potential of a Kähler potential on a projective variety, introduced by Witt Nyström, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus-invariant Kähler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a curve in the space of Kähler potentials is linear in the time variable if and only if the latter curve is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nyström established the sufficiency. The goal of this article is to disprove this conjecture. More generally, we characterize the Fubini--Study geodesics for which the conjecture is true on projective space. The proof involves explicitly solving the Monge--Ampère equation describing geodesics on the subspace of Fubini--Study metrics and computing their Chebyshev potentials.
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