Extremal metrics and stabilities on polarized manifolds
Extremal metrics and stabilities on polarized manifolds
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DOI：
10.4171/022-2/39
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发表时间：
2006-03
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影响因子：
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通讯作者：
T. Mabuchi
中科院分区：
文献类型：
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作者：
T. Mabuchi
由 Donaldson、Kobayashi、Lubke、Uhlenbeck 和 Yau 建立的向量丛的希钦·小林对应关系指出，紧卡勒流形上的不可分解全纯向量丛在 Takemoto·Mumford 意义上是稳定的，当且仅当向量丛承认埃尔米特-爱因斯坦度量时。它的流形类比丘氏猜想源于卡拉比猜想，它询问极化流形的“稳定性”和“极值度量的存在性”是否等价。在这篇文章中，将讨论唐纳森、田和我们小组在这一主题上的最新进展，以及它与代数几何的关系。
TheHitchin�Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Lubke, Uhlenbeck andYau, states that an indecomposable holomorphic vector bundle over a compact Kahler manifold is stable in the sense of Takemoto�Mumford if and only if the vector bundle admits a Hermitian-Einstein metric. Its manifold analogue known as Yau�s conjecture, which originated from Calabi�s conjecture, asks whether �stability� and �existence of extremal metrics� for polarized manifolds are equivalent. In this note the recent progress of this subject, by Donaldson, Tian and our group, together with its relationship to algebraic geometry will be discussed.