Large time limit and local $L^2$-index theorems for families
Large time limit and local $L^2$-index theorems for families
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DOI：
10.4171/jncg/203
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发表时间：
2013-06
影响因子：
0.9
通讯作者：
Sara Azzali;S. Goette;T. Schick
中科院分区：
文献类型：
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作者：
Sara Azzali;S. Goette;T. Schick
我们在没有任何额外规律性假设的情况下明确计算 L-2 设置中 Bismut-Lott 型超连接的纤维热算子的大时间限制。这是由某些非紧空间（具有协紧群作用的流形族）的指数理论推动的，其中热算子在很大程度上的收敛意味着精炼的 L-2 指数公式。作为应用，我们证明了签名算子族的局部 L-2-指数定理和 L-2-Bismut-Lott 定理，用 Kamber-Tondeur 类表示平丛的 Becker-Gottlieb 传递。凭借稍强的规律性，我们获得了各自的精炼版本：我们将 L-2-eta 形式和 L-2-torsion 形式构造为海侵形式。
We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L-2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L-2-index formulas. As applications, we prove a local L-2-index theorem for families of signature operators and an L-2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L-2-eta forms and L-2-torsion forms as transgression forms.