扭转形式的可加性公理与更高的Cheeger-Müller/Bismut-Zhang定理

《Journal of the Institute of Mathematics of Jussieu》：ADDITIVITY AXIOM FOR THE TORSION FORMS AND HIGHER CHEEGER-MüLLER/BISMUT-ZHANG THEOREM

　　摘要：我们研究了一个带有平坦复向量丛的平滑纤维化结构，以及一个将纤维化分成两部分的超曲面。我们的主要成果是一个粘合公式，它将整个纤维化的Bismut-Lott解析挠率形式与每一部分的挠率形式联系起来。这一结果解决了2003年在哥廷根会议上提出的一个猜想。该结果还引出了更高阶的Cheeger-Müller/Bismut-Zhang定理。我们的方法结合了沿超曲面法向的绝热极限以及平坦向量丛上的Witten型变形。参考文献：[1] Atiyah, MF, Patodi, VK和Singer, IM (1975) 光谱不对称性与黎曼几何I。Math. Proc. Cambridge Philos. Soc. 77, 43–69.10.1017/S0305004100049410CrossRefGoogle Scholar [2] Badzioch, B, Dorabia?a, W, Klein, JR和Williams, B (2011) 高阶挠率不变量的等价性。Adv. Math. 226, 2192–2232.10.1016/j.aim.2010.09.017CrossRefGoogle Scholar [3] Berezans’kii, JM (1968) 自伴算子的特征函数展开，数学专著译本，第17卷（罗德岛普罗维登斯：美国数学会）。10.1090/mmono/017CrossRefGoogle Scholar [4] Bismut, J-M (1986) Dirac算子族的Atiyah-Singer指数定理：两种热方程证明。Invent. Math. 83, 91–151.10.1007/BF01388755CrossRefGoogle Scholar [5] Bismut, J-M (1997) 全纯浸入族与高阶解析挠率形式。Astérisque 244, viii+275页。Google Scholar [6] Bismut, J-M和Cheeger, J (1989) $\unicode{x3b7}$-不变量及其绝热极限。J. Amer. Math. Soc. 2, 33–70.Google Scholar [7] Bismut, J-M和Freed, DS (1986) 椭圆族的分析II。Dirac算子、eta不变量与全纯性定理。Comm. 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