Heat kernel asymptotics, local index theorem and trace integrals for CR manifolds with $S^1$ action
Heat kernel asymptotics, local index theorem and trace integrals for CR manifolds with $S^1$ action
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发表时间：
2015-10
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通讯作者：
Jih-Hsin Cheng;Chin-Yu Hsiao;I. Tsai
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作者：
Jih-Hsin Cheng;Chin-Yu Hsiao;I. Tsai
在由Atiyah和Singer提出的横截椭圆算子中，Kohn在具有$S^1$作用的CR流形上的$Box_b$算子对于复杂分析者来说是一个自然的具有几何意义的算子。我们的第一个主要结果建立了这样一个算子的热核的渐近展开式，它的值在傅立叶分量中，它涉及到从$S^1作用的低维地层的距离函数方面的一个前所未有的贡献。我们的第二个主要结果计算了包括弦理论中感兴趣的{Sasakian流形}在内的这种流形上的局部指数密度，它以emph{切向}特征形式表示，这表明热核展开中来自地层的某些非平凡贡献最终将通过将Getzler的重标度技术应用于非对角线估计来抵消。这导致了一个局部结果，它可以被认为是这些CR流形上的一种局部指标定理。作为我们的CR指标定理的应用，我们可以证明Grauert-Riemenschneider判据的CR形式，并且在具有横截$S^1$作用的弱伪凸CR流形上产生许多CR函数，并且在某类CR流形上产生多个CR截面，回答了(关于这类流形)多复变量和CR几何中的一些长期存在的问题。我们给出了这些CR流形的例子，其中一些是由Brieskorn流形产生的。此外，在某些情况下，不使用等变上同调方法，也不像前人所做的那样保留来自低维地层的贡献，我们可以将Kawasaki的Hirzebruch-Riemann-Roch公式重新解释为具有奥尔布洛德全纯线丛的复流形的Hirzebruch-Riemann-Roch公式，作为在光滑CR流形上通过单一积分得到的指数定理，该流形本质上是该线丛的圆丛。
Among those transversally elliptic operators initiated by Atiyah and Singer, Kohn's $\Box_b$ operator on CR manifolds with $S^1$ action is a natural one of geometric significance for complex analysts. Our first main result establishes an asymptotic expansion for the heat kernel of such an operator with values in its Fourier components, which involves an unprecedented contribution in terms of a distance function from lower dimensional strata of the $S^1$-action. Our second main result computes a local index density, in terms of \emph{tangential} characteristic forms, on such manifolds including \emph{Sasakian manifolds} of interest in String Theory, by showing that certain non-trivial contributions from strata in the heat kernel expansion will eventually cancel out by applying Getzler's rescaling technique to off-diagonal estimates. This leads to a local result which can be thought of as a type of local index theorem on these CR manifolds. As applications of our CR index theorem we can prove a CR version of Grauert-Riemenschneider criterion, and produce many CR functions on a weakly pseudoconvex CR manifold with transversal $S^1$ action and many CR sections on some class of CR manifolds, answering (on this class of manifolds) some long-standing questions in several complex variables and CR geometry. We give examples of these CR manifolds, some of which arise from Brieskorn manifolds. Moreover in some cases, without use of equivariant cohomology method nor keeping contributions arising from lower dimensional strata as done in previous works, we can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a complex orbifold with an orbifold holomorphic line bundle, as an index theorem obtained by a single integral over a smooth CR manifold which is essentially the circle bundle of this line bundle.