标题： 关于奇异叶状结构的 Lefschetz 数的 Atiyah-Bott 公式 显示英文标题

标题： An Atiyah-Bott formula for the Lefschetz number of a singular foliation

摘要： 本文给出了一个关于几何自同态的Lefschetz数的公式，其形式类似于Atiyah-Bott定理。 基础数据包括：首先，一个紧流形和一个处处非零的光滑实向量场$\mathcal{T}$，它保持某个黎曼度量；其次，一个定义在带有联络的Hermite向量丛截面上的首阶算子序列，其曲率被$\mathcal{T}$消没，并且沿$\mathcal{T}$的积分曲线的平行移动是酉的。 假设定序中的算子与各种协变导数$\mathcal{L}_{\mathcal{T}}=\nabla_{\mathcal{T}}$交换，并且它们限制到被$\mathcal{L}_{\mathcal{T}}$消没的截面空间上的子复形满足椭圆性条件，则可得到有限维的等变上同调空间。 经过调整的Atiyah-Bott框架，仅适用于$\mathcal{L}_{\mathcal{T}}$-平行截面的复形，并结合上同调的有限性，可以定义Lefschetz数。 用一个关于波前集的条件（这是阿蒂亚和博特所依赖的条件）取代与自同态相关的等变映射$f$的不动点为简单的条件，可以得出由$\mathcal{T}$左不变且由$f$固定的轨道闭包集合是有限的，并由此得到一个类似于他们的公式，现在这个公式将李夫希茨数与这些轨道上的迹线联系起来。 显示英文摘要

摘要： This paper presents a formula for the Lefschetz number of a geometric endomorphism in the style of the Atiyah-Bott theorem. The underlying data consist, first, of a compact manifold and a nowhere vanishing smooth real vector field $\mathcal{T}$ that preserves some Riemannian metric, and second, a sequence of first order operators on sections of Hermitian vector bundles with connection whose curvature is annihilated by $\mathcal{T}$ and for which parallel transport along integral curves of $\mathcal{T}$ is unitary. Assuming that the operators of the sequence commute with the various covariant derivatives $\mathcal{L}_{\mathcal{T}}=\nabla_{\mathcal{T}}$ and that their restriction to the spaces of sections annihilated by $\mathcal{L}_{\mathcal{T}}$ form a complex, an ellipticity condition gives finite-dimensionality of the resulting equivariant cohomology spaces. The Atiyah-Bott framework, adapted to give a geometric endomorphism only for the complex of $\mathcal{L}_{\mathcal{T}}$-parallel sections, together with the finiteness of cohomology allows for the definition of a Lefschetz number. Replacing the condition that the fixed points of the equivariant map $f$ associated with the endomorphism be simple by a condition on wave front sets, which is the underlying condition of Atiyah and Bott, yields that the set of closures of orbits by $\mathcal{T}$ left invariant by $f$ is finite, and then a formula similar to theirs, now relating the Lefschetz number with traces along these orbits.